Relativity

Contents

Classical Mechanics

where G is the gravitational constant, m1 and m2 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 (see Figure 01). If one of the objects is much heavier than the other, e.g., m1 >> m2 like the Sun / Earth system, then m1 can be placed in the origin of the coordinate system and Eq.(1) can be solved as a one-body problem. In case the two masses are similar, the problem can be reduced to a one-body problem with a fictitious object moving around the center of mass, and Eq.(1) is still applicable. The equation of motion becomes rapidly un-manageable 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

Figure 01 Gravitational Interaction [view large image]

solution is usually obtained by some kind of approximation and by numerical computation using large computers.

	dx2 + dy2 + dz2 = 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 Eq.(4) the velocity of light c would be different with c' = c - V.
Figure 02 Galilean Transformation [view large image]

Special Relativity

• 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 = to. According to the 1st formula in Eq.(8a), the relation is given by: L = x2 - x1 = L' (1 - V2/c2)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[view large image]	Figure 04a Time Dilation [view large image]

• 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 = t₂ - t₁ =
  T' / (1 - V²/c²)^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 04a (where = (1 - V2/c2)-1/2), show the muon decay as experienced in its own frame and from the view point of a stationary observer on Earth. The twins paradox is another consequence of time dilation. As shown in Figure 04b, one of a pair of twins (a1) leaves on a space journey during which he travels close to the speed of light (c), while his brother (b1) remains on Earth. Because of (a1)'s motion, time runs more slowly in the space-craft as seen by the earthbound twin. So on his return the space traveler (a2) will find that his brother (b2) has aged more than himself. Although it seems against common sense, a number of experiments have implied that in this scenario the traveling twin would indeed be younger.

  Figure 04b Twins Paradox [view large image]

  Note that the formula T = T' is valid even when the body is accelerated, and the two twins are not interchangeable, as there is no inertial frame in which the traveling twin is always at rest.

  Events (a point in the four dimensional space-time as shown in Figure 05) that take place simultaneously to an observer in S, e.g., t = to at x = x1 and x = x2, are separated by a time interval t'2 - t'1 = (V(x1 - x2)/c2) / (1 - V2/c2)1/2 in S' according to the 4th formula in Eq.(8a) (see Figure 06). For an infinitesimal interval of space-time, Eq.(6) can be written in the form: dx2 + dy2 + dz2 - c2 dt2 = 0 ---------- (9)

  Figure 05 Reference Frame [view large image]

  This is called the null line applicable to object moving at the speed of light. It can be shown that for object moving under the speed of light the space-time interval
  
  ds² = dx² + dy² + dz² - c² dt² ---------- (10)
  
  is also invariant under the Lorentz transformation. ds is called the proper time since it is the interval for an object at rest with dx = dy = dz = 0. In general, if [(dx/dt)² + (dy/dt)² + (dz/dt)²] = v², Eq.(10) can be re-written into the form:

  ds2= - (1-v2/c2)c2dt2. The interval is called time-like if ds2 < 0 for v < c. As shown in Figure 05, 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 ds2 > 0, then it is called space-like, which implies Event 3 is entirely unrelated to Event 1. Alternatively it can be interpreted that two events joined by a space-like interval can never influence each other, since that would imply a flow of information at speeds faster than the speed of light. The interval ds plays the role of the time parameter in Newtonian mechanics to keep track of the development of events such as in the generalized equation of motion in Eq.(12b).

  Figure 06 Rotation in Minkowski Space-time[view large image]

  Note that the velocity of light c is constant in all Lorentz frame of references in Figure 06 as originally envisioned.

