

Time Travel Research Center
© 2005 Cetin BAL  GSM:+90 05366063183  Turkey/Denizli
Classical Mechanics, Relativity, and Time
Contents
Classical Mechanics
Special Relativity
General Relativity
Time
Classical mechanics describes the way objects move and interact in
accordance with Newton's laws of motion. The basic assumptions involve a
frame of reference (x,y,z) with respect to which the objects move, there is
an independent time variable t to record the sequence of the movement, the
gravitational or electromagnetic interaction between objects is
instantaneous, and objects with geometric extent are often idealized as a
point (with the justification that the size is much smaller than the
distance involved). The basic equation is:
F = m a  (1)
This formula is known as the equation of motion and looks deceptively simple.
However, the force F and acceleration a are vectors, which
have to be resolved into the x, y, z components. The acceleration a
is the derivative of the velocity v, which is also a vector and the
derivative of the positional vector r, i.e., v = dr/dt.
The mass m is a proportional constant according to the Newton's second law.
If the positional vector r is decomposed into r = x i +
y j + z k, where i, j, and k are unit
vectors along the x, y, z axes respectively, then Eq.(1) can be written in
its component form:
F_{x} = d^{2}x/dt^{2}, F_{y} = d^{2}y/dt^{2},
and F_{z} = d^{2}z/dt^{2}  (2)
which are essentially three separate differential equations. The force F
can be a sum of many forces acting together; if the resultant is zero then
the object is said to be in equilibrium and would not experience
acceleration. This is the Newton's first law. The Newton's third law is : in
his own words, "To every action there is always opposed an equal reaction;
or, the mutual actions of two bodies upon each other always equal, and
directed to contrary parts."
The Newton's law of universal gravitation is in the form:
F = (Gm_{1}m_{2} / r^{2}) (r/r)
 (3)
where G is the gravitational constant, m_{1} and m_{2} are
the masses of the two objects interacting via gravitation, r is the distance
between these two objects, and (r/r) is an unit vector along the
direction of r as shown in Figure 01 below:
m_{1} r m_{2}
oo
r/r =>
Figure 01 Gravitational Interaction
If one of the objects is much heavier than the other, e.g., m_{1} >>
m_{2} like the Sun / Earth system, then m_{1} can be placed
in the origin of the coordinate system and Eq.(1) can be solved as a onebody
problem. In case the two masses are similar, the problem can be reduced to a
onebody problem with a fictitious object moving around the center of mass,
and Eq.(1) is still applicable. The equation of motion becomes rapidly unmanageable
for system of three bodies and beyond. Eq.(1) would include accelerations
for all the objects and the force on one object would involve the
interaction with all the others. This is the situation often encountered in
celestial mechanics with spacecraft flying among planets. The solution is
usually obtained by some kind of approximation and by numerical computation
using large computers.
Now consider two frames of reference S and S'. The S' system is coincided
with S at t = 0 and moving with a constant velocity V along the x axis as
shown in the diagram below:
S_____x_{1}_____x_{2}_____ x S'________________
x' ===> V
Figure 02 Equivalent Inertial Frames of Reference
The transformation between these two coordinate systems (known as inertial
frames of reference) can be expressed by the Galilean transformation:
x' = x  Vt, y' = y, z' = z, and t' = t  (4)
It is obvious that the length L = x_{2}  x_{1} = L' = x'_{2}
 x'_{1}, i.e., it remains unchanged in the two coordinate systems.
In general, the invariant form of the infinitesimal length of an object can
be expressed as:
dx^{2} + dy^{2} + dz^{2} = dx'^{2} + dy'^{2}
+ dz'^{2}  (5)
It can be shown that the gravitational force in Eq.(3) together with the
equation of motion in Eq.(1) are also invariant under the Galilean
transformation. However, according to the formula in Eq.(4) the velocity of
light c would be different with c' = c  V.
It had been demonstrated by Michelson and Morley in 1887 and others that
the speed of light in free space is the same everywhere, regardless of any
motion of source or observer. The
Maxwell's equations
in electromagnetism also indicates that the speed of light is an absolute
constant regardless of the relative motion of the person measuring that
speed. But as just mentioned above, the velocity of light is different in
different inertial frame according to the Galilean transformation. The
theory of special relativity was postulated to reconcile the inconsistency
from these kinds of observations. Mathematically, the statement about the
constant velocity of light in different inertial frames can be expressed as:
x^{2} + y^{2} + z^{2} = c^{2} t^{2}
 (6)
x'^{2} + y'^{2} + z'^{2} = c^{2} t'^{2}
 (7)
if a spherical light wave is generated at the origin of the S and S'
inertial frames when they are coincided at t = 0. This situation is possible
only when t is not equal to t'. It can be shown that the Lorentz
transformation below would satisfy the requirement in Eq.(6) and (7):
x' = (x  Vt) / (1  V^{2}/c^{2})^{1/2}, y' =
y, z' = z, and t' = (t  Vx/c^{2}) / (1  V^{2}/c^{2})^{1/2}
 (8a)
The inverse transformation is:
x = (x' + Vt) / (1  V^{2}/c^{2})^{1/2}, y =
y', z = z', and t = (t' + Vx'/c^{2}) / (1  V^{2}/c^{2})^{1/2}
 (8b)
Eq.(8a) reduces to the Galilean transformation in Eq.(4) when V/c << 1. In
this way, x, y, z, and ct form a fourdimensional space known as the
Minkowski spacetime. It revolutionizes highenergy physics when velocity of
the particles is close to the velocity of light (relative to the observer's
inertial frame). All the basic formulae such as the field equations and the
Lagrangians have to be invariant under the Lorentz transformation, and the
definition for many physical entities changes form at high speed as listed
in the followings:
 The independent time variable t is now treated on the same footing as
the other spatial dimensions.
 The length L' = x'_{2}  x'_{1} in the moving inertial
frame is now different with respect to a stationary observer in the S
system


at t = t_{o}. According to the 1^{st}
formula in Eq.(8a), the relation is given by: L = x_{2}  x_{1}
= L' (1  V^{2}/c^{2})^{1/2}, which is known
as the Lorentz contraction. The length of a moving rod is shorter
according to a stationary observer. The phenomena is demonstrated in
high energy collision, when the round shape of atom becomes a pancake
with the flatten face perpendicular to the direction of motion (see
Figure 03). 
Figure 03 Lorentz Contraction

Figure 04 Time Dilation


 The time interval T' = t'_{2}  t'_{1} of a clock
sitting in the S' system at x' = x'_{o} appears to take longer to
tick with respect to a stationary observer in the S system. According to
the 4^{th} formula in Eq.(8b), the relation is given by T = t_{2}
 t_{1} =
T' / (1  V^{2}/c^{2})^{1/2}. The effect is known
as time dilation and T' is referred to as the proper time  the time
experienced in the clock's rest frame. The phenomena is demonstrated in
the decay of unstable particle moving at near the speed of light. The
lifetime of such particle appears to be much longer than the one measured
in a stationary lab. The diagrams in Figure 04 (where
= (1  V^{2}/c^{2})^{1/2}),
show the muon decay as experienced in its own frame and from the view
point of a stationary observer on Earth.

 Events (a point in the four dimensional spacetime as shown in
Figure 05) that take place simultaneously to an observer in S, e.g., t
= t_{o} at x = x_{1} and x = x_{2}, are
separated by a time interval t'_{2}  t'_{1} = (V(x_{1}
 x_{2})/c^{2}) / (1  V^{2}/c^{2})^{1/2}
in S' according to the 4^{th} formula in Eq.(8a) (see Figure
06).
 For an infinitesimal interval of spacetime, Eq.(6) can be written
in the form:
dx^{2} + dy^{2} + dz^{2}  c^{2} dt^{2}
= 0  (9)

Figure 05 Reference Frame


This is called the null line applicable to object moving at the speed
of light. It can be shown that the spacetime interval
ds^{2} = dx^{2} + dy^{2} + dz^{2}  c^{2}
dt^{2}  (10)
Figure 06 Rotation in Minkows ki spacetime
is invariant under the Lorentz transformation. ds is
called proper interval because it is related to the interval for an
object at rest with dx = dy = dz = 0. The interval is called timelike
if ds^{2} < 0. In this case, Event 2 can be related causally
in some way to Event 1 provided that a signal (traveling slower than
the speed of light) is available. If ds^{2} > 0, then it is
called spacelike, which implies Event 3 is entirely unrelated to
Event 1. Alternatively it can be interpreted that two events joined by
a spacelike interval can never influence each other, since that would
imply a flow of 
information at speeds faster than the speed of light (see
Figure 05).

 It is obvious from Eq.(10) that it is actually ict, which becomes the
4^{th} dimension in special relativity. This is the reason for the
unusual appearance of the spacetime axes when subject to a rotation in
the Minkowski spacetime (Figure 06).
 In Galilean transformation the velocity in the S' system is v' = v V,
which is now replaced by v' = (v  V) / (1  vV/c^{2}) in special
relativity.
 The mass for a particle moving at velocity V is given by m = m_{o}
/ (1  V^{2}/c^{2})^{1/2}, where m_{o} is
the rest mass (relative to the observer's inertial frame).
 Mass and energy are equivalent. They can be converted to each other.
The total energy E of a particle is given by the formula E = m c^{2}
= m_{o} c^{2} + T. The first term is the rest mass energy,
and the second term is the kinetic energy.
 In term of the momentum p, E = (m_{o}^{2} c^{4}
+ p^{2} c^{2})^{1/2}.


The inertial frames of reference in both classical
mechanics and special relativity move with a constant velocity related
to each others. Such arrangement seems to become impossible in the
presence of gravity, which produces acceleration (change of velocity).
However, there is a class of frames of reference that can be obtained
locally by letting it freely falling. This kind of frames would generate
an opposite force, which exactly nullifies the acting force. The local
region (such as in an 
Figure 07 Freefalling Frame

