The Theory of Relativity Lee Chul Hoon

chulhoon@hanyang.ac.kr

Copyright © 2001 by Lee Chul Hoon

Table of Contents 1. Introduction2. The Special Theory of Relativity

The Galilean Transformation and the Newtonian RelativityElectromagnetism, the Speed of Light and the Michelson-Morley ExperimentThe Lorentz Transformation and The Postulates of Special Relativity Derivation of the Lorentz transformation equationsRelativistic Kinematics Simultaneity Comparison of Time Interval Comparison of Lengths Proper Time Interval How the Velocity of a Particle Transforms Alternation and Doppler Effect in RelativityRelativistic Dynamics Relativistic Momentum The Equivalence of Mass and Energy Relativistic Force Law The Compton EffectRelativity and Electromagnetism The Invariance of Maxwell's Equations and the Transformation of Electric and Magnetic Fields Equation of Motion of a Charged Particle in an Electromagnetic Field The Fields of a Uniformly Moving Charge Forces Between Moving Charges Electromagnetic Energy-Momentum Tensor

3. The General Theory of RelativityThe Equivalence PrincipleThe Einstein Field EquationsThe Consequence of General Relativity

Chapter 1. Introduction The two pillars of Modern Physics that began to be developed at the turn of the century from the 19th to the 20th are Quantum Mechanics and the Theory of Relativity. Classical Physics, represented by Newton's Mechanics, turned out to be inadequate to explain the pnenomena in microscopic scale or those involving fast-moving bodies whose speed is colse to the speed of light. The totally new concepts of Quantum Mechanics were needed to account for the microscopic scale pnenomena. On the other hand, the fact that the light travels with the same speed in all directions, regardless of the motion of the source or the observer, led A. Einstein (1879-1955) to develop the Special Theory of Relativity in 1905. The Galilean coordinate transformation between the reference frames moving with respect to each other cannot be compatible with the constancy of the speed of light and had to be replaced by the Lorentz transformation. Einstein further postulated that all laws of physics are the same in all inertial reference frames whose coordinates are related to each other by Lorentz transformations.

The first of Newton's laws of motion, that a body moves with a constant velocity unless acted on by forces, is really a statement about a particular reference frame, an inertial frame, in which the second law holds. The second law, that a body is accelerated in proportion to and in the direction of the force acted on it, cannot be justified without the specification of the reference frame provided by the first law. Is the inertial frame unique? Obviously not. Any frame that is moving uniformly with respect to a given inertial frame also satisfies the first law and is an inertial frame. In Classical Mechanics, the coordinates of the inertial frames are related to each other by the Galilean transformations. Newton's second law can be seen to be invariant under the Galilean transformation if one only considers those forces that depend on the relative positions of the interacting bodies. Newton's laws of motion appear the same in all inertial frames and therefore, as far as Newton's Mechanics is concerned, there can be no physical criteria for a perferred inertial frame. The idea of the 'rest frame' loses its absolute meaning, and word 'velocity' can only retain the relative meaning. This fact may be called the 'Newtonian Relativity'.

The theory of Electromagnetism, summarized and completed by J. Maxwell (1831-1879), appeared to alter the situation. Maxwell's equations allow a wave solution which represents electromagnetic waves propagating with the same speed in all directions. The Galilean transformation cannot retain the constancy of the speed of light and there could be only one inertial frame in which the light travels with the same speed in all directions. One could designate this frame the absolute rest frame. However, all experimental searches for this frame have failed. Einstein accepted the constancy of the speed of light in all inertial frames as a postulate and showed that the coordinate transformation between inertial frames has to be the Lorentz transformation to retain the invariance of the speed of light. All inertial frames are on the same footing again and the concept of relativity can remain. But the problem is that Newton's equation of motion, being invariant under the Galilean transformation, is not invariant under the Lorentz transformation. In the process of rescuing the concept of relativity in Electromagnetism, one in turn faces the possibility of ruining the concept of relativity in Mechanics.

Einstein, postulating that all laws of physics are the same in all inertial reference frames, chose to modify Newton's equation of motion and invented the Relativistic Dynamics which is invariant under the Lorentz transformation.