• It is obvious that Eq.(10) is not a sum of squares of the coordinate differentials. One of these is associated with an opposite sign. This is called pseudo-Euclidean geometry and is the reason for the un-usual appearance of the space-time axes when subject to a rotation in the Minkowski space-time as shown in Figure 06. Mathematically, the Minkowski space-time rotation can be expressed in a form resembling to the 2-dimensional rotation by re-writing Eq.(8b) in term of the hyperbolic functions:
  
  x = x' cosh(A) + c t' sinh(A)
  ct = x' sinh(A) + c t' cosh(A)
  
  where tanh(A) = (v/c), sinh(A) = (v/c) , cosh(A) = are the hyperbolic functions, and = 1 / (1 - v² / c²)^1/2. As v approaches c, tanh(A) ~ 1, the x' and ct' axes merge together at the null line (see 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/c²) in special relativity.
• The mass for a particle moving at velocity V is given by m = m_o / (1 - V²/c²)^1/2, where m_o is the rest mass (relative to the observer's rest 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 c² = m_o c² + 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_o² c⁴ + p² c²)^1/2. The three-momentum and energy together form a four-vector
  (p_x, p_y, p_z, E/c) in the Minkowski space. The negative of its magnitude - m_o² c² = p² - E² / c² is an invarant quantity under Lorentz transformation.

General Relativity

		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 07a Free-falling Frame [view large image]	Figure 07b Equivalence Principle [view large image]	elevator) would experience zero gravity as shown in Figure 07a. Figure 07b shows the similar kind of situation in producing gravity with acceleration. The inter-changeable nature of gravity and acceleration is known as the principle of equivalence.

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/(c2a(1 - e2)) ---------- (12c) where M is the mass of the Sun, a is the length of the semi-major axis, and e is the eccentricity of the ellipse. In Figure 08, 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 08 Perihelion Advance [view large image]

Schwarzschild's Solution and Black Hole

• As r , the space-time metric reduces to the expression for flat space-time:
  
  ds² = - c²dt² + dr² + r² (sin² d² + d²) ---------- (14)
  
  
• At the non-relativistic limit where v²/c² << 1, an expression for the space-time metric can be derived from classical mechanics:
  
  ds² = - (1 + 2V/c²) c²dt² + dr² + r² (sin² d² + d²)
  
  where V is the gravitational potential.
  Comparing with the space-time metric g₄₄ in Eq.(13) in the region where r > 2GM / c², the Newton's law of gravitation: V = - GM / r is recovered from general relativity. Thus, at the non-relativistic limit only the time dilation effect (relating to g₄₄) is important. The effect of spatial curvature (relating to g₁₁) is negligible at this limit.
  
  
• At the distance where r = r_s = 2GM/c², 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 rs is outside the central object. For the Earth, and the Sun, rs is well inside the physical boundary (rs 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 [view large image]	Figure 09b Embedding Diagram [view large image]	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 - rs/r)1/2 At the Schwarzschild radius rs, the redshift becomes infinity. This is the effect that makes light invisible at rs.
Figure 09c Redshift [view large image]

• 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
  
  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 rs. 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 rs is equal to 2GM / rs3 = c6/(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 rs from the center of a black hole.
  Figure 09d Tidal Force	Figure 09e Time Dilation [view large image]

  Inside rs, 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 g44 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 [view large image]

  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 (co-moving objects are stationary in that frame), then

  Figure 09g White Hole [view large image]

  c2 d2 = -ds2 = -g44 c2 dt2 where g44 is negative according to the convention. Thus,

  d = ± (-g₄₄)^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 space-time metric for a static wormhole (also known as the Einstein-Rosen Bridge) can be expressed in the form:
  
  

  ---------- (15a)

  The lapse function defines the proper time between consecutive layers of spatial hyper-surfaces; 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 / c2 = rs. The throat of the wormhole is located at r = b(r) = rs 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 [view large image]

  Another way to conceptualize a wormhole topology is to have the spatial part of the space-time metric in Eq.(15a) imbedded in a flat hyperspace with the extra-dimension denoted by W:

  dl² = dW² + dr² + r² (sin² d²) = dr² / (1 - r_s/r) + r² (sin² d²) ---------- (15b)
  
  

  		Eq.(15b) can be used to equate dW = (r/rs - 1)-1/2dr. Integration of the equation gives W2 = 4rs(r - rs), which is a parabola function with vertex at W = 0, r = rs. 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.
  Figure 09i Worm- hole in Hyperspace [view large image]	Figure 09j Worm- hole Throat [view large image]

  Inside the wormhole, g₁₁ and g₄₄ change sign as shown in Eq.(13). The region becomes space-like, 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(M-m)G/c2), 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
  Figure 09k Space Travel [view large image]	Figure 09l Time Travel [view large image]	two distant points (see Figure 09k), or even for the possibility of time travel (Figure 09l).