Figure 08 Equivalence Principle

elevator) would experience zero gravity as shown in
Figure 07. Figure 08 shows the similar kind of situation in producing
gravity with acceleration. The interchangeable nature of gravity and
acceleration is known as the principle of equivalence. 
The spacetime interval in Eq.(10) is valid as long as the observer is
confined to the freefall frame of reference (inside the elevator). Since
the gravitational field is not uniform in general, the frame of reference
will follow a curved world line (a line traced out by 4 dimensional events,
it is a straight line for constant velocity motion) as shown in Figure 05,
which depicts the path of a particle accelerating from rest up to the
velocity of light. In order to describe this kind of motion globally, Eq.(10)
is now replaced by a more general form taking into account the unevenness of
the field:
ds^{2} = g_{ik} dx^{i} dx^{k} 
(11)
where the notations have been simplified such that x = x^{1}, y = x^{2},
z = x^{3}, ct = x^{4}; the indices i, k run from 1 to 4 and
denote a summation whenever it is paired (in the subscript and the
superscript). The g_{ik} is known as spacetime metric, which is a
second rank tensor and a function of the spacetime. In an inertial frame (flat
spacetime), g_{11} = g_{22} = g_{33} = 1, g_{44}
= 1, g_{ik} = 0 for i ¬= k. In general the spacetime metric g_{ik}
is determined by the nonlinear differential equations as postulated by
Einstein:
R_{ik} = (8G/c^{4})
(T_{ik}  g_{ik}T)  (12)
where R_{ik} is a second rank tensor related to the curvature of
space; it involves the first and second derivatives of g_{ik}; T_{ik}
(and T = T_{i}^{i}) is the energymomentum tensor of matterenergy.
Thus, gravity is geometrized and the geometry of the spacetime is
ultimately determined by matterenergy.
The equations of gravitational field can be solved exactly for the case of a
centrally symmetric field in vacuum with mass m at the center. In terms of
spherical coordinates and ct, it has the form:
ds^{2} = dr^{2} / (1  2Gm/c^{2}r) + r^{2}
(sin^{2} d^{2}
+ d^{2})  (1  2Gm/c^{2}r)
c^{2}dt^{2}  (13)
This is known as the Schwarzschild solution. It is a useful example for
illustrating the many properties in general relativity:
 At large distance from the center, where r approaches infinity, the
spacetime metric reduces to the flat spacetime.
 At sufficiently large distances, where r >> 2Gm/c^{2}, the
solution assumed the characteristics of the classical inversesquare law
of gravitation in the following form:
ds^{2} = (1 + 2Gm/c^{2}r) dr^{2} + r^{2}
(sin^{2} d^{2}
+ d^{2})  (1  2Gm/c^{2}r)
c^{2}dt^{2}  (14)
 At the distance where r = r_{s} = 2Gm/c^{2}, the
escape velocity v_{e} = (2Gm/r)^{1/2} = c. It implies that
even light cannot escape the grip of gravity at this point. This is
Schwarzschild radius separating the black hole from the rest of the world.
 A black hole can form only when the Schwarzschild radius r_{s}
is outside the central object. For the Earth, and the Sun, r_{s}
is well inside the physical boundary (r_{s} for the Earth is about
1 cm anf for the Sun is about 3 km). Even the neutron star (with similar
mass to the Sun) has a physical radius of about 10 km. Only collapsing
stars or galactic center have the necessary condition to form a black
hole.
 Gravitational redshift is generated as the photons loss energy in
overcoming the pull of the gravitational field. For the case of spherical
mass at the center, the shifted wavelength is computed by the formula:
=
_{o} (1  r_{s}/r)^{1/2}
 (15)
At the Schwarzschild radius r_{s}, the redshift becomes infinity.
This is the effect that makes light invisible at r_{s}.
 The gravitational field also induces time dilation as experienced by a
far away stationary observer. It is in a similar form as Eq.(15):
T = T_{o} (1  r_{s}/r)^{1/2}  (16)
where T_{o} is the time interval recorded by an observer plunging
toward the black hole and T is the corresponding time perceived by a
stationary observer. Thus, for the stationary observer it takes an
infinite long interval for the other observer approaching the
Schwarzschild radius r_{s}. However, the adventurer is not aware
this curious effect on the time interval. His journey may be interrupted
only by the tidal force, which is much more ferocious for black hole with
smaller size (the tidal force at r_{s} is proportional to 1 / m^{2}).
 Inside r_{s}, all matter must fall towards the center, even
light, which keeps coming from outside (but cannot get out from inside).
After a finite time the inward falling object will end up at r = 0, where
the density of matter is infinite. The metric g_{44} changes sign
inside the black hole, i.e., time has become just another dimension of
space. It is suggested such change has the effect of smoothly rounded off
the singularity at r = 0.
Figure 09 Schwarzschild Solutions
 White holes are similar to black holes except white holes are
ejecting matter while black holes are absorbing matter. The existence
of white holes is implied by a negative square root solution to the
Schwarzchild metric. If we define the proper time d
to be the time associated with the moving frame, in which the spatial
variation vanishes (comoving objects are stationary in that frame),
then


c^{2} d^{2}
= ds^{2} = g_{44} c^{2} dt^{2}
where g_{44} is negative according to the convention. Thus,
d = ± (g_{44})^{1/2}
dt
where the positive sign corresponds to the black hole solution while the
solution with negative sign is interpreted as the with hole with
time running backward (see Figure 09).
 Inside the Schwarzschild radius, g_{11} and g_{44}
change sign as shown in Eq.(14). It is possible to create another geometry,
which prevents the formation of a singularity and forms a tunnel known as
the EinsteinRosen Bridge or wormhole.
Even though mathematically this bridge is allowed to exist, physically it
is doomed to collapse; the gravitational forces are completely
overwhelming. However if there is a large amount of negative energy to
sustain the structure, then a wormhole may remain open. This would allow
for travel within the wormhole between two distant points, or to even
allow for the possibility of "time" travel (see Figure 09).


The concept of black hole has its origin in a solution
of Einstein's General Relativity for a spherical object with mass M and radius R.
If the mass collapses to a radius less than R = 2xGxM/c^{2},
where G is the gravitational constant and c is the speed of light, then
nothing (including light) can escape from inside this radius. It is
called the event horizon or the Schwarzschild radius
(named after the astrophysicist who solved the equation). Figure 0506a
shows a schematic diagram of the Schwarzschild
geometry. 
Figure 0506a Black Hole

Figure 0506b Worm Holes

It is known as the embedding diagram. The two dimensional circles are
slices of three dimensional spheres (of the same radius)  the hyperspace.
The verticle axis denotes the "stretch" of space in the radial direction.
The slope of the curve can be considered as representing the curvature of
the space. It is flat (or zero) at the outer edge and becomes infinity at
the Schwarzschild radius. This pictorial representation is very similar to a
rubber sheet stretched by
a rock. The shape of the region inside the horizon is somewhat arbitrary.
It is only known that everything plunges inevitably to the central
singularity once passing over the horizon. In a more realistic drawing the
event horizon would be placed far below the diagram at infinity. The
complete Schwarzschild geometry consists of a black hole, a white hole, and two singularities connected at their horizons by a
worm hole
as shown in Figure 0506b. A white hole is a black hole running backwards in
time. Just as black holes swallow things irretrievably, so do white holes
spit them out. White holes cannot exist, since they violate the second law
of thermodynamics by allowing some time reversal events such as reassembling
a broken glass back to its original whole. The white hole geometry outside
the horizon represents another Universe. The worm hole joining the two
separate singularities is known as the EinsteinRosen bridge, but even if it
can somehow be generated, it would be unstable and pinch off immediately.
Therefore, only the black hole geometry is applicable to the physical world.

It is believed that every quasar, active galactic
nucleus, and even normal galactic nucleus contains a black hole with a
mass of between ten million to several billion solar masses at its core.
The difference in appearance is related to the intensity of the activity.
Since galaxies rotate, matter falling toward the central black hole will
form a rapidly spinning disk of gas  an accretion disk  rather than
falling directly into the hole. Kinetic energy released by infalling
matter, and frictional effects within the accretion disk, raise the
temperature of the nner parts of the disk to enormous values and provide
plenty of energy to power AGN's on all scales from Seyferts to quasars.
By a process that is still not fully understood, the central engine
accelerates streams of charged particles to very high speeds. The inner
rim of the accretion disk, together with surrounding gas and magnetic
fields, forms a nozzle that confines the outward flow of energetic
particles into narrow streams that shoot out perpendicularly to the
plane of the disk. The lower half of Figure 0507a shows a model of the
black hole. 
Figure 0507a AGN Model

Figure 0507b is a HST (Hubble Space Telescope) image of NGC4261, which
is a radio galaxy. The image strongly suggests that it is a black hole
fitting the description of the theoretical model. Infrared observation of
NGC1068 in 2004
was able to resolve the inner region down to a few parsec. Figure 0507c
penetrates the dusty central region and shows the structures on arcsec
scales. The picture on the right is a model for the nucleus of NGC1068. It
contains a central hot component (dust temperature > 800K, yellow)
marginally resolved along the source axis (dashed line). Its finite width
and the dashed circle indicate the currently undetermined extent. The much
larger warm component (T=320K, red) is well resolved. The arrows


indicate the projected orientation of the two
interferometer baselines and the angular resolution L/2B, where L is the
wavelength and B is the projected baseline. The image shows that the
active galactic nuclei are arranged like a thick doughnut. This model
requires a continuous injection of kinetic energy to maintain such cloud
system. The mechanism is currently unknown; thus a better understanding
of the physics of these spectacular objects is needed. 
Figure 0507b NGC4261

Figure 0507c NGC1068


The quasar 3C273 is a 2billionsolarmass black hole encircled by a
doughnut of gas (accretion disk) and with two gigantic jets shooting out
along the spinning axis. The Schwrzschild radius for this object is about
6x10^{9} km. Such supermassive black hole can be created while
matter is still at quite low density (~ 10^{3} gm/cm^{3}).
Since the tidal force at the event horizon of a black hole is inversely
proportional to the square of its mass, its effect on a space visitor would
be unnoticeable, although he would soon be in dire trouble as he plunges
irrevocably toward the central singularity. However for a stationary
observer, it takes an infinitely long time for the asronaut to approach the
event horizon (due to gravitational time dilation) and the view of the
asronaut would gradually disappear (due to gravitational red shift of light).
The effect on the astronaut visiting a stellar black hole (miniquasar) would be more violent due to the drastic
increase of the tidal force.
Another example that offers exact solution is the homogeneous and
isotropic space filled with pressureless dust. It is applicable to the case
of the cosmic expansion, if each dust point presents a galaxy. Universes of
this type are variously known as Friedman universes, FriedmanRobertsonWalker
universes, ... etc. The effect of radiation energy, which is important in
the very early universe and the cosmological constant, which may be
responsible for the cosmic repulsion are neglected in the following
consideration.
 The spacetime interval is associated with the RobertsonWalker metric.
It has three forms depending on the curvature of the 3
dimensional space:
ds^{2} = R(t)^{2} (dw^{2} + s^{2} (sin^{2}
d^{2} + d^{2})
 c^{2}dt^{2}  (17)
where s = sin(k^{1/2} w) / k^{1/2}, and R(t) is called the
scale factor.
For flat space with zero curvature, k = 0, and s = w.
For closed space with positive curvature, k = +1, s = sin(w), where w
ranges from 0 to 2.
For open space with negative curvature, k = 1, s = sinh(w), where w
ranges from 0 to infinity.
 In term of the RobertsonWalker metric and with the energymomentum
tensor T_{00} = , the
gravitational field equation becomes:
(dR/dt)^{2} + kc^{2} = 8GR^{2}
/ 3  (18a)
It can be rearranged into the form:
k^{2} c^{2} = H^{2} R^{2} (
 1)  (18b)
if we define the Hubble parameter H(t) = (dR/dt) / R, the critical density
_{c} = (3 H^{2})
/ (8G), and
=
/
_{c}.
By inspecting Eq.(18b), it is obvious that
for flat space, =
_{c},
= 1, and k = 0;
for closed space, >
_{c},
> 1, and k = +1;
for open space, <
_{c},
< 1, and k = 1.
 If we choose = M / (4R^{3}/3),
where M may be interpreted as the total mass of the universe, Eq.(18a)
becomes:
(dR/dt)^{2} + kc^{2} = 2GM / R  (18c)
The solution for this equation is:
for k = 0, R = (9GM/2)^{1/3} t^{2/3}  (19a)
for k = +1, R = R_{o} (1  cos(µ)), ct = R_{o} (µ 
sin(µ))  (19b)
where R_{o} = GM / c^{2}, and µ is defined by c dt = R(t)
dµ.
for k = 1, R = R_{o} (cosh(µ)  1), ct = R_{o} (sinh(µ) 
µ)  (19c)

 Figure 10 shows the three types of universe  open, flat, and
closed. Universes that are too far above the critical divide expand
too fast for matter to condense into stars and galaxies; such
universes therefore remain devoid of life. Those that fall too far
below the critical divide collapse before stars have a chance to form.
The shaded area indicates the range of cosmological expansions and
epochs in which observers could evolve. Recent type Ia supernova data
seems to indicate an open universe with acceleration.