After inventing the Relativistic Dynamics, Einstein started modifying Newton's theory of gravitation to invent a new theory of gravitation which is compatible with his Special Relativity. His path to a new theory took an unexpected turn, eventually leading to a completely new concept of gravitation. The final result which he published in 1916 is the General Theory of Relativity. In this theory, the effect of gravity is manifested as the spacetime curvature. Matter affects the geometry of spacetime and the spacetime curvature in turn affects the motion of matter. Spacetime is no longer just a background stage where physical phenomena occur as in Classical Physics. It has its own dynamics in General Relativity.

This note is intended to include the contents of Relativity that can be covered in one semester of a two-hour-a-week undergraduate course. The details of the kinematics and dynamics of Special Relativity are given in Chapter 2. The full coverage General Relativity is not intended, but only a brief introduction of it is given in Chapter 3.

Chapter 2. The Special Theory of Relativity Table of Contents The Galilean Transformation and the Newtonian RelativityElectromagnetism, the Speed of Light and the Michelson-Morley ExperimentThe Lorentz Transformation and The Postulates of Special RelativityRelativistic KinematicsRelativistic DynamicsRelativity and Electromagnetism

The Galilean Transformation and the Newtonian Relativity An event is something that happens at a point in space and at an instant in time. To specify an event, we need four coordinates in a particular reference frame (S), say t,x,y,z. Consider another reference frame (S') which is moving with a constant speed v in the x-direction with respect to S. To specify the same event in S' one needs a different set of coordinates, t',x',y',z'. How are the two sets (t,x,y,z) and (t',x',y',z') related to each other? According to the Galilean coordinate transformations

and these transformations leave the time interval (of the two events) and the length (space interval) unchanged. [1]

An initial reference frame is one in which the law of inertia (Newton's 1st law) holds. [2] If S is an intertial frame, so is S'. [3]

The special theory of relativity deals only with the description of events by observers in inertial reference frames. The acceleration of a particle is the same in all reference frames which move relative to one another with constant velocity.

If the mass of a particle (m) and the force (F) acting on it are the same in S and S', then Newtonian laws of motion is the same in both systems (invariant under the Galilean coordinate transformations).

The laws of mechanics are the same in all inertial frames. No inertial frame is preferred over any other; no physically definable absolute rest frame. =>Newtonian Relativity.

The electromagnetic force depends not only on the relative positions of the charged particles but also on their velocities. The velocity of a particle is not the same in S and S', and therefore the electromagnetic force on a charged particle is not the same in S and S'. => Electrodynamics is not invariant under the Galilean coordinate transformations.

Notes

[1]

[2]

[3]

Electromagnetism, the Speed of Light and the Michelson-Morley Experiment Electromagnetism is summarized by Maxwell's equations. Maxwell's equations are not invariant under the Galilean coordinate transformations and therefore one can choose as an absolute rest frame the one in which Maxwell's equations hold. The speed of light is derived by Maxwell's equations to be

regardless of the motion of the source. The medium of light propagation was given the name "ether" and the absolute rest frame was considered to be the one in which "ether" is at rest.

In a frame S' moving at a constant speed v with respect to this ehter frame, an observer would measure a different speed for the light ( ) according to the Galilean transformations. The Michelson-Morley experiment was designed as an attempt to locate the ether frame (the absolute rest frame).

Rotate the instrument through

The change in the path difference of the beams by one wave length <=> a shift of one fringe.

The result of the experiments showed no such shifts (null result). One way to interpret the null result of the Michelson-Morley experiment is to conclude that the speed of light is the same in every inertial system. Then there could be no experimental evidence to indicate the existence of a unique inertial system (the one in which the "ether" is at rest). The other attempts to interpret the null result of the Michelson-Morley experiments preserving the concept of a preferred "ether" frame have not been successful.

The Lorentz Transformation and The Postulates of Special Relativity Einstein postulated that

1. the laws of physics are the same in all inertial systems- no preferred inertial systems,

2. the speed of light in free space has the same value C in all inertial systems.

To accomodate the above assumptions, we must

1. obtain equations of transformation between two inertial systems moving uniformly with respect to each other which will keep the speed of light invariant. (=> Lorentz transformation)

2. examine the laws of physics to check whether or not they keep the same form (i.e. invariant) under this new transformation. Those laws that are not invariant will need to be generalized so as to obey the principle of relativity. We expect the generalization to be such that the new laws will reduce to the old ones when v/c << 1.

Derivation of the Lorentz transformation equations

Let's consider two inertial frames S and S' with a common x (x')- axis and y', z' axes parallel to y, z axes respectively. To simplify the algebra we choose the relative velocity to be along the x (x')-axis (without loss of generality).