Kerr's Solution and Rotating Black Hole

---------- (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 space-time. 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 space-time (same as Eq.(14)):
  
  ds² = - c²dt² + dr² + r² (sin² d² + d²) ---------- (15g)
  
  

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 non-rotating black hole such special trajectory occurs at r = 3rs/2 = 3GM/c2. The sphere with such a radius is

Figure 09m Bending of Light [view large image]

called the photon sphere. However, the orbit is unstable; it can be disrupted with very small perturbation. Mathematically, such precarious orbit is defined by d2r / d2 = 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 co-rotating light ray.


• As r descends further toward the center, the space-time metric g₄₄ (or g_tt) vanishes at
  
  r = [GM + (G²M² - a²c²cos²)^1/2] / c² ---------- (15i)
  
  

  This is called static limit. It can be intuitively characterized as the region where the rotation of the space-time is dragged along with the velocity of light. Within this region, space-time is warped in such a way that no observer can maintain him/herself in a non-rotating orbit, but is forced to become co-rotating (Figure 09n). The surface of this region is elliptical with its major axis at = /2 (the equator), and r = 2GM / c2 (= the non-rotating Schwarzschild's radius). The minor axis is in the directions of = 0, and (the poles), and r = [GM + (G2M2 - a2c2)1/2] / c2 (see Figure 09o).

  Figure 09n Frame Dragging [view large image]

• 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 first event horizon occurs at
  
  r = [GM + (G²M² - a²c²)^1/2] / c² ---------- (15j)
  
  The inner horizon (sometimes called the Cauchy horizon) is located at
  
  r = [GM - (G²M² - a²c²)^1/2] / c² ---------- (15k)
  
  

  The space-time metric g11 (or grr) changes sign when crossing over the first event horizon, and then reverses back again at crossing over the second horizon. Therefore, at the outer event horizon, it allows objects to move only towards the center. However, when an object passes the inner event horizon, it is able to move in directions away from the center, pass through another set of inner and outer event horizons, and emerge out of the black hole into another universe or another part of this universe. The separation between the two horizons is 2 (G2M2 - a2c2)1/2 / c2. As the spin increases, the two horizons move toward each other and merge at r = GM / c2 when a / c = GM / c2. In case a / c > GM / c2, there will be no event horizon, the black hole becomes a "naked

  Figure 09o Kerr's Solution [view large image]

  singularity", i.e., it is not covered by an event horizon. Note: 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.

  
  
  
• 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:
  
  ds² = - (1 - 2GM / c²r) c²dt² + (2GMa / c²r) dt d + (a²/c²) (1 + 2GM / c²r) d² ---------- (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:
  
  ds² = - c²dt² + cos² dr² + (a²/c²) (cos² d² + sin² d²) ---------- (15o)
  
  Interesting theoretical physics can take place around this ring singularity. Since 1 g₁₁ = cos² 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.




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 disk-shaped 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 09p).

Figure 09p Disk and Jets [view large image]

Hawking Radiation

		the event horizon may allow one member of the virtual particle / anti-particle 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 09q); 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
Figure 09q Hawking Radiation [view large image]	Figure 09r Blackhole Evaporation [view large image]	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)

This phenomenon of Hawking radiation also occurs in the event horizon created by an accelerating observer. Figure 09s 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:

Figure 09s Event Horizon of an Accelerating Observer [view large image]

T = a (/2kBc). 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.