Figure 10 Types of Universe


The equation of motion in relativity is the geodesic (shortest path) for
a particle moving in spacetime:
d^{2}x^{i}/ds^{2} +
^{i}_{kl} (dx^{k}/ds)
(dx^{l}/ds) = 0  (20)
where ^{i}_{kl}
is in terms of the metric tensor and its first derivative. It is known as
the Christoffel symbols. For
^{i}_{kl}
= 0 Eq.(20) reduces to the usual equation of motion of a free particle.

Using the gravitational field equation and the equation
of motion, Einstein presented a calculation on the effect of GR on the
advance of the perihelion of Mercury:
= 6GM/(c^{2}a(1
 e^{2}))  (21)
where M is the mass of the Sun, a is the length of the semimajor axis,
and e is the eccentricity of the ellipse. In Figure 11, the amount of
the advance is greatly exaggerated. The actual advance due to the effect
of GR is only 0.43 seconds of arc per year. The most recent and most
accurate results seem to be converging towards a value that makes the GR
predictions agree well with observation. 
Figure 11 Perihelion Advance


A brief history of time and beyond:

For 10 billion years, the universe has been in existence
without bothering with the definition of time. It started about 3.5
billion years ago when unicellular organisms took up residence on Earth.
They had to adjust their activities according to the daily and yearly
cycles. Since then all living beings including human come equipped with
biological clocks within to adopt to these rhythms. For thousands
of years, protohumans probably had only dim notions of time: past,
present, and future. Beginning around 2500 BCE, systemic definitions of
time were developed in the form of calendars. The Egyptian pioneers
first divided a day into 24 units. Other calendars were linked to
religion and the need to predict days of ritual significance, such as
the summer solstice. All calendars had to resolve the incommensurate
cycles of days, lunations and solar years, usually by intercalating
extra days or months at regular intervals. The Julian calendar was
established at 46 BCE. The first mean of measuring daily time was
probably the Egyptian shadow stick, dating from about 1450 BCE. It was
soon followed by the water clock or clepsydra (Figure 12) and the
sandglass or hourglass, in which time is measured by the change in level
of flowing water or sand. The 
Figure 12 Clepsydra

earliest mechanical clocks containing movable parts were
built about 700 years ago. It had no minute hand.

When Newton published the three natural laws in 1686, time is no longer
confined to record the daily and yearly rhythms. It had become a
mathematical entity  a parameter to keep track of motions in a fixed,
infinite, unmoving space. Einstein changed all this with his relativity
theories, and once wrote, "Newton, forgive me." In the new theories, time is
treated almost on the same footing as the other spatial dimensions with some
minor differences. Recently, theory in quantum gravity considers time
and space to be discrete at the Planck scale with a minimum size of about 10
^{43} sec and 10^{33} cm respectively. At this scale, they
are useless as framework for the motion of other objects. It is suggested
that time and space are the active participants in the dynamics of this
world.
In place of space and time as parameters to describe the evolutions of
systems, the modern approach is to introduce the artificial time (T)
and artificial spatial coordinates (X_{1}, ..., X_{n}),
which have nothing whatsoever to do with real time and real space. Since
nobody has ever seen or detected the artificial spacetime, they have to be
unobservable in the mathematical formulation. Such theories are said to be
reparametrization invariant because they are not affected by the change to
another set of artificial spacetime parameters. This is sometimes referred
to as the unobservability principle. It then more or less leads to the
string theory, which is reparametrization and
conformally (change of scale)
invariant.
Instead of trying to define time or provide an answer to the
philosophical question of "What is time?", some of the characteristics of
time are listed in the followings:
Figure 13 Arrow of Time
 The arrow of time  The laws of physics do not care about the
direction of time, i.e., the mathematical formulations are invariant
if the direction of time is reversed. Yet there is a big difference
between the forward and backward directions of real time in ordinary
life. We have no difficulty to tell the sequence of events in Figure
13 (from 1 to 9), because it is highly improbable for the crumbling
dust to return to the structured building in the

reversed direction. This is usually explained by the
second law of thermodynamics, which states that in the macroscopic
world there is a tendency for a closed system moving toward greater
disorder. 
 It is strongly suspected that general relativity also
preserves causality and forbids agents from changing the past, although
this has not been rigorously demonstrated. The idea of using time machine
to visit the past is still a hot topic.
 The psychological time  The psychological perception of time is
affected by such things as medications, time of day, level of happiness,
external stimuli, and even the temperature. Einstein once remarked, "When
you spend two hours with a nice girl, you think it's only a minute. But
when you sit on a hot stove for a minute, you think it's two hours." Thus,
the time line experienced by the conscious brain is often quite different
from the "objective" time line of events occurring in the real world.
 Cosmological arrow of time  Since volume expansion tend to increase
the entropy of a system, the cosmic expansion seems to be a good indicator
for the direction of time (Figure 12b). But what would happen if and when
the universe stops expanding and begins to contract? According to Hawking,
he used to believe that disorder would decrease when the universe
recollapse. Now he realizes that the no boundary condition implies
that disorder would in fact continue to increase during the contraction.
He attributes the association between the thermodynamic and cosmolgical
arrows to the no boundary condition instead of volume expansion.
Figure 12b Arrows of
Time


Asymmetry between past and future  The asymmetry between past
and future has two aspects. On the one hand, we know events happened
in the past by memory recall, history books, fossil records, and
astronomical observations etc., but we know nothing about the future.
The direction of time in which we remember the past and not the
future is referred to as the psychological arrow of time (Figure
12b). On the other hand, we can only move forward into the future
and can never go backward to the past physically. This kind of
asymmetry is related to the principle of cause and effect (causality),
which is an important concept in physical theories. For example, the
notion that events can be ordered into causes and effects is
necessary to prevent contradictions such as the
grandfather paradox,
which asks what happens if a timetraveller kills his own
grandfather before he ever meets his grandmother. Within special
relativity, causality can be preserved by forbidding information
from traveling faster than the speed of light  the worldline is not
allowed to loop back to the past. It is strongly suspected that
general relativity also preserves causality and forbids agents from
changing the past, despite the possibility of developing a
closed timelike curve.

Figure 12c Time Machine

The idea of using time machine to visit the past is
still a hot topic in science fictions. Figure 12c shows a fictitious
time machine.





Rate of change  Time is often equated to the change of a variable
dR. Mathematically, it can be expressed as dR / dt = constant, where
dt denotes the change in time. One revolution of the Earth around the
Sun defines a year. One complete rotation of Earth defines a day. The
moving hands of clocks define hour, minute, and second. In these cases,
the variable R is the angle of rotation. The accuracy in clocks has
improved over the last 700 years until today, an atomic clock known as
NIST7 is accurate to 10^{9} second per day. Modern quartz
clocks use the piezoelectric properties of the quartz crystal, which
vibrates at a specific frequency when placed in an alternating
electric current circuit. The induced "crystal current" is amplified
and used to operate an LED display or electrically actuated hands (see
Figure 14).

Figure 14 Clocks

Figure 15 Atomic Clock

Atomic clocks use the frequency of vibration of atoms
to regulate a quartz crystal clock (see Figure 15). One second is now
defined as the duration of 9,192,631,770 periods of the radiation from
the hyperfine transition of the cesium atom. 
 End of time  Recently, there is suggestion that time is just an
illusion. Time as such does not exist. There is no invisible river of
time. But there are things that can be called "instants of time", or "Nows".
As we live, we seem to move through a succession of Nows. The idea is that
when we think we are seeing actual motion, the brain is interpreting all
the simultaneously encoded images and playing them as a movie  frame by
frame. Mathematically, it is similar to plot points in the phase space,
where the element of time is left out. It has to rely on either color
codes (for example, blue for earlier and red for later moment) or
animation to display the evolution of the system. In this way, time is
just replaced by something else.
 Imaginary time  The imaginary time T is defined by: t
it'. This
simple substitution is rather controversial in the community of
theoretical physicists. Some argue that it is merely a mathematical trick,
or a convenient tool devoid of any physical significance. Others suggest
that imaginary time is the true physical quantity. Whatever its merit,
imaginary time does provide an alternate computational technique and
conceptional viewpoint as shown in the following examples (Note that
imaginary and real are just mathematical terminologies in this context. It
has nothing to do with the usual connotation of mental perspective.).
Quantization of particles and fields is most elegantly prescribed by the
method of path integral. The mathematical formula in term of the real time
t for the probability amplitude to go from q(t_{1}) to q(t_{2})
is proportional to:
where the sum is over all paths and L is the Lagrangian (a function of the
position and velocity). The oscillating factors in this formula are very
difficult to manipulate. However by substituting the real time with the
imaginary time, Eq.(22a) is changed into:
which become the more manageable exponentially decreasing functions. At
the end of the computation, the imaginary time can be switched back to the
real time.
Figure 15a The "No Boundary" Proposal
Meanwhile, if the imaginary time is substituted intoEq.(10),
it changes into a form familiar to Euclidean geometry:
ds^{2} = dx^{2} + dy^{2} + dz^{2} + c^{2}
dt'^{2}  (23)
While the real time in Eq.(10) restricts the time direction within
the light cone, for the imaginary time there is no difference between
the time direction and directions in space. Thus according to Hawking,
it is possible for such spacetime to be finite in extent and have no

singularities that formed a boundary or edge. As shown
in Figure 15a, spacetime would be like the surface of the earth, only
with two more dimensions. Such Euclidean sphere has zero points at the
North and 
South Poles, but these points would not be any more singular than these
Poles on earth. The corresponding universe in real time would have a
minimum size, which corresponds to the maximum radius of the history in
imaginary time. At later real times, the universe would expand at an
increasing inflationary rate to a very large size (see lower right diagram
in Figure 15a or Figure 10n). By combining the two examples as illustrated
above, a path integral similar to Eq.(22b) can be used to calculate the
probability amplitude for the no boundary universes. The sum over
histories is actually
performed backward in time from the current state of
the universe such as threedimensional and flat, then construct the
set of all possible histories that would end up like ours (with
variables such as inflation, big crunch, ...), and finally sum them up
by assigning a weighting factor to each to produce the probability
amplitude for the history of a certain universe (see Figure 15b).
Although the answer matches observations (i.e., our universe is the
most probable) and incur no 
singularity, many physicists argue that this is just
giving up on the problem of explaining why our universe is the way it
is  it is not, they say, science. Figure 15b
Sum over Histories

It can be shown that the probability amplitude for a
particular history of the universe is related to the WheelerDeWitt
Equation (which looks like the zero energy timeindependent
Schrodinger equation):
[ d^{2} / d R^{2}  U(R) ]
= 0
 (23)
where R is the scale factor, U is the potential (a function of R, the
curvature, the cosomological constant, the densities of matter, and
radiation), and
is the wave

function of the universe, in which the particle
position is replaced by the radius R for a multitude of universes. The
result confirm that the expanding isotropic universe similar to our
own is the most probable one (see Figure 15c).
Figure 15c Wave Function of the Universe

 End of time (Figure 17)  Recently, there is suggestion that time is
just an illusion. Time as such does not exist. There is no invisible river
of time. But there are things that can be called "instants of time", or "Nows".
As we live, we seem to move through a succession of Nows. The idea is that
when we think we are seeing actual motion, the brain is interpreting all
the simultaneously encoded images and playing them as a movie  frame by
frame. Conceptually, it is similar to plot points in the phase space,
which is also known as configuration space where the element of time is
left out. Figure 18 depicts a very simple configuration space for an
universe of three particles A, B, and C. The lengths of each side (of the
triangle) form the three grid axes AB, BC, and CA. The seven triangles
represent several possible arrangements or NOWs of the model universe. The
black diamond indicates the position of each of these triangles or NOWs in
the configuration space. In general if there are n particles, the
configuration space will be constructed with n grid axes. We have no
access to the past and future. The past is only in our memories or some
sort of records. They are actually a present phenomena. The future is the
not yet realized events equally inacessible. We only experience a
continuous sequence of NOWs one after the other. Time was invented for the
convenience of human to keep track of
the changing NOWs. Note that the representation in
phase space is background independent. The spatial coordinates x, y, z
are absent in the picture, and so is the time. Essentially, this is
the foundation of the socalled relational theory in which all that
matters is the relationships or links between the events. It plays a
crucial role in the formulation of loop quantum
gravity, in which space and time are discrete quantities and
evolve dynamically like the atoms.
Figure 18 Configuration Space
Figure 17 End of Time