The space-time coordinates of an event are t,x,y,z in S and t',x',y',z' in S'.

The differential form of the above equations is

Now if we accept the assumption of the space-time homogeneity, the coefficients

cannot depend on coordinates t,x,y,z. Therefore we can integrate the above equations to obtain

We simplify the situation once more by setting t=t'=0 at the origin at the instant the origins O and O' coincide. Then b0 = b1 = b2 = b3 = 0. Since the x-axis coincides continuously with the x'-axis, it follows that y'=z'=0 for y=z=0, x, t arbitrary. =>

Also the x-y plane should transform over to the x'-y' plane. Then it follows that z'=0, y arbitrary. => a32 = 0 => z' = a33z. Likewise y' = a22y.

Now consider a rod lying along the y-axis extending from y=0 to y=1. The length of the rod measured by an S-observer is 1, and to an S'-observer the length is a22. If we fix the same rod along the y'-axis, then an S'-observer should measure the length to be 1 and an S-observer should measure it to be 1/a22. The symmetry between the two inertial frames requires that

Likewise a33-1. Also from the symmetry (or isotropy) argument we conclude that t' and x' cannot depend on y and z.

For the origin O', x'=0 and x=vt

Summarizing the results obtained so far,

Now we require that the speed of light is the same C in all directions in both S and S'. Then for a light wave originated at the origin at t=t'=0,

The inverse transformation equations are

Note that the above two sets of transformation equations are identical in form except that v changes to -v. For the more general case of an arbitrary diretcion of ,

Relativistic Kinematics Simultaneity

The concept of absolute simultaneity no longer exists in the theory of special relativity. Consider two events A and B are observed by an S-observer to occur simultaneously

(tA=tB) at different space points xA and xB( ) respectively. To an S'-observer the same events A and B do not appear to occur simultaneously according to the Lorentz transformation.

Comparison of Time Interval

Consider two events A and B which occur at the same space point in the S'-frame

(x'A=x'B), at different instants in time ( ). The time interval of the same events

measuring by an S-observer (tB-tB A) is longer by the factor of than that measured by an S'-observer (t'BB -t'A)

Comparison of Lengths

Consider a rod stationary in S'-frame and extending along the x'-axis. If the length of the rod measured by an S'-observer is L' (= x'B-x'A), then the length of the same rod

measured by an S-observer is shorter by the factor of

( )

Since one must measure the positions of the both edns of the rod simultaneously to measure its length (tA=tB),

The length of a rod transverse to the relative motion is measured to be the same by all inertial observers (y'=y,z'=z).

Proper Time Interval

The differential form of the Lorentz transformation is

The proper time interval ( ) defined by

is invariant under the Lorentz transformation i.e. the same in all inertial frames.

How the Velocity of a Particle Transforms

Suppose a particle is observed to be moving with velocity in the S-frame. What is the velocity of the same particle observed in the S'-frame?

Alternation and Doppler Effect in Relativity

Consider a train of plane monochromatic waves of unit amplitude emitted from a source at the origin of the S'-frame propagating in the x'-y' plane making an angle with the x'-axis.

The equation describing the propagation is

where

The above result indicates that the wave observed by an S-observer is also a plane monochromatic wave with a different wavelength (Doppler effect) and a different propagation angle (aberration) [1] .

Notes

[1]

Relativistic Dynamics Now that we have obtained he new (Lorentz) transformation equations which keep the speed of light invariant, the next thing to do is to modify the laws of classical (Newtonian) mechanics so that the new mechanics is consistent with relativity. The new laws must reduce to the classical ones when the velocities involved are sufficiently small compared to c.

Relativistic Momentum

In classical mechanics the linear momentum of a particle is defined by

and in a two-body collision the total momentum

is conserved.

If the collision is elastic the total kinematic energy ( ) is also conserved. Now let us consider the elastic collision problem of two particles of the same mass (m) described below.

The interaction is assumed to take place in a negligibly short time-interval and the velocities of the particles before and after the interaction observed in the S- and S'-frames are given below.

Although the total momentum is conserved in the S-frame, it is not in the S'-frame. If we want the conservation of momentum in the collision problem as a physical las we must modify the equation.

and redefine the linear momentum.