Since information is defined as the opposite of entropy - an increase of entropy implies a decrease of information and vice versa - a question arises about what happens to the information in a black hole. While relativity seems to suggest that information about matter falling into a black hole would be lost, quantum mechanics seemed to be suggesting it would eventually escape. Hawking claimed the random nature of Hawking radiation meant that while energy could escape, information could not. But in the summer of 2004, he changed his mind. Much of the impetus for the rethinking comes from the superstring theory, which presents the black hole as a "fuzzball" (see Figure 09t). The modified black hole does not possess a sharp event horizon; information can be stored in the strings and imprinted on outgoing Hawking radiation. Other scheme suggests that information might escape by means of quantum teleportation. Models of black holes from superstring theory also cast doubt on the idea of the singularity (at the center of 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. Thus, the question has not been answered to everybody's satisfaction, and Hawking may still be able to recover his losing bet (on disappearing information).

Figure 09t Embedding Diagrams of BH and Fuzzball [view large image]

Standard Cosmology

• The space-time interval is associated with the Robertson-Walker metric. It has three forms depending on the curvature of the 3 dimensional space:
  
  ds² = R(t)² (dw² + s² (d² + sin² d²)) - c²dt² ---------- (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 Robertson-Walker metric and with the energy-momentum tensor T₀₀ = , the gravitational field equation becomes:
  
  (dR/dt)² + kc² = 8GR² / 3 ---------- (18a)
  
  which can be re-arranged into the form:
  
  k² c² = H² R² ( - 1) ---------- (18b)
  
  where H(t) = (dR/dt) / R is the Hubble's parameter, _c = (3 H²) / (8G) is the critical density, 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 / (4R³/3) ---------- (18c)
  
  where M may be interpreted as the total mass of the universe, Eq.(18a) becomes:
  
  (dR/dt)² + kc² = 2GM / R ---------- (18d)
  
  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 / c², 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)




From the observed value of the Hubble constant H = 71 km/sec-Mpc for the current epoch, and by substituting the speed of light c for v in the Hubble's law: v = H d, it can be shown that the rate of cosmic expansion would be exceeding the speed of light beyond a distance of 1.27x1028 cm. However, there is no contradiction, since the cosmic expansion is about the expansion of space - a general relativity phenomenon , while the limit imposed on the speed of light is a special relativity effect related to motion through space. Note that this so-called Hubble distance d is exactly equal to the distance to the event horizon (Figure 10a). This is not accidental. For the event horizon is defined as the distance do covered by light during the age of the universe to, i.e., do = c to; since H = 1 / to, d and do are given by the same formula, although for different

Figure 10a Event Horizon [view large image]

reasons. It is also clear from the formula that the boundary will expand with the age of the universe bringing more objects into our view.




• The redshift caused by the cosmic expansion can be expression in terms of the Robertson-Walker metric R:
  
  z = (₂ - ₁) / ₁ = R(t₂) / R(t₁) - 1 ---------- (19d)
  
  where t₂ denotes the current epoch, ₂ is the observed wavelength, and ₁ is the wavelength that was emitted at time t₁ in the distant past.
  
  

  Figure 10b 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 10b Types of Universe [view large image]

• 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. It was originally introduced by Einstein to address the failure of constructing a static universe. Thus Eq.(18a) becomes:
  
  

  (dR/dt)2 + kc2 = 8GR2 / 3 + R2 c2 / 3 ---------- (20) where is the cosmological constant. It can be expressed in term of the corresponding density : = 8G / c2. The effect of the cosmological constant on the cosmic expansion is summarized in Figure 10c. Assuming a flat universe, current observations of distance supernovae, the cosmic microwave background radiation, and the dynamics of galaxies favor a value of = 0.7 c. Past attempts to identify the cosmological constant with the vacuum

  Figure 10c Cosmological Constant [view large image]

  energy of the various quantum fields was not very successful. The fact that is constant during the cosmic expansion, makes it more similar to the hypothetical inflaton field at the inflation phase close to the moment of the Big Bang.