 Another attempt to do away with time asserts that it is all an
illusion. The thermal time hypothesis (Figure 19) suggests that a
statistical effect gives rise to the "appearance" of time.
Similar to temperature, which is the average of the momentum of each
molecule, the same applies to the thermal time but including many more
constituents such as space (which is expanding on the average). It
predicts that the ratio of the observer's proper time to the thermal
time is the surrounding temperature. A toy model has been constructed
successfully from the CMBR data to explain the cosmic expansion as
described by standard cosmology. It can also reproduce the temperature
of a black hole associated with the Hawking radiation. The idea
follows the rework of quantum mechanics without time. The evolution in
time is replaced by the variation of correlations between things as
mentioned above. Instead of "collapsing" the wave function of an
electron, both the electron and the measuring device are described by
a single wave function, and a single measurement of the

entire setup causes the collapse. It is anticipated
that combining quantum mechanics with general relativity would become
less daunting when it is rewritten in timefree form. The loop quantum
theory is an example adopting such idea.
Figure 19 Thermal Time


Two Times  Instead of playing with "no time", how about
constructing theory with 2 dimensions of time? It was found long time
ago that such theory leads to negative probability. Worse still, it
admits the weird case of time travel backward as shown in Figure 20.
Thus, the twodimensional time gives every appearance of being a nonstarter.
However, it is shown in a research work in 2007 that the problems
disappear if some kind of symmetry is imposed on position and momentum,
and the theory involves one more spatial dimension, i.e., our 4D spacetime
is generalized to 6D spacetime with 4 spatial dimensions and two time
dimensions. It is suggested that the world we see around us is merely a
"shadow" of a 6D world. According to this idea, the
standard model is just one shadow (projection) of this 6D world. There are
other shadows that include gravity, finally uniting it with the standard
model. Although the effects of an extra time dimension are subtle, it is
purported to be very real out there.

Figure 20 Two Dimensional Time

For example, the problem associated with the
axion (a
not yet observed entity) can be resolved by the application of 2Dtime
physics without such hypothetical particle.

Figure 10r Warped 5D
An interesting application of warped 5^{th}
dimension has been developed by Lisa Randall. In this model, the 5^{th}
dimension is located in between two 3D branes. It is found that the
extra dimension is severely warped in the form of de Sitter space with
positive curvature by the presence of bulk and brane energy even
though the branes themselves are completely flat (see Figure 10r). The
strength of gravity depends on the position of the 5^{th}
dimension. As shown in Figure 10r (in term of graviton's probability
function), it can be very strong on the Gravitybrane but becomes
feeble on the Weakbrane where all the forces and particles in the
Standard Model are confined. Only the gravitons can move anywhere in
the branes and in the bulk. This model explains why gravity is weak in
our world although it can be 
very strong in another brane. It also introduces a new
way to solve the hierarchy problem of huge difference in mass
between the Planck scale and the Electroweak scale. If the Planck
scale mass is set at the Gravitybrane, then the mass of particles on
the Weakbrane would be reduced by a factor of 10^{16} to the
Tev range as expected. 
In the weak field limit where the spacetime metric tensors g_{ik}
deviate only a small amount from flat spacetime, the gravitational field
equation (12a) is reduced to the form:
Figure 10s EW and GW
where the h_{k}^{i} is small correction to g_{k}^{i},
T_{k}^{i} is the energymomentum tensor, and
is the d'Alembertian
operator in four
dimensional spacetime. This equation looks similar to
the electromagnetic wave equation except that it is now a second rank
tensor field (with 10 components) instead of the more familiar vector
field. It is responsible for many different characteristics in these two
kinds of field. Figure 10s shows the differences in polarization and
radiation pattern. There are two polarization states in gravitational
wave. They alternatively squeeze and stretch the interacting particles
shown as white circle in the diagrams (with direction of propagation
perpendicular to the viewing page). Table 02 compares the properties of
these two kinds of wave. 

Property 
Electromagnetic Wave 
Gravitational Wave 
Field 
Vector 
Second Rank Tensor 
Wave 
Transversal 
Transversal 
Polarization 
One State 
Two States 
Radiation Pattern 
Dipole 
Quadrupole 
Source 
Accelerating Charge 
Accelerating MassEnergy 
Interaction 
With Charges 
With MassEnergy 
Quantum Particle 
Spin 1 Photon 
Spin 2 Graviton 
Rest Mass 
Massless 
Massless 
Table 02 Electromagnetic and Gravitational Waves
Figure 10t GW Detector
Gravitational wave have never been observed because of
low radiation power and weak interaction strength. A rod about 1 meter
long spun at the verge of breaking would radiate perhaps 10^{30}
erg/sec. The cross section for the interaction between gravitational
wave of ~ 10^{4} cycles and an ammonia molecule is roughly 10^{60}
cm^{2}. Figure 10t is the schematics of a gravitational wave bar
detector. The impinging gravitational wave excites the fundamental
longitudinal resonance (at ~ 1000 Hz) of the bar, kept at low
temperatures. The induced vibration of the bar end face is amplified
mechanically by the resonant 
transducer, which also converts the signal into an
electromagnetic one. The signal is then amplified and acquired (see
Figure 10t). It is suggested that largescale astronomical motions of
matter could generate appreciable gravitational energy flux.
Figure 10u GW from Binary Pulsars

The binary pulsar PSR1913+16 was discovered in 1975.
This system consists of two compact neutron stars orbiting each other
with a maximum separation of only one solar radius. The rapid motion
means that the orbital period of this system should decrease on a much
shorter time scale because of the emission of a strong gravitational
wave. The change predicted by general relativity is in excellent 
agreement with observations as shown in Figure 10u. Thus,
the observation indirectly confirms the phenomena of gravitational
radiation.
Figure 10v Black Hole Merger

Figure 10v illustrates the merger of two orbiting black
holes. Initially, the gravitational signals from such an event would
show oscillation with increasing amplitude and decreasing wavelength as
the black holes spiral toward each other. A chaotic pattern of
gravitational waves may be given off at the moment of merger. Finally,
the resultant single black hole is expected to 
"ring", creating waves with diminished amplitude. This
event will emit no xray burst, not even a flash of light.

Gravitational waves may be viewed as coherent states of many gravitons,
much like the electromagnetic waves are coherent states of photons. Since
gravitational wave has evaded detection for over 50 years, it seems even
harder to find the individual gravitons. However, it is suggested that in
highenergy colliders such as the LHC, it is possible to produce gravitons,
which can then disappear into the extra dimensions. This would lead to a ‘missing energy’ signature, with
unbalanced events. Such signatures are routinely used in particle
experiments to detect the production of neutrinos (difficult to detect). The
exchange of gravitons in the extra dimensions would also affect the dynamics
of other scattering processes.
A leading cosmological model, known as inflation, predicts that our universe is just one part of a greater multiverse and that
our Big Bang may have been one of many. In this model, our universe expanded
extremely rapidly during the period of 10^{35}10^{32}
second after the Big Bang.
Figure 10w LISA
Another model, rooted in string theory, envisions a scenario in
which the Big Bang occurred as a
result of the collision between two parallel universes
floating in higher dimensional space. Each of these models predicts a
specific pattern of gravitational waves emitted from the Big Bang. NASA
and ESA plan to launch the Laser Interferometer Space Antenna (LISA) to
detect gravitational wave by 2013. It consists of three satellites
orbiting the sun (Figure 10w). They will be linked by three laser beams,
forming a triangle of light. They are designed to detect a change in
their spacing as small as 1/10 the diameter of an atom. With such
sensitivity LISA might be able to detect gravitational waves created
immediately after the birth of the cosmos. It offers a chance to select
between the contesting cosmological models, and also provides an
opportunity to test the string theory. 

The equations of gravitational field can be solved exactly for the case
of a centrally symmetric field in vacuum with mass M at the center. In terms
of spherical coordinates and ct, the spacetime metric has the form:
ds^{2} =  (1  2GM/c^{2}r) c^{2}dt^{2} + dr^{2}
/ (1  2GM/c^{2}r) + r^{2} (sin^{2}
d^{2}
+ d^{2})
 (13)
This is known as the Schwarzschild solution. It is a useful example for
illustrating the effect of gravity in general relativity:
 As r
,
the spacetime metric reduces to the expression for flat spacetime:
ds^{2} =  c^{2}dt^{2} + dr^{2} + r^{2}
(sin^{2}
d^{2}
+ d^{2})
 (14)
 At the nonrelativistic limit where v^{2}/c^{2} << 1,
an expression for the spacetime metric can be derived from classical
mechanics:
ds^{2} =  (1 + 2V/c^{2}) c^{2}dt^{2}
+ dr^{2} + r^{2} (sin^{2}
d^{2}
+ d^{2})
where V is the gravitational potential.
Comparing with the spacetime metric g_{44} in Eq.(13) in the
region where r > 2GM / c^{2}, the Newton's law of gravitation:
V =  GM / r is recovered from general relativity. Thus, at the nonrelativistic
limit only the time dilation effect (relating to g_{44}) is
important. The effect of spatial curvature (relating to g_{11}) is
negligible at this limit.
 At the distance where r = r_{s} = 2GM/c^{2}, the
escape velocity v_{e} = (2GM/r)^{1/2} = c. It implies that
even light cannot escape the grip of gravity at this point. This is
Schwarzschild radius separating the black hole from the rest of the world.

 A black hole can form only when the Schwarzschild radius r_{s}
is outside the central object. For the Earth, and the Sun, r_{s}
is well inside the physical boundary (r_{s} for the Earth is
about 1 cm and for the Sun is about 3 km). Even the neutron star (with
similar mass to the Sun) has a physical radius of about 10 km. Only
collapsing stars or galactic center have the necessary condition to
form a black hole. Figure 09a is an embedding diagram (Figure 09b) of
a black hole. The two dimensional circles are the projection of three
dimensional spheres  the

Figure 09a Black Hole

hyperspace. The vertical axis denotes the "stretch" of
space in the radial direction. The slope of the curve can be
considered as representing the curvature of the space.


 Gravitational redshift is generated as the photons loss energy in
overcoming the pull of the gravitational field (see Figure 09c). For
the case of a black hole, the shifted wavelength is computed by the
formula:
=
_{o}
/ (1  r_{s}/r)^{1/2}
At the Schwarzschild radius r_{s}, the redshift becomes
infinity. This is the effect that makes light invisible at r_{s}.

Figure 09c Redshift


 The gravitational field also induces time dilation as experienced by a
far away stationary observer. It is in a similar form:
T = T_{o} / (1  r_{s}/r)^{1/2}
Figure 09d Tidal Force
Figure 09e Time Dilation
where T_{o} is the time interval recorded by an observer
plunging toward the black hole and T is the corresponding time perceived
by a stationary observer. Thus, for the stationary observer it takes an
infinite long interval for the other
observer approaching the Schwarzschild radius r_{s}.
However, the adventurer is not aware this curious effect on the time
interval. His journey may be interrupted only by the tidal force,
which is much more ferocious for black hole with smaller size (the
tidal force at r_{s} is equal to
2GM / r_{s}^{3} =
c^{6}/(2GM)^{2}, the + and  signs represent the
stretch and squeeze parallel and perpendicular to the radial direction
respectively, see Figure 09d). Figure 09e shows the time dilation for
a space traveler located at a distance of 1.1 r_{s} from the
center of a black hole. 