For compactness of notation, let us define Then the Lorentz transformation equations can be written

and the proper time interval is

The 4-velocity is defined by

The space components of reduce to the classical definition of velocity

(ux, uy,uz) when u/c << 1. Since is invariant under the Lorentz transformation, is transformed in the same form as does

Any quantity which transforms in the same way as above is called a 4-vector. Check if you can get the same result as above from the transformation equations of u , u ,u .x y z

The relativistic momentum (4-momentum) is defined by

and the transformation equations for the 4-momentum are

Therefore, obviously, if is conserved in the S-frame, so is in the S'-frame.

The four components of the relativistic momentum are

The spatial part of the 4-momentum (Pi, i=1,2,3) can be considered as (rest mass)*(spatial part of the 4-velocity) or (relativistic mass)*(ordinary velocity) depending on whether we connect the factor with m0 or with ux, uy,uz.

m0 : rest mass or proper mass

m :relativistic mass

The time part of the 4-momentum (P0) reducesto 1/c * (rest mass energy+ ordinary kinetic energy) in the limit when u/c <<1 ).

From the above equation one can see that the four components of are not all independent ; P0 is determined once Pi's are determined.

The Equivalence of Mass and Energy

In every collision of particles the momentum is conserved, but the kinetic energy is not unless it is an elastic collision. Consider a completely inelastic collision of two identical particles of rest mass m0 approaching each other with the same speed u (in the -direction) before the collision and sticking together (coming to rest) after the collision.

Before the collision the components of the total momentum are

After the collision,

From the conservation of the momentum,

We see that the rest mass of the combined body (M) is greater than the total rest mass of the two separate particles before the collision. What happens is that all the kinetic energy disappears on collision and in its place after the collision, there appears some form of internal energy (such as heat energy or excitation energy) which results in the increase of the rest mass.

We also see that what is conserved is not the kinetic energy but the total energy (E) which is equal to c2 * (total relativistic mass). Simply by multiplying or dividing by c2 one obtains energy from rest mass or rest mass from energy. Indeed, that mass we associate with various forms of energy really has all the properties that are given to mass heretofore such as inertia, weight and so forth.

Relativistic Force Law

A natural extension of Newton's 2nd law which reduces to Newton's 2nd law when u is sufficiently small compared with c and which is covariant under the Lorentz transformation is

In the frame where the particle is momentarily at rest becomes

Now consider another frame in which the particle has velocity . This frame is moving

with velocity with respect to the original frame and, since is a 4-vector, we can

obtain from through the Lorentz transformation.

The spatial part of the relativistic dynamic equation ( )

corresponds to the three equations of Newton's 2nd law, and the time part ( ) is the extension of the energy-work theorem of the classical mechanics.

The Compton Effect

Here, we apply the relativistic momentum conservation law to the collision of a photon and a free electron. For a photon, one cannot use the equation

, because u=c and the denominator vanishes. Therefore it is meaningless to speak of the rest mass of a photon, or it has to be zero. Instead of the above equation P0 of a photon is given by

. Using the equation, , we obtain

where ki is the ith component of the unit vector along the direction of the photon. Now let us consider the collision between a photon and a free electron initially at rest.

Before the collision,

After the collision,

From

Solve the above equations to obtain

Relativity and Electromagnetism

The Invariance of Maxwell's Equations and the Transformation of Electric and Magnetic Fields

The invariance of the speed of light suggests that Maxwell's equations should have the same form in all inertial frames. We will first see that Maxwell's equations

are indeed invariant under the Lorentz transformation. From Eq. (3),

Putting this into Eq. (4) one obtains,

Now put Eq's (5) and (6) into Eq's (1) and (2) to obtain after some manipulation,

Notice that, for given and , the scalar and vector potentials ( and ) are not uniquely determined. The transformation

with an arbitrary function leaves and unchanged. Therefore, using this freedom, one can always make

and then Eq.'s (7) and (8) become

The charge density times c ( ) and current ( ) together form a 4-vector ( ),

being . Consider a volume element containing charge . In the frame where

the volume element is momentairly at rest, and . Now, observed from the frame in which the volume element is moving with velocity , the volume is

reduced to and the charge density is increased accordingly,

. The current is

If we take , we can write

So, is a 4-vector. The differential operator in Eq.'s (10) and (11) is invariant under the Lorentz transformation, and therefore and should form a 4-

vector Since Eq.'s (10) and (11) combine together to form a 4-vector equation

they are invariant under the Lorentz transformation. The Transformation equations for the electric and magnetic fields can be easily obtained from the transformation equations for the 4-vector potential . Particularly, for the case the S'-frame is moving with speed v in the x-direction with respect to the S-frame with the corresponding axes being parallel with each other,

