View with images and charts Preliminary Concepts 1.1 Spacetime [13] Matter changes the geometry of spacetime, this (curved) geometry being interpreted as gravity. Spacetime is usually interpreted with space being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions. According to certain Euclidean space perceptions, the universe has three dimensions of space and one dimension of time. 1.2 Concept with Dimensions [13] The concept of spacetime combines space and time to a single abstract "space", for which a unified coordinate system is chosen. Typically three spatial dimensions (length, width, height), and one temporal dimension (time) are required. Dimensions are independent components of a coordinate grid needed to locate a point in a certain defined "space". In spacetime, a coordinate grid that spans the 3+1 dimensions locates events (rather than just points in space), i.e. time is added as another dimension to the coordinate grid. This way the coordinates specify where and when events occur. 1.3 Spacetime Intervals [13] In a Euclidean space, the separation between two points is measured by the distance between the two points. A distance is purely spatial, and is always positive. In spacetime, the separation between two events is measured by the interval between the two events, which takes into account not only the spatial separation between the events, but also their temporal separation. The interval between two events is defined as: (Spacetime interval), Where c is the speed of light, and Δt and Δr denote differences of the time and space coordinates, respectively, between the events. 1.4 Inertial and Non-Inertial Frames [3] According to Newtonian’s first law, “A body at rest remain at rest and a body in motion continues with steady speed in a straight line until an external force is applied on the body.” This law may also regarded as the definition of force, i.e., ‘Force is the source by which the state of the body whether in motion or at rest may be changed, ‘Now we know the motion of a body has no meaning unless it is described with some well defined co-ordinate system or frame of reference with respect to which the velocity of a body is measured i.e., we must choose a meaningful co-ordinate system by which the motion of a body may be described. For this as we have already discussed. Newton introduced the idea of ‘absolute space’. In any case a frame of reference must be chosen in such a way that the laws of nature may become fundamentally simpler when expressed in terms of such frames of reference. There are generally two types of reference system:
1. The frames with respect to which an accelerated body in un-accelerated. This also includes the state of rest. 2. The frames with respect to which an un-accelerated body is accelerated. 1.4.1 Inertial Frames The frames with respect to which an un-accelerated body appears un-accelerated are inertial frames. In other words the frames which are at rest or in uniform translator motion relative to one another are inertial frames. Let us consider any co-ordinate system relative to which a body in motion has co-ordinates of the body relative to the assumed co-ordinate system is function of time, so that Newton’s first law can be stated in mathematical from, since the body is not being acted by a force.
Where
are three components of velocity of x, y and z directions respectively.
From these equations the components of the velocity are constant, that is, we may say that without the application of an external force, a body in motion continues in motion with uniform velocity in a straight line, which is Newton’s first law. Thus we may always choose a frame of reference or co-ordinate system with respect to which the body is at rest or in uniform motion. Thus when a body is not subjected to any force, there exists a frame of reference with respective to which the body is at rest or moving with constant linear velocity. i.e., with respect to which the body is unaccelerated. Such a frame may say ‘An inertial frame is one in which Newton’s first law is true.” Or an unaccelerated frame is inertial frame. 1.4.2 Non-Inertial Frames The frames relative to which an unaccelerated body appears accelerated are called noninertial frames. In other words the accelerated frames are non-inertial. Experiments give an inference that Newton’s frame of reference is fixed in stars in an inertial frame. A co-ordinate system fixed in earth is not an inertial frame since earth rotate about its axis and also about the sun. In fact any thing which is capable of turning is not an inertial frame, since though no force act on the body, but it is neither at rest or moving in a straight line with constant speed with respect to such frame of reference.