 Inside r_{s}, all matter must fall towards the center,
even light, which keeps coming from outside (but cannot get out from
inside). After a finite time the inward falling object will end up at
r = 0, where the density of matter is infinite. The metric g_{44}
changes sign inside the black hole, i.e., time has become just another
dimension of space. It is suggested such change has the effect of
smoothly rounded off the singularity at r = 0. There is speculation
that quantum effect will remove the singularity; it might provide a
gateway leading to another universe. Figure 09f shows a collapsing
star, and the behavior of light near and inside the black hole. The
effect of the black hole on light is indicated by the direction of the
light cone. It shows that far away from the black hole the light cone
points to all directions around. The directions are tilted toward the
black hole as it moves closer. At the event horizon it cannot escape
to outside. The directions of the cone only point to the center once
inside the black hole, there is no turning back.

Figure 09f Black Hole, Inside



 White holes are similar to black holes except white holes are
ejecting matter while black holes are absorbing matter. The existence
of white holes is implied by the negative square root solution to the
Schwarzchild metric. In particular, if we define the proper time d
to be the time associated with the moving frame, in which the spatial
variation vanishes (comoving objects are stationary in that frame),
then

Figure 09g White Hole

c^{2} d^{2}
= ds^{2} = g_{44} c^{2} dt^{2}
where g_{44} is negative according to the convention. Thus, 
d
= ± (g_{44})^{1/2} dt
where the positive sign corresponds to the black hole solution while the
solution with negative sign is interpreted as the white hole with time
running backward. Figure 09g depicts an embedding diagram of a white hole,
which is just a black hole turned upside down to give an illusion that
matter is spilling out instead of pouring in.
 The spacetime metric for a static wormhole (also known as the
EinsteinRosen Bridge) can be expressed in the form:

 (15a)


The lapse function defines the proper time between
consecutive layers of spatial hypersurfaces; while the shape function
determines the shape of the worm hole. The shape function takes on a
very simple form for the case of the Schwarzschild's metric, i.e.,
b(r) = 2GM / c^{2} = r_{s}. The throat of the wormhole
is located at r = b(r) = r_{s} in this case. Figure 09h is a
computer generated embedding diagram of a blackhole, a wormhole, and a
whitehole. The surface of the diagram measures the curvature of space.
Color scale represents rate at which idealized clocks measure time;
red is slow, blue fast. 
Figure 09h Wormhole

Another way to conceptualize a wormhole topology is to
have the spatial part of the spacetime metric in Eq.(15a) imbedded in
a flat hyperspace with the extradimension denoted by W: 
dl^{2} = dW^{2} + dr^{2} + r^{2}
(sin^{2}
d^{2})
= dr^{2} / (1  r_{s}/r) + r^{2} (sin^{2}
d^{2})
 (15b)
Figure 09i Worm hole in Hyperspace
Figure 09j Worm hole Throat
Eq.(15b) can be used to equate dW =
(r/r_{s}  1)^{1/2}dr. Integration of the equation
gives W^{2} = 4r_{s}(r  r_{s}), which is a
parabola function with vertex at W = 0, r = r_{s}. Sweeping
the curve around the W axis to include all values of
from 0 to 2
results in a paraboloid surface as shown in Figure 09i. 

Inside the wormhole, g_{11} and g_{44} change sign as
shown in Eq.(13). The region becomes spacelike, which means two events cannot be linked unless the signal propagates with
greater than light speed. This peculiar property is also related to the
instability of the wormhole. However it is suggested that if there is a
large amount of negative mass/energy m (in the forms of a thin spherical
shell, which appears in the embedding diagram as the yellow circle as
shown in Figure 09j) to sustain the structure, then creation of a wormhole
may become feasible. The negative mass ensures that the throat of
the wormhole lies outside the horizon (since the new
event horizon is now 2(Mm)G/c^{2}), so that travelers can
pass through it, while the positive surface pressure of such exotic
material would prevent the wormhole from collapsing. This would allow
for shortcut in space travel within the wormhole between two distant
points (see Figure 09k), or for the possibility of time travel
courtesy of LHC (Figure 09l). 

Figure 09l Time Travel
Figure 09k Space Travel
The spacetime metric generated by a rotating mass M with angular velocity
was found by Roy Kerr in
1963:

 (15c)

Since most collapsing massive objects would have some initial angular
momentum, the Kerr's metric seems to be a more realistic representation for
the black hole spacetime. However, it is argued that the collapsing object
would lost its rotational energy in the form of gravitational wave and ends up in a perfect spheroidal shape. Whichever to be the case,
followings are some comments on the Kerr's metric and its modification to the
Schwarzschild's black hole:
 At r , the Kerr's
metric reduces to the one for flat spacetime (same as Eq.(14)):
ds^{2} =  c^{2}dt^{2} + dr^{2} + r^{2}
(sin^{2} d^{2}
+ d^{2}) 
(15g)
 The Kerr's metric reduces to the Schwarzschild's metric in Eq.(13)
naturally when a = 0, i.e., with no rotation.
Figure 09m Bending of Light
 The trajectory of light ray can be computed from the equation of
motion Eq.(12b), or from the equation of energy conservation. It was shown theoretically
that light ray is deflected by gravitational field, e.g., binding of
distant star light near the sun (see Figure 09m). This prediction was
first confirmed by Sir Arthur Eddington's 1919 solar eclipse expeditions.
If the light ray comes close enough to a dense object, the path will be
bent so much that it runs around in a circle. For nonrotating black hole
such special trajectory occurs at r = 3r_{s}/2 = 3GM/c^{2}.
The sphere with such a radius is

called the photon sphere. However, the orbit is unstable;
it can be disrupted with very small perturbation. Mathematically, such
precarious orbit is defined by d^{2}r / d^{2}
= 0. It is from this formula that the radius of the photon sphere is
deduced. There are two 
photon spheres for the rotating black hole  an outer one for light ray
traveling in a direction opposite to the spin of the black, and an inner one
for corotating light ray.
 As r descends further toward the center, the spacetime metric g_{44}
(or g_{tt}) vanishes at
r = [GM + (G^{2}M^{2}  a^{2}c^{2}cos^{2})^{1/2}]
/ c^{2}  (15i)
Figure 09n Frame Dragging
This is called static limit. It can be intuitively
characterized as the region where the rotation of the spacetime is
dragged along with the velocity of light. Within this region, spacetime
is warped in such a way that no observer can maintain him/herself in a nonrotating
orbit, but is forced to become corotating (Figure 09n). The surface of
this region is elliptical with its major axis at
=
/2 (the equator), and r =
2GM / c^{2} (= the non 
rotating Schwarzschild's radius). The minor axis is in the
directions of = 0, and
(the poles), and r = [GM
+ (G^{2}M^{2}  a^{2}c^{2})^{1/2}]
/ c^{2} (see Figure 09o).

It is found in 2006 that a spinning superconductor generates a much larger
drag than that from a regular rotating mass. The effect is attributed to some
sort of "gravitomagnetism" (Figure 09n, no relation to electromagnetism)
produced by massive gravitons similar to the Meissner effect produced by massive photons. The discovery may fulfill the "antigravity
device" perpetrated by science fictions.
Figure 09o Kerr's Solution
 A rotating black hole is very different from a Schwarzschild black
hole in that the spin of the black will cause the creation of two event
horizons instead of just one (see Figure 09o). The outer event horizon
occurs at
r_{+} = [GM + (G^{2}M^{2}  a^{2}c^{2})^{1/2}]
/ c^{2}  (15j)
The inner horizon (sometimes called the Cauchy horizon) is located at
r_{} = [GM  (G^{2}M^{2}  a^{2}c^{2})^{1/2}]
/ c^{2}  (15k)

The Ergosphere is the region between the static limit and
the outer event horizon. Since this region is outside the event horizon,
particles falling within the ergosphere may escape the black hole
extracting its spin energy in the process. 
The separation between the two horizons is 2 (G^{2}M^{2} 
a^{2}c^{2})^{1/2} / c^{2}. For a = 0, r_{}
=0, hence for a nonrotating black hole, the inner event horizon can be
considered as fallen into the center. As the spin increases, the two horizons
move toward each other and merge at r = GM / c^{2} when a / c = GM / c^{2}.
In case a / c > GM / c^{2}, there will be no event horizon, the black
hole becomes a "naked singularity", i.e., it is not covered by an event
horizon.
Note 1: The only physical part of a black hole is the singularity. The static
limit, and event horizon are not physical barrier; they only mark the
imaginary boundaries between types of space.
Note 2: A proposal in 2007 suggests that naked black hole should be detectable
via the "gravitational lenses" effect. Existing telescopes should have
sufficient spatial resolution to spot naked singularities in the center of the
Milky Way.
 As r , a singularity
develops in the form of a ring at the equator (see Figure 09p), where
/2, and cos
0 in such a way that r >
(a / c) cos. The Kerr's
metric then becomes:
ds^{2} =  (1  2GM / c^{2}r) c^{2}dt^{2} +
(2GMa / c^{2}r) dt d
+ (a^{2}/c^{2}) (1 + 2GM / c^{2}r) d^{2}
 (15n)
The severity of the singularity depends on the angular momentum per unit mass
a. If a is sufficiently large it would be very mild; on the other hand it
becomes a point singularity at the limit a
0. Away from the ring of
singularity in the region where r ~ 0, the Kerr's metric has the form:
ds^{2} =  c^{2}dt^{2} + cos^{2}
dr^{2} + (a^{2}/c^{2}) (cos^{2}
d^{2} + sin^{2}
d^{2}) 
(15o)
Interesting theoretical physics can take place around this ring singularity.
Since 1 g_{11} =
cos^{2} in Eq.(15o),
the spatial curvature is negative, which acts like a repulsive force. One
consequence is that nothing can actually fall into this region unless
approaching in a trajectroy along the ring's side. Any other angle and the
negative spatial curvature actually produces an antigravity field that repels
matter. It could be the mechanism that produces the jets observed in many
black holes.
Figure 09p Penrose Diagram
 The Penrose diagram in Figure 09p summarizes all regions of spacetime
associated with a rotating black hole. A penrose diagram is not meant to
accurately portray distances, it describes only the causal structure. Only
radial and time directions are represented while angular directions are
suppressed. The units of space and time are scaled in such a way that any
object moving at the speed of light will follow a path at an angle of 45º
to the vertical. All possible paths for physical objects must stay closer
than 45º to the vertical. The green diamond represents our entire Universe
over its entire history and destiny, from the infinite past to the
infinite future (it takes an infinite amount of time to cross the outer
event horizon from the view point of a distant observer). The purple
diamond is the region between the outer and inner event horizons, where
everything plunges inward. The red area denotes the space between the
inner event horizon and the ring singularity, where the spacetime reacquires
its normal characteristic with rising curvature toward the ring
singularity. The other half in yellow is the region

inside the ring singularity, where the gravity becomes
repulsive. The blue diamonds can be interpreted as another universes or
another part in our own universe. This concept is similar to the wormhole
in the Schwarzchild's solution. 