For the more general case of arbitrary relative velocity between the S and S'-frames,

Equation of Motion of a Charged Particle in an Electromagnetic Field

Suppose at some moment a charged particle is moving with velocity with respect to the S-frame. S' is another inertial frame which is moving with velocity relative to the S-frame so that the particle is momentarily at rest in the S'-frame. We know that the force acting on a non-moving charged particle is given by

Then the force vector in the S-frame is given by

Now the transformation equations between and (Eq. (19)) are put into Eq. (21) to obtain

Then,

The Fields of a Uniformly Moving Charge

Consider a particle of charge q moving with uniform velocity in the S-frame. We wish to calculate the electric and magnetic fields in the S-frame caused by this moving charge. If we choose another inertial frame (S') in which the particle is at rest at the origin, the field in this frame is merely the static electric field of a point charge,

We obtain the fields and in the S-frame by using Eq. (18)

Nothe that and is not spherically symmetric while is in the S'-frame. Forces Between Moving Charges

Let us consider two particles of the same charge q separated by the distance r along the y-axis. If both particles are moving with constant velocity along the x-axis, there exists magnetic attraction as well as electric repulsion. However, the repulsion force is always greater than the attraction force so that the net result is repulsion. The difference form the static case is of the order of u2.

If both particles are moving with the same constant speed u but in different directions, particle A in the x-direction and particle B in the y-direction, Newton's 3rd law (action-reaction law) does not hold. (The discrepancy is of the order of (u/c)2). Therefore the total momentum of the particles is not conserved.

Electromagnetic Energy-Momentum Tensor

However, if we include the momentum of the electromagnetic field, the total momentum is still conserved. The momentum density of the electromagnetic field is given by

Now let us show that the total momentum of the charged matter and the electromagnetic field in the whole space is conserved. From

,

[1]

[2]

Notes

[1]

[2]

Chapter 3. The General Theory of Relativity Table of Contents The Equivalence PrincipleThe Einstein Field EquationsThe Consequence of General Relativity

The Equivalence Principle After formulating the special theory of relativity, Einstein started the attempt of modifying Newton's theory of gravity to find a new theory of gravity that fits into his special relativity. The final destination that this attempt led Einstein was a completely new concept of gravity that could not have been anticipated at the beginning. In the general theory of relativity, published in 1916, Einstein put forward the idea that the effects of gravity are manifested in the curvature of spacetime. The distribution and motion of matter determines the curvature of spacetime, and the spacetime curvature in turn affects the motion of matter. Spacetime is no longer just the background stage on which physical phenomena happen, but it plays the roles of both the background stage and the actor in the drama and has its own dynamics. In the process of developing the general theory of relativity, Einstein used the "Equivalence Principle", published in 1907, as an important guide. The equivalence principle asserts that the local effects of the gravitational field and the acceleration of the reference frame are the same. Its primitive version is about the equivalence of the gravitational and inertial masses. The concept of mass appears in Physics in two different cases, one is as the source of the gravitational force and the other as the proportionality constant between the force and the acceleration. To distinguish them, the term gravitational mass (mg) is used in the former cae and the inertial mass (mi) in the latter. The gravitational force acted on a body in the gravitational field is , where is the gravitational acceleration. The acceleration of the body under this force is, according to Newton's 2nd law,

If different bodies have different values for the ratio between the gravitational and inertial masses (mg/ mi), they would have different accelerations when free-falling in the same gravitational field. However, since Galileo Galilei first stated that they all bodies fall with the same acceleration in Earth's gravitational field, many experimental tests have been performed to confirm the statement. The experiment by Brazinski and Panof in 1971 confirmed it to the accuracy of 1/1012. Let us consider a reference frame that is being accelerated with the acceleration in the situation without the gravitational field. In this frame, all bodies on wich no forces are acted will appear to be accelerated with the acceleration . Therefore, if all bodies appear to move with the same acceleration,