1.5 Space and Time in Newtonian Mechanics [6] If we consider the path and the velocity of the particle, we always have to assume the existence of co-ordinate system (or frame of reference) in which the position of the particle may be specified by some co ordinates from instant to instant and a mean of measuring time like a clock which is capable of marking the intervals of time at which the position of the particle is to be recorded. A co-ordinate system or frame of reference may be considered to
be the wall of a room or the position of the stars along with the direction of the plumb line. The time may be also be measured by the period of rotation of the earth, i.e., subdivision of the day. The `laws of mechanics or the Newtonian law may be verified by using such frame of reference and the sources to measure the time at least to a very good approximation. According to Newton’s second law “The rate of change of momentum is proportional to the net force impressed when the changes take place in the direction in which the force acts.” i.e.,
(1)
Where m is the mass of the body and f is the acceleration (or rate of change of velocity). In such frame of reference this law holds. But there are certain frame of reference in which this law does not hold. 1.6 The New Concept of Space and Time [2] The negative result of Michelson Morley’s experiment was explained by Lorentz and Fitzgerald. These mathematicians changed the concept of space and time considered under Newtonian mechanics. According to them, material body moving with velocity v through ether is contracted by the factor
in the direction of motion of body. The failure of
Michelson and Morley’s experiment leads Einstein to formulate the new concept of space and time. He ruled out the ether hypothesis. He said that no experiment could detect the velocity of earth through ether. His conclusion was that the motion relative to material body has physical significance while the motion through ether is meaningless. In other words, there is no such thing as an absolute motion and all motions are relative. The physical laws are independent of the motion of the observer.
Fig-1.1.1 Suppose O is the middle point of the straight line
and
that the two clocks giving absolute time are kept at events occurring at
and
and
is the velocity of earth. Suppose . In any coordinate system, the
are said to be simultaneous, if the clocks placed at
and
indicate the same time when the events occur. It can not be so. For, if a light signal is given at and
simultaneously, then the light signals will not reach simultaneously the observer at
O, on account of the fact that the time
, taken by the light signal from
time
to O is
taken by the light signal from
such that,
to O is
and the
As
, but
Therefore the time is not absolute. It varies from one inertial system to another inertial system, i.e.,
Therefore, Einstein further ruled out the new concept of absolute time. He
said that the theory of relativity having new concept of space and time is not valid for mechanical phenomenon and also for all optical and electromagnetic phenomenon. 1.7 Galilean Frame [3] Newton’s first law of inertia states that a body at rest remains at rest and a body in motion continues with constant velocity in a straight line unless an external force is applied to it. The above statement has no meaning , i.e., the motion of a body has no meaning unless it is described with respect to some well defined co-ordinate system of frame of reference relative to which the velocity of the body is measured. This led Newton to introduce the idea of absolute space which represents the system of reference relative to which the motion of a body can be defined. The question naturally arises how to find an absolute system of reference. The choice of frame of reference should be such that the laws of nature may become simpler when they are expressed in terms of such frames of reference. There are two types of reference (i)
Accelerated frame of reference,
(ii)
Unaccelerated frame of reference.
We shall have to choose unaccelerated frame of references. Since such frames of references all the laws of mechanics preserve the same form when they are expressed in terms of any of these frames of references. For these frames are moving with uniform velocity. Consider a co-ordinates system relative to which the co ordinates of a body in motion are ordinates
The co-
are functions of time . Since the body is not acted by any force, i.e., it is
moving with constant velocity,
being velocity components in
directions respectively. This is Newton’s first law
of inertia. Such a co-ordinate system is called inertial frame. Thus an inertial frame of reference is one in which Newton’s first law of motion holds good. 1.8 Galilean Transformations [3] The consequences of research work of Galileo on the motion of the projectile led to formulate transformations which later on, were called after his name ‘Galilean transformations’. These are used to describe the motions which are observed by two observers in two different inertial frames. His two main results are as follows
(i)
The motion of a particle projected at any angle may be derived from the motion of the particle thrown vertically upward.
(ii)
If a particle is thrown straight up from a cart which is moving with uniform speed, the observer on the cart may see the particle moving up and down but the motion observed by an observer on the ground may be described by superimposing the motion of the cart into that of the projectile.
Consider two frames velocity
and
of references one at rest and other is moving with uniform
. Let O and O’ be the observers situated at the origins of S and S’ respectively.
They are Observing the same