As depicted in the Penrose diagram, the spacetime metric
g_{11} (or g_{rr}) changes sign when crossing over the
first event horizon, and then reverses back again at crossing over the
second horizon. Thus between the two horizons, space and time exchange
places. Instead of time always moving inexorably onward, the radial
dimension of space moves inexorably inward to the Cauchy horizon. After
that the Kerr solution predicts a second reversal, which implies no more
plunging inward. In this strange region inside the Cauchy horizon the
observer can, by selecting a particular orbit around the ring singularity,
travel backwards in time and meet himself, in violation of the principle
of causality (cause must precede effect). It is surmised that a closed
timelike curve (CTC, Figure 09q) to loop back to the past is possible in
a heavily curved spacetime (Line A in Figure 09p). Another possibility
admitted by the equations for the observer in the central region is to
plunge through the hole in the ring into an antigravity region (Line B).
Or he can 
Figure 09q Closed Timelike Curve (CTC)

travel through two further horizons (or more properly
antihorizons), to emerge at coordinate time t = –
into some other universe (Line C). These exotic properties of rotating
black holes have inspired several science fiction stories. 
Figure 09ra Disk and Jets
 A more realistic rotating black hole would have an accretion disk
around the equator and a pair of jets ejecting along the poles. An
accretion disk is matter that is drawn to the black hole. As matter is
gradually pulled into the center, it gains speed and energy. It can be
heated to temperatures as high as 3 billion K by internal friction, and
emit energetic radiation such as gamma rays. Some of the particles that
has funneled into the diskshaped torus by the hole's spin and magnetic
fields, can escape the black hole in the form of high speed jets along the
poles (Figure 09ra).

Figure 09rb Black Hole Detections

 Black Holes are very difficult to detect because they are black. It
would be a definitive proof of their existence if we can see the "shadow"
in the form of a dim spot in the centre of the glowing accretion disc (Diagram
c in Figure 09rb). Most of the circumstantial evidences come from
observing high velocity objects moving tightly in small space as shown in
Diagram a, Figure 09rb. Other evidences involve detecting gravity waves
from merging black hole(s) or the double image from a spinning black hole
(see Diagram b and d in Figure 09rb), which throws out huge jets of debris
to advertise its presence. The final answer to whether our black hole
ideas are correct would be sending a probe to a nearby candidate and have
it transmit data as it makes its final plunge.


Recently in 2006, a scenario for time travel has been
proposed without relying on rotating black holes or exotic wormhole tunnels.
The idea is based heavily on the Superstring theory, according to some
versions of which, our universe is a fourdimensional membrane or "brane"
embedded in a higherdimensional hyperspace called the bulk. Almost all
matter and forcecarrying particles are trapped on the 3D brane, where they
are contrained to travel at the speed of light or lower. However, sterile
neutrinos and gravitons are particles that can access the hidden dimensions
and travel faster than light (Figure 09s). From some view points or frames
of reference, this is equivalent to time travel (see Figure 05, event 4). 
The main ingredients of this theory are summarized in the
followings: Figure 09s Time Travel

 For many years, physicists believed that extra dimensions were
acceptable, but only if they were finite in size, either curled up
or bounded between branes. An infinite extra dimension was supposed to
destablilize everything around us. It is now known that infinite extra
dimension from a single brane (like our 3D brane) is possible if that
extra dimension is heavily warped so that the gravitons are localized near
the brane in a small region no more than the Planck length scale of 10^{33}
cm (see Figure 09t).
 Thus, instead of the requirement for a heavily curved spacetime in
our 3D universe, or a folding one such as shown in Figure 09k, a model can
be conceived such that the extra dimension in the bulk is seriously warped.
It is found that the extra dimensions can be distorted by exotic matter in
such a way that anything moving through the bulk can travel faster than
the speed of light. Such offbrane short cuts can appear as "closed timelike
curves" (CTC) as shown in Figure 09q.

Figure 09t Extra Dimension

 This has dramatic consequences for inhabitants stuck on the 3D universe.
To them, any entity that takes a short cut through the bulk appears to vanish
and then pops up again at some point far sooner than it could have had it kept
to the 3D universe, and in some other frames of reference, it has travelled
backward in time (see Figure 05, event 4).
 However, according to the Superstring theory, only closed strings such as
the sterile neutrinos and gravitons can wander into the bulk. We may not be
able to do time travel ourselves, but we can study time travel experimentally
by manipulating these particles. The problem is that no one has ever spotted a
graviton or a sterile neutrino.
 Nevertheless, experiment to test time travel has been proposed via the
mechanism of neutrino oscillation, which could change ordinary neutrinos into
sterile neutrinos and vice versa. The odds of this happening increase whenever the density of
the material the neutrinos are travelling through changes abruptly. Thus, it
is suggested that a beam of ordinary neutrinos would be sent through the Earth,
from South Pole to a detector located at the equator. Some of them flipped
into sterile neutrinos going through the bulk and emerge as ordinary neturino
(via another flipping), which may appear as moving at greater than light speed
or arrived before setting off.
 Critics point out that even if future technology can manipulate sterile
neutrino in the next 50 years or so, the problem with the hypothetical exotic
matter (to produce the warped spacetime) is only shifted from our universe to
unknown dimensions. Concealing the problem in the bulk is little better than a
scenario in which it is out in the open. However, if the experiment is
successful, it will verify many conjectures such as higher dimensions, some
versions of the Superstring theory, sterile neutrinos, exotic matter in higher
dimensions, and time travel albeit only for sterile neutrinos and gravitons.
According to Stephen Hawking himself, an inspiration came to him before
going to bed one evening in 1970 (getting into bed is a rather slow process with
his disability). He suddenly realized that since nothing can escape from a black
hole, the area of
the event horizon might stay the same or increase with time but it could never
decrease. In fact, the area would increase whenever matter or radiation fell
into the black hole. This nondecreasing behavior of a black hole's area was
very reminiscent to that of entropy, which measures the degree of disorder in a system. One can create order out of
disorder, but that requires expenditure of effort or energy such that there is
an overall increase in disorder. In simple mathematical terms these statements
can be expressed in differential forms as:
m dm = k dA  (16a) for the black hole (since A
r_{s}^{2}
m^{2} from the
Schwarzschild's solution),
where m is the mass of the black hole, A is the area of the event horizon, and k
is a proportional constant;
dE = T dS  (16b) for the entropy,
where E is the energy, T is the temperature, and S is the entropy.
Since mass and energy are equivalent, we can equate Eqs.(16a) and (16b) to
obtain:
dS = K dA / ( m T )  (16c)
where K is a new proportional constant. This equation implies that the area of
the event horizon A is a measure of the entropy S of the black hole. Furthermore,
the black hole is associated with a temperature T, and should emits radiation as
any hot body. Thus, the black hole is not completely closed to the universe
outside. It turns out that vacuum fluctuations at the edge of
the event horizon may allow one member of the virtual
particle / antiparticle pair to fall inside with negative energy; while the
other escapes as a real particle with a positive energy according to the law
of energy conservation. This is known as Hawking radiation (see Figure 09u);
it is the first successful attempt to combine general relativity and quantum
theory. The flow of negative energy (or mass) into the black hole would
reduce its 
mass. As the black hole loses mass, the area of its event
horizon gets smaller, but this decrease in the entropy of the black hole is
more than compensated for by the entropy of the emitted radiation, so that
the second law of thermodynamics is never violated. If we demand that in Eq.(16c)
Figure 09u Hawking Radiation
Figure 09v Blackhole Evaporation

dS dA as stated
originally (actually, it can be shown that S = (k_{B}c^{3}/4G)
x A), then T 1 / m, and the
rate of radiation L can be expressed as L
r_{s}^{2}T^{4}
1 / m^{2}. Therefore,
as the black hole loses mass, its temperature and rate of emission increase,
then it lose mass even more quickly (Figure 09v). What happens when the mass of
the black hole eventually becomes extremely small is not quite clear, but the
most reasonable guess is that it would disappear completely in a tremendous
final burst of emission.
It can be shown that the temperature T associated with the thermal radiation for
a black hole is:
T = 0.6 x 10^{7} m_{sun} / m (in degrees Kelvin)
where m_{sun} is the mass of the Sun. If the Sun is reduced to a black
hole, its temperature would be just about 10^{7} ^{o}K. On the
other hand, there might be primordial black holes with a very much smaller mass
that were made by the collapse of irregularities in the very early stages of the
universe. Those with masses greater 10^{15} gm could have survived to
the present day. They would have the size of a proton (~ 10^{13}cm) and
a temperature of 10^{11} ^{o}K. At this temperature they would
emit photons, neutrinos, and gravitons in profusion; they would radiate
thermally at an ever increasing rate, and sending out X rays and gamma rays to
be discovered. The lifetime of a black hole is roughly equal to
= m / L = 10^{35} m^{3}
year, where m is in gm. This makes an ordinary mass black hole (m ~ 2x10^{33}
gm for the Sun) live for a long time and its radiation unobservable.
Figure 09w Event Horizon of an Accelerating Observer
This phenomenon of Hawking radiation also occurs in the
event horizon created by an accelerating observer. Figure 09w shows that
light ray emitted at certain distance can never catch up with the observer
and thus an event horizon exists beyond which the observer cannot
communicate. Theoretical arguement suggests that even in empty space, the
observer will be able to detect radiation from the event horizon. A simple
formula is derived to express the relationship between the acceleration a
and the temperature T: 
T = a (/2k_{B}c).
It is suggested that members of the correlated virtual photon pairs are
separated by the event horizon. As a result part of the information is
missing, the observer detects random motion associated with the temperature.
In this case the energy is extracted from the acceleration, which according
to general relativity, is equivalent to gravitation. 
Meanwhile, other research in superstring theory finds the
black hole to be a "fuzzball" (Figure 09y). The modified black hole does not
possess a sharp event horizon; information can be stored in the strings and
imprinted on outgoing Hawking radiation. Models of black holes from
superstring theory also cast doubt on the idea of the singularity (at the
center of the black hole). Yet another scheme suggests that information
might leak out by means of quantum teleportation between the entangled pair
of virtual particles (one of which has escaped while the other is trapped
inside the black hole). However in the theory of loop quantum gravity, it has been shown that the information trapped in a black hole
will be unable to escape via Hawking radiation. But it will survive,
eventually rejoining the rest of the universe when the black hole evaporates.