there is no way to determine whether it is the effect of the gravitational field or the accelerated motion of the reference frame itself. When the equivalence of the effects of the gravitational field and the accelerated motion of the reference frame is applied to the motions of bodies restrictively, it is called the weak equivalence principle. Einstein assumed this equivalence to apply to all physical phenomena without restriction, and in this case we say the strong equivalence principle is applied. According to the strong equivalence principle, an observer who is using the reference frame that is free-falling in a uniform gravitational field will see all physical phenomena follow the same physical laws as in the case without gravity. However, the gravitational field at each point in space can have different strength and direction when the gravitational field is not uniform, and then the free-falling reference frame at each point in space can be different from each other. Therefore, in the reference frame that is free-falling with respect to one particular point in space, the effect of vanishing gravity will appear only in a small region around that point in which the gravitational field can be approximated to be uniform. Such a reference frame that is free-falling with respect to one particular point is called a local inertial frame with respect to that point. In the general case with non-uniform gravitational field, there only exist local inertial frames but no global inertial frames. Les us denote a point in a 4-dimensional spacetime by the coordinate

. According to the equivalence principle there exists a local inertial frame with respect to an arbitrary point and, using the local inertial coordinate system with respect to that point, the proper time ( ) at that point can be written in the same form as in the special relativity;

where is the Minkowski metric. In the x-coordinate system this becomes

and therefore the metric at point X in the x-coordinate system is

The function is determined by the relationships between the x-coordinate system and the local inertial coordinate systems and those relationships are in turn determined by the form of the gravitational field. Therefore, gravity determines the metric, and all the information about the gravitational field is recorded in the metric. In conclusion, it can be said that the effects of gravity are manifested in the geometry of spacetime. The coordinate transformation induces the metric transformation,

If there exists a coordinate transformation that transforms the metric to the Minkowski metric at all spacetime points, the spacetime is intrinsically flat. Otherwise the spacetime is curved. A free-falling body in a curved spacetime follows a geodesic satisfying the geodesic equation,

where is the affine connection defined by

The Newtonian limit is the limit where the gravitational field is sufficiently weak and the bodies are moving sufficiently slowly. In that limit one can find a coordinate system in which the metric is nearly Minkowski so that

and

If only the terms of the first order in those small quantities are kept, the geodesic equation reduces to

Here t=x0, and the time derivatives of the already small quantities h0i are also ignored. The zeroth component of the geodesic equation ( ) reduces to the trivial result 0=0. Comparing the above limit equation with the equation of motion in Newton's theory of gravity

where is the gravitational potential, one can identify

so that

The Einstein Field Equations In Newton's theory of gravity, the gravitational potential contains all the information about the gravitational field. The field equation that governs is

where is the mass density. In the general theory of relativity is replaced by the metric , and the field equation that governs is the following Einstein field equation;

where is the energy-momentum tensor, the Riemann-Christoffel curvature tensor, and R the curvature scalar. This equation of course reduces to the above Newtonian gravitational field equation in the Newtonian limit. Being a highly nonlinear coupled differential equation, the Einstein field equation is quite difficult to be solved exactly and only in some situations with high symmetry exact solutions are known.

The Consequence of General Relativity Newton's theory of gravity has been proven to be quite adequate to describe such gravitational phenomena as orbital motions of the planets in the solar system and trajectories of objects in Earth's gravitational field. What the general relativity predicts ofr these usual phenomena differently from Newton's theory have to be quantitatively small. In his paper of 1916 Einstein presented the quantitive predictions for the following phenomena; the deflection of starlight passing near the Sun, the precession of Mercury's orbit around the Sun and the redshift of radiation from distant stars. All these predictions have been observationally confirmed within the experimental errors. One can consider two situations where the general relativistic predictions can be significantly different from those of Newton's theory; one is the situation with an unusually strong gravitational field and the other is when one deals with very large scales in time and space so that, although the gravitational field is not very strong, small

general realtivistic effects are accumulated to give the final results significantly different from those of Newton's theory. The former is the case of the phenomena around a compact stellar objects and the latter is the case of cosmology. In the rest of this section we will consider the case of compact stellar objects and then describe the standard cosmology briefly. The geometry in the vacuum around a spherically symmetric source is given by the Schwarzschild metric which is the solution of the

Einstein field equation with ,

This metric can represent the geometry around a spherically symmetric nonrotating stellar object and M is identified as the mass of the object. Suppose a source sitting on the surface of the object at r=r0 emits radiation with the proper frequency . When on observer stationed sufficiently far away from the object ( ) receives this radiation, the observed frequency of the radiation is given by