Gravitational Wave

dimensional space-time. 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 10d 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 01 compares the properties of these two kinds of wave.

Figure 10d EW and GW [view large image]

Table 01 Electromagnetic and Gravitational Waves

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 ~ 104 cycles and an ammonia molecule is roughly 10-60 cm2. Figure 10e 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

Figure 10e GW Detector [view large image]

transducer, which also converts the signal into an electromagnetic one. The signal is then amplified and acquired (see Figure 10e). It is suggested that large-scale astronomical motions of matter could generate appreciable gravitational energy flux.

	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
Figure 10f GW from Binary Pulsars [view large image]	agreement with observations as shown in Figure 10f. Thus, the observation indirectly confirms the phenomena of gravitational radiation.

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 10g). 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.

Figure 10g LISA [view large image]

Time

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 11) and the sandglass or hourglass, in which time is measured by the change in level of flowing water or sand. The

Figure 11 Clepsydra [view large image]

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

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:


• Thermodynamic 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 12a (from 1 to 9), because it is highly improbable for the crumbling dust to return to the structured building in the reversed direction. It 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. This is the

Figure 12a Entropy - Thermodynamic Arrow [view large image]

thermodynamic arrow of time in Figure 12b.

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 (see imaginary time) 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 [view large image]

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 time-traveller 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. 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 (Figure 12c).

Figure 12c Time Machine [view large image]

• 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. In short, the units of time are often defined by cyclic motions and their subdivisions. The accuracy in clocks has improved over the last 700 years until today, an atomic clock known as NIST-7 is accurate to 10-9 second per day. Modern quartz clocks use the piezo- electric 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 13). Atomic clocks use the frequency of vibration of atoms to regulate a quartz crystal clock (see Figure 14). One second is now defined as the duration of 9,192,631,770
  Figure 13 Clocks [view large image, top, bottom]	Figure 14 Atomic Clock [view large image]	periods of the radiation from the hyperfine transition of the cesium atom.

• 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 alternate computational technique and conceptional viewpoint as shown in the following examples.
  
  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(t₁) to q(t₂) 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.
  
  

  Meanwhile, if the imaginary time is substituted into Eq.(10), it changes into a form familiar to Euclidean geometry: ds2 = dx2 + dy2 + dz2 + c2 dT2 ---------- (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 space-time to be finite in extent and yet to have no

  Figure 15a The "No Boundary" Proposal [view large image]

  singularities that formed a boundary or edge. As shown in Figure 15a, space-time would be like the surface of the earth, only with two more dimensions. The universe has zero size at the North and South Poles,

  but these points would not be singularities, any more than the Poles on earth. The corresponding universe in real time would have a minimum size (at about 14x10⁹ years ago), which corresponds to the maximum radius of the histroy in imaginary time. At later real times, the universe would expand at an increasing inflationary rate to a very large size (see Figure 15a). In short, the evolutionary sequence is reversed in each of these versions, i.e., the large size is mapped into the small size and vice versa. It is found that the South Pole in Figure 15a should be slightly flattened in order to recover the observed behaviour of cosmic expansion (from a singularity to a brief inflationary phase followed by long period of deceleration) when the imaginary time is translated back to real time. To avoid the singularity, it is suggested that the so-called imaginary time is really the real time, and that what we call real time is just a figment of our imaginations.

  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. It can be shown that it is related to the Wheeler-DeWitt Equation (which looks like the zero energy time-independent Schrodinger equation): [ d2 / d R2 - 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

  Figure 15b Wave Function of the Universe [view large image]

  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 15b).

  The psychological time (subjective 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." (Figure 16) 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.

  Figure 16 Einstein

  and Friend [view large image]

• 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. Note that this kind of representation 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 so-called 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
Figure 17 End of Time [view large image]	Figure 18 Configuration Space [view large image]	space and time are discrete quantities and evolve dynamically like the atoms.