Figure 09y Fuzzball

Another example that offers exact solution is the homogeneous and isotropic
space filled with pressureless dust. It is applicable to the case of the cosmic
expansion, if each dust point presents a galaxy. Universes of this type are
variously known as Friedman universes, FriedmanRobertsonWalker universes, ...
etc.
 The spacetime interval is associated with the RobertsonWalker metric. It
has three forms depending on the curvature of the 3
dimensional space:
ds^{2} = R(t)^{2} (dw^{2} + s^{2} (d^{2}
+ sin^{2} d^{2}))
 c^{2}dt^{2}  (17)
where s = sin(k^{1/2} w) / k^{1/2}, and R(t) is called the
scale factor.
For flat space with zero curvature, k = 0, and s = w.
For closed space with positive curvature, k = +1, s = sin(w), where w ranges
from 0 to 2.
For open space with negative curvature, k = 1, s = sinh(w), where w ranges
from 0 to infinity.
 In term of the RobertsonWalker metric and with the energymomentum tensor
T_{00} = , the
gravitational field equation becomes:
(dR/dt)^{2} + kc^{2} = 8GR^{2}
/ 3  (18a)
which can be rearranged into the form:
k^{2} c^{2} = H^{2} R^{2} (
 1)  (18b)
where H(t) = (dR/dt) / R is the Hubble's parameter,
=
/
_{c}, and
_{c} = (3 H^{2})
/ (8G) is the critical density,
which corresponds to the total energy density for a flat universe. Eq.(18b)
shows that
for flat space, =
_{c},
= 1, and k = 0;
for closed space, >
_{c},
> 1, and k = +1;
for open space, <
_{c},
< 1, and k = 1.
 More generally, when several types of matter coexist in the universe, the
following consistency relation must be satisfied:
_{j}_{j}
+ _{k} = 1
where the sum is over the various types of matter and
_{k} =  kc^{2}/(R_{0}H_{0})^{2}
with the subscript "0" standing for present quantities. Present cosmological
observations yield:
_{b} ~ 0.04 for
brayons (ordinary visible and nonluminous matter),
_{d} ~ 0.26 for
exotic dark matter,
_{}
~ 0.7 for dark energy,
_{}
~ 5 x 10^{5} for photons (radiation), and
_{k} = 0.
 If we choose = M / (4R^{3}/3)
 (18c)
where M may be interpreted as the total mass of the universe, Eq.(18a) becomes:
(dR/dt)^{2} + kc^{2} = 2GM / R  (18d)
The solution for this equation is:
for k = 0, R = (9GM/2)^{1/3} t^{2/3}, t = 2/3H
 (19a)
for k = +1, R = R_{o} (1  cos(µ)), ct = R_{o} (µ 
sin(µ))  (19b)
where R_{o} = GM / c^{2}, and µ is defined by c dt = R(µ) dµ.
for k = 1, R = R_{o} (cosh(µ)  1), ct = R_{o} (sinh(µ)
 µ)  (19c)
For the radiation (relativistic matter) dominated universe, the radiation
pressure p = /3. Starting
from the thermodynamic relation dU = pdV, where U =
V, and V is the volume of the
system (the universe in this case), it can be shown that
1/R4, Eq.(18a) (for
the case k = 0) becomes dR/dt
1/R, the solution for
which is R t^{½}.
Incidentally, since the temperature T is related to the energy density as
T^{4}, thus T
1/R.
 The red shift caused by the cosmic expansion can be expression in terms of
the scale factor R [see Figure 09z, which shows the red shift of the Balmer
series of the hydrogen line spectra (dark lines, denoted by H_{},
...) superimposed on the continuous spectra (colored), the arrows indicate the
direction of increasing values for the corresponding variables]:
z = _{1} /
_{2}  1 = R(t_{2})
/ R(t_{1})  1  (19d)
Figure 09z Cosmic Red Shift
where t_{2} denotes the current epoch,
_{2} is the
original wavelength (as measured on Earth),
_{1} is the
wavelength of a spectral line that was emitted at time t_{1} (red
shifted) in the distant past, R(t_{2}) is often taken to be 1 for
convenience, then R(t_{1}) is less than 1 in the past and is often
just denoted as R(t). The quantity (z + 1) can be considered as the amount
of stretching during the intervening time for the light to travel from
there to here. It is a curious relativistic effect that the velocity and
distance remain finite approaching the velocity of light and the event
horizon respectively even though 
the amount of stretching becomes infinity. The
relativistic expression for z+1 (In terms of the recession velocity v) is:
z + 1 = [(1+ v/c) / (1 v/c)]^{1/2}.

 From the observed value of the Hubble constant Ho = 71 km/secMpc for the
current epoch, and by substituting the speed of light c for v in the Hubble's
law: v = H_{o} d, it follows that the rate of cosmic expansion would be
exceeding the
speed of light beyond a distance d_{h} = c/H_{o}
= 1.27x10^{28} cm. However, there is no contradiction, because
special relativity states that no signal can propagate faster than the
speed of light, but the expansion of the universe cannot be used to
propagate a signal although it can move with superluminal speed. We will
not be able to see this region anyway because the event horizon is located
right at the boundary (Figure 10b), if we take the Hubble time T = 1/H_{o}
= 13.7 billion years (from WMAP data) to be the the age of the observable 
universe. Figure 10b illustrates the Hubble's law
with a spacetime diagram. Consideration of kinematics alone shows
that a subtle inconsistency 
is hidden in the original form of the formula.
Figure 10a Event Horizon
Figure 10b Spacetime Diagram
Since the Hubble's law is an
empirical formula, so the distance should be referred to the lookback
distances, which is measured some time in the past. Analysis of the spacetime
diagram shows that the formula is actually related to the comoving distance d,
which can be considered as measured by a ruler moving
together with our frame of reference. In Figure 10b, all
the objects emanated from the Big Bang are represented by the red solid
lines with different velocity v. They are straight lines because the
Hubble's law assumes constant velocity in cosmic expansion without
considering the effects of inflation, gravitational retardation, and dark
energy acceleration. The corresponding lookback distance is marked as s_{1},
s_{2}, s_{3}, ... until the lookback event horizon s_{h}
for object moving with the speed of light. The comoving distance in the

current epoch corresponding to different v is labeled by d_{1},
d_{2}, d_{3}, ... until the event horizon d_{h}.
The past light cone (in black solid line, the cone is made by rotating

this line around the time axis as shown by the small insert) indicates
light signal from objects with different rate of recession reaching the
observer at "Here and Now". It can be shown that d = vT = s (1+v/c).
For Hubble's original data in 1929 s ~ 1 Mpc and v ~ 500 km/s, the difference between the two
definitions of distance is about 0.001 Mpc.
Figure 10c Cosmic Distance
Figure 10d Quasar Data
The difference for higher redshifted objects would be considerably more (~ 10% for the furthest data point),
and up to cT/2 for v = c (Figure 10c). Thus, the original Hubble's law (associated
with observational data) in the form s = vT is valid only for v << c. The
green lines in Figure 10b depict the situation at a future time T'; it shows
the expanding lookback event horizon s'_{h}. The unobservable universe
occupies the region outside the future light cone at the Big Bang.
Quasars possess
very high red shift up to z = 2.5 corresponding to v/c = 0.85. Figure 10d
shows the observational data relating red shift to the apparent visual
magnitude. The red shift can be translated to the recession velocity by the
formula v/c = [(z+1)^{2}1]/[(z+1)^{2}+1], while the apparent
visual magnitude
m can be expressed in terms of the absolute magnitude M and the lookback
distance s (in cm) as m = M  97.5 + 5xlog(s). The deviation from the standard
Hubble's law has prompted argument against the association of the quasar red
shift to the cosmic expansion. However, if we compare the observational data
in Figure 10d with the modified Hubble's law (s) for the lookback
distance in Figure 10c, they are in general agreement. For example, if the
absolute magnitude for the quasar is M = 25.5, then a point such as
(v/c=0.8,s=0.45) on the lookback distance in Figure 10c would correspond to a
point (z=2.0,m=16) in Figure 10d. The deviation to the left may be related to
the original estimate of M by assuming the comoving distance (as in the
Hubble's law d = v/H_{0}). The intrinsic luminosity would be brighter
if it is further away.
Note that the modern Hubble time is close to T = 13.7x10^{9} years.
The age of the universe as derived by Eq.(19a) from the standard cosmology for
flat space is 2T/3 = 9.13x10^{9} years, which is considerably shorter
than the ages of many astronomical objects. The difference arises because the
standard cosmology takes into account the slowing down of the expansion.
Nevertheless, the Hubble time seems to yield a more sensible age by a lucky
coincidence that the total amount of slowing down due to gravity and the total
amount of speeding up due to dark energy almost exactly compensate each other.
Figure 10e shows the three types of universe  open, flat, and closed.
Universes that are too far above the critical divide expand too fast for
matter to condense into stars and galaxies; such universes therefore
remain devoid of life. Those that fall too far below the critical divide
collapse before stars have a chance to form. The shaded area indicates the
range of cosmological expansions and epochs in which observers could
evolve. By measuring the sum of the angles in a huge narrow triangle with
one vertex on Earth and the other two at the points of emission of CMBR (see
insert in Figure 10e), the WMAP data yields a value of 180^{o} indicating that the universe is

flat. Such conclusion is further supported by the
acceleration of Type Ia supernovae, which imply a precise amount of dark
energy needed to make the universe flat (with the total matterenergy
density equals to the critical density).
Figure 10e Types of Universe

Note: For closed space the sum of the angles in a triangle would be greater
than 180^{o}; it is smaller than 180^{o} for open space; it
equals to 180^{o} exactly for flat space.
It is found lately that the cosmic expansion may be accelerating. An
additional term with the cosmological constant is added to the gravitational
equation (18a) as a possible candidate for the dark energy to drive the
acceleration. It was originally introduced by Einstein to address the failure of
constructing a static universe. Thus Eq.(18a) becomes:
The de Sitter universe is devoid of matter energy containing only vacuum
energy. It follows that Eq.(20a) is simplified to:
(dR/dt)^{2} + kc^{2} =
R^{2} c^{2}
/ 3 or (dR/cdt)^{2} 
R^{2} / 3 = k
 (20b).
For k = 0, the solution is: R(t) = R_{0} e^{Ht}
where H = (/3)^{1/2}c
= (dR/dt)/R is the Hubble constant, and R_{0} is a constant.. This
solution shows the weird property of a constant matterenergy density
_{},
i.e., vacuum energy is continuously being created to fill up the void in the
wake of the expansion. The idea is similar to the steady state universe, which requires the continuous creation of matter. Since the event horizon is
d_{h} = c / H = 1 / (/3)^{1/2}
= constant; therefore, like the earth's horizon, the de Sitter horizon can
never be reached  it is always a finite distance away. The de Sitter
universe with k = 0 starts at t

from a singularity. It reaches a size of R_{o} at t = 0, and expands
to infinity as t +.
Alternatively, R_{o} at t = 0 can be taken as the initial condition.
The universe then can either grow or decay exponentially. Note that Eq.(20b)
is time reversal invariant. For some reason, this universe chose to grow
exponentially in the positive time direction. 
For k = +1, R(t) = (c/H) cosh(Ht)
(c/2H) e^{Ht}
as t _{
}. Figure 10g de Sitter Universe with k = +1

Figure 10g depicts a de Sitter universe with k = +1 from t

to t +.
It shrinks to a minimum size of (c/H) at t = 0.
For k = 1, R(t) = (c/H) sinh(Ht)
(c/2H) e^{Ht} as t
. Its size shrinks to
zero at t = 0. There is no solution for negative t as R also becomes negative.
All de Sitter universes expand forever exponentially. Such behaviour is similar
to the situation at the very beginning and very end of our universe, either when
energy matter had not been created or they have been diluted so much at the end.
The antide Sitter (AdS) universe is also described by Eq.(20b) except that is now negative.
For k = 1, R(t) = (c/H) sin(Ht).
where H = (/3)^{1/2}c.
There is no solution for k = 0, and k = +1. Since a negative implies attraction, the AdS
undergoes a cycle of expansion and contraction with a time scale of
/H (see Figure 10f).
The simple de Sitter, antide Sitter models suggest that for large positive
, everything flies apart so
quickly that there is no chance for matter to assemble itself into sturctures
like galaxies, stars, planets, atoms or even nuclei. On the other hand, with
large negative , the
expanding universe quickly turns around and terminates the evolution of life
before it can arrive at the present state. The estimated value of about 0.7x10^{29}
gm/cm^{3} seems to be fine tuned for the existence of life. Nobody is
able to find an explanation for such coincidence yet.
^{¶} The steady state universe requires continuous creation of matter in
order to keep the expanding universe uniform everywhere and anytime. The
refutation of the theory came with the observation that quasars were found only
at large distances. The discovery of microwave background radiation, which
indicates a much denser state further back at the time of Big Bang, finally
ruled it out completely.
Astronomical observations over the last 20 years have
indicated that the universe has experienced two periods of acceleration (see
Figure 10h). The first one was a very rapid expansion soon after the Big
Bang. It is usually referred to as inflation. The dark energy
accelerated expansion is gentler, and has occurred only recently in the
current epoch. The inflation is explained by an as yet unidentified scalar
field (inflaton) undergoing a phase transition. While there are three
categories of theory for the accelerated expansion: modifications to general
relativity perhaps with extra dimensions, a cosmological constant, and a
universal evolving scalar field. None of these offers a satisfactory
explanation. 
The followings provide a mathematical description of cosmic
inflation and expansion without invoking a detailed mechanism. Figure 10h Cosmic Inflation