Then the value of the redshift factor is

When This value for the Sun for example is calculated to be very small, O(10-6). It is about 10-4 for a typical white dwarf and a few tenths for a typical neutron star. We can see that, when dealing with as compact an object as a neutron star, general relativistic effects become quite large and Newtonian treatment of gravity becomes inadequate. As r0 approaches 2GM, the value of z grows to infinity. For an object so compact that its surface retreats inside r=2GM, no light emitted from it can reach an outside receiver. This object is called a black hole and the surface at r=2GM, the Schwarzschild radius, is called the event horizon. Let us now review cosmology. After publishing his general theory of relativity in 1916, Einstein in 1917 presented a new cosmological model based on the concept of space in the Riemann geometry. The cosmological space in this model is a 3 dimensional space corresponding to the surface of a 4 dimensional sphere in a 4 dimensional Euclidean space. The radius of the sphere determines the size of the Universe which is finite and without a boundary anywhere. Matter distribution in the space is assumed to be uniform. Einstein also assumed the Universe to be static with fixed size. When this model is put into the Einstein field equation, it turns out that the energy density or pressure has to take a negative value. Einstein considered this unacceptable and , in order to remedy the problem, he modified his field equation by inserting a new term, called "cosmological

term", into it. However, with the help of observations by Hubble in the later part of 1920, it is eventually etablished that the Universe is not static but actually expanding. Acknowledging this new development, Einstein in 1931 erased the cosmological term and restored his equation back to the original form. Many studies in cosmology followed eventually leading to the establishment of the "standard model cosmology". In the standard model cosmology, a very high degree of symmetry is assumed. The so-called "cosmological principle" assumes that the Universe, on a large scale to include several clusters of galaxies in one unit, is uniform and isotropic. Under this symmetry, the spacetime of the Universe can be described by the Robertson-Walker metric;

where R(t) is the cosmological scale factor and k is some constant. At an instant of a given t, the spatial geometry of the Universe is determined by the sign of k; it is a close (finite) space corresponding to the 3 dimensional surface of a sphere in a 4 dimensioanl Euclidean space if k>0, an open (infinite) space with a negative curvature if k<0 and a flat space if k=0. In the spacetime described by the Robertson-Walker metric, it can be

checked that the orbit with constant space coordinates is a free-fall orbit, and objects in the cosmological space is interpreted to follow such orbits on the average. Then the coordinate t is the time that the clocks attatched to such objects would indicate. Even though the spatial coordinates of objects remain fixed, the distances between objects increase with time as the value of the cosmic scale factor R(t) increases with time. The proper distance between an object at r=0 and another at r=r1 is

Then the speed at which the two objects move away from each other is calculated as follows;

The speed is proportional to the distance D and the proportionality constant is

which is the Hubble constant. The Hubble constant is spatially constant but changes with time. Its present value is usually expressed as

where h, known to have the value between about 0.5 and 1.0, represents the uncertainty in our knowledge about its exact value. Puting the Robertson-Walker metric into the Einstein field equation and assuming the perfect-fluid form for the energy-momentum tensor.

with , one obtains the following equations

The second of the above equations is the energy conservation equation. For non-relativistic matter with , the solution of the equation is (the

subscript m represents matter), while for radiation with , it is (the

subscript r represents radiation). Since one finds , that is, the temperature of the Universe is inversely proportional to the cosmic scale factor. From the first of the above equations, one can see that the future of the Universe depends critically on whether the present value of the energy density is greater or smaller

than the critical value If , k is positive and R will stop growing in some finite future and start decreasing , that is, the Universe will stop expanding in some finite future and contract again. If , k is negative and R will keep growing forever. The case is the boundary between the above two cases. Which of the three is the actual case for our Universe is not definitely known yet. However, in all three cases, if one goes back to the past, R keeps decreasing until it reaches zero in some finite past. Universe started in some finite past in an infinite energy density and temperature, somehow exploded to begin expanding. The expansion still continues although the rate of expansion has been decreasing. Another name for the standard model cosmology is the "Big-Bang cosmology". We conclude this section with a remark about the big bang. It is a point of singularity at which the laws of physics cannot be applied. We reach that point only if we assume that the general theory of relativity and other usual physical laws are valid under such extreme conditions as those near big bang. It is generally believed that, under those extreme conditions, the current physical laws including the general theory of relativity have to be replaced by some more fundamental laws that are not yet known.