 If we assume a homogeneous and isotropic universe, with the total mass to
be constant as expressed in Eq.(18c), then Eq.(18a) can be recast into a form that shows the acceleration /
deceleration explicitly:
(d^{2}R/dt^{2}) / R =  4G
/ 3  (20c)
This formula shows that the matter dominated universe in general would
experience deceleration as the term on the right is always negative.
 In the presence of other forms of matterenergy, a pressure term p has to
be added into Eq.(20c):
(d^{2}R/dt^{2}) / R =  (4G
/ 3c^{2}) (c^{2}
+ 3p)  (20d)
 The matterenergy density
and the pressure p are usually related by the equation of state, which can be
expressed as p = wc^{2},
thus Eq.(20d) can be rewritten as:
(d^{2}R/dt^{2}) / R =  (4G
/ 3) (1 + 3w)  (20e)
For relativistic matter, w = 1/3. For nonrelativistic
matter, w = 0. For the de Sitter universe, w = 1. The expansion of the universe will accelerate
if
w < 1/3, when the righthand side in Eq.(20e) becomes positive. The
negative pressure is a characteristic of expansion under constant density
as shown by a simple example. It is also known as false vacuum. In the simplest version of
the inflationary paradigm a single scalar field (inflaton) dominates the
energy density of the universe. To achieve the acceleration condition
w < 1/3 and the observed properties of our universe, the inflation must
evolve slowly such that the potential energy dominates over the kinetic
energy for a sufficient part of the inflation. Figure 10i shows a theory (the
old one) that doesn't work because the scalar field (the Higgs field)
evolves rapidly. 
While the newer theory is just right creating an universe
as we see it today. This scenario remains ambiguous as the precise form of
the scalar field is unknown, and there is still nagging doubt about the
occurrence of inflation. Figure 10i Theories of Inflation

 Another way to achieve acceleration can be derived from a constant energymatter
density term (the cosmological constant) as shown in Eq.(20a), which can be recast in the form:
(d^{2}R/dt^{2}) / R =  4G
/ 3 + c^{2} / 3
 (20f)
This equation shows that at some point in the evolution of the universe the
righthand term would become positive, and the expansion would switch from
deacceleration to acceleration. This condition yields the change over epoch at
z = 0.72 for _{m} =
0.3, and
= 0.7. Effort to identify the cosmological constant with the vacuum energy of
the various quantum fields is not very successful. The simplest versions of
quantum theory predict far too much energy  10^{120} higher than the
observed value by one estimate.
 The explanation of the current acceleration in term of an evolving scalar
field presents a tremendous hierarchy problem. The very high energy (10^{16}
Gev) associated with the scalar field in the inflation period requires an
unreasonable amount of finetuning to account for the extraordinarily low
energy (10^{48} Gev) of the scalar field in the current epoch.
 A different explanation was proposed by a research team in 2005, they
links the dark energy acceleration to the gigantic ripples in spacetime
created in the epoch of inflation. In effect, it is like hobbling amid a huge
undulating wave in the ocean, you don't actually see the wave (the wavelength
is too long)  all you feel is the swell moving up and down.
As z
0, d
(c/H_{o}) z , and
1/z
while for z , d
(c/H_{o}) / q_{o}z,
and
z .
These limiting cases clearly demonstrate the curious effect
that the angular size of an object becomes larger as it is further away from
Earth. It appears to decrease with distance only for nearby objects. Figure
10k shows the infrared blobs produced by the first stars (high z objects).
It is suggested that the appearance of the puffy blobs with large angular
size is caused by the expansion of the universe with z > 1.6 as shown in
Figures 10i and 10k. Figure 10k Infrared Blobs 

The Minkowski spacetime in relativity has the signature (,+,+,+). The
negative signature in the time component has obscure properties that make
visualization very difficult as well as creating novel features not easily
comprehensible. In many instances, the problem can be resolved partially by
substituting t it, which
transforms the Minkowski spacetime to the Euclidean fourdimensional space with
signature (+,+,+,+). Sometimes, the transformation is reversed back to the
original Minkowski spacetime at the end of the computation. However, some
physicists prefer to treat the Euclidean space as the ultimate reality, while
the Minkowski spacetime is considered just a figment of our imagination  a
point of view not yet validated by observation. Actually, even a fourdimensional
Euclidean space is very difficult to visualize. So let us start with a threedimensional
space as we experienced all our life.
In the fourdimensional Euclidean space, the 3D hypersphere is defined by the
formula:
x_{1}^{2} + x_{2}^{2} + x_{3}^{2}
+ x_{4}^{2} = a^{2}
and ds^{2} = dx_{1}^{2} + dx_{2}^{2} +
dx_{3}^{2} + dx_{4}^{2} = a^{2} (d^{2}
+ sin^{2} d_{2}^{2})
 (20h).
where d_{2}^{2}
= (d^{2} + sin^{2}
d^{2}) is an unit 2D
sphere equivalent to d^{2},
and d^{2} assumes the
role of d^{2} in Eq.(20g).
The formulation can be generalized further to 5D Euclidean space with the 4D
hypersphere metric:
ds^{2} = a^{2} (d^{2}
+ sin^{2} d_{3}^{2})
 (20i).
where the 3D d_{3}^{2}
= (d^{2} + sin^{2}
d_{2}^{2}) is
equivalent to d_{2}^{2}
in Eq.(20h).

The no boundary proposal makes use of Eq.(20i) and suggests that at the
beginning the universe was running with
as the time component from
= 0 to
/2 in the Euclidean space.
At /2 the character of the
time component underwent a transformation, e.g.,
/2 + ict/a. Since
then the 4D hypersphere has changed into a Minkowski hyperboloid described
by the metric:
ds^{2} = a^{2} [c^{2}dt^{2} + cosh^{2}(ct/a)
d_{3}^{2}]
Such a hybrid universe is shown in Figure 10n. Thus, there is no singularity
at the beginning of time. That particular region becomes as smooth as the
North (South) pole on Earth. It is 
Figure 10n Hybrid Universe

suggested that the (red) region in Euclidean time represents
the moment of nucleation via tunneling through some sort of energy barrier
from nothing (in the form of vacuum fluctuation for example).

The expression for d_{2}^{2}
is just the metric on 2D unit sphere (an ordinary sphere with radius equal to
1) in a 3D space as shown by Eq.(20g). Similarly, d_{3}^{2}
is the metric on 3D unit hypersphere in a 4D space. In general, d_{n}^{2}
denotes the metric on ndimensional hypersphere in a (n+1)dimensional space. It
is always possible to create ndimensional hypersphere and other geometrical
shapes in a (n+1) dimensional space including the Euclidean and many others. For
example, a 4D hypersphere can be embedded in a 5D antide Sitter space with
metric in the following form:
ds^{2} = d^{2}
+ sinh^{2}(/l) d_{4}^{2}
 (20j).
where l = (/3)^{1/2},
and d_{4}^{2}
= a^{2}(d^{2} +
sin^{2} d_{3}^{2})
is the 4D hypersphere.
This 5D AdS space has been used to apply the holographic principle to physical theories or objects (such as superstring theory or black hole) by
encoding them from such 5D space to a 4D hypersphere.
It is found that when 4dimensional theory is formulated in
a 5dimensional spacetime, novel features emerge as the theory is reduced
back to 4D. The KaluzaKlein theory introduced in the 1920s is the most
famous example, which unifies the theories of gravitation and
electromagnetism. The technique of compactification to hide the extradimension
is now used extensively in the superstring theory. It is from the process of compactification that
produces various particle properties. 
Figure 10o 5D Spacetime

Let us first consider the simple example of the massless KleinGordon
equation in quantum field theory. In 4D spacetime, this equation has the form:
(x_{})
= 0  (20k)
where _{
}
,
and the index _{} runs from 1 to 4
(with the 4th index identified to the time component).
The extension to 5D spacetime can be accomplished very easy by including an
extra variable x_{0} in the formulation, and let _{
}runs from 0 to 4. Then Eq.(20k)
becomes:
_{5}(x_{},x_{0})
= 0  (20l)
where an additional second partial derivative in x_{0} is inserted into
the equation. Such 5D spacetime is shown in the left of Figure 10o. Since the
real world is three dimensional in spatial coordinates, we can consider the
unobservable extra dimension to curlup into very small circle (at every point
in the 3D space) as shown in the right of Figure 10o. Under this circumstance,
the extra dimension would become periodic with x_{0} = x_{0} + 2R,
where R is the radius of this space. Consequently, the scalar field can be
expressed in terms of periodic functions:
(x_{},x_{0})
= (x_{})
e^{ip0x0}
where p_{0} = n / R to satisfy the periodic condition, and n is an
arbitrary integer. Thus, Eq.(20l) can be reduced to the form:
_{5}
=
 (n / R)^{2} = 0
 (20m)
which is just the 4D spacetime Klein Gordon Equation with effective mass = (n
/ R). This is a very simple example of KaluzaKlein tower (also known as KK
particles, KK modes), which in effect is the energy of the standing waves in the
extra compactified dimension (Figure 10p). In the n = 0 mode, the KK particle
will be indistinguishable from the known particles. The lightest KK particle
corresponds to n = 1; it would have the same charge as the known particles but
different mass.
If experimenters discover new heavy particles with the same
charges as familiar ones and masses that are similar to one another, those
particles will be strong evidence of extra dimensions. If such particles
also occur at regular intervals of mass, it would very likely mean that a
simple curledup dimension has been discovered. But more complicated extradimensional
geometries will yield more complicated patterns of masses. If enough such
particles are discovered, the KK particles would then reveal not only the
existence of extra dimensions, but also the their sizes and shapes. Since no
KK particle has been detected so far, it seems to indicate that the size of
the curledup dimension could be very small. 
In unit of ev, the mass of the KK tower M_{n} =
(n/R) x 1.2 x 10^{4} ev. Thus, the new LHC collider (to be
operational in 2007) with available energy up to 14 Tev can probe the curledup
dimension down to the size of about 10^{17} cm.
Figure 10p KK Tower

Similarly in the KaluzaKlein theory, the gravitational field equation in 5D
spacetime can be reduced to:
Nowadays, we know that electromagnetism and gravity are far
from being the whole story. A satisfactory unified theory must accommodate a
good deal more like the weak and strong interactions. In fact, five
dimensions are not enough; we might just manage with ten (such as in the
string theory). The KaluzaKlein method has been generalized to YangMills
theory in (4+N)dimensional spactime. If supersymmetry is added to the
formulation, the new components of the super curvature tensor emerge as 
quarks and leptons. Such formulism is called supergravity.
The decomposition is illustrated in a much simplified form in Figure 10q.
Figure 10q Supergravity

It seems that everything is there. The most serious problem with quantum
KaluzaKlein theory is that it is nonrenormalizable; and the fact that certain
particles are missing in this picture eventually forces physicists to develop a
more powerful formalism: the superstring theory.
But the central theme of KaluzaKlein theory remains: the physical laws depend
on the geometry of hidden extra dimensions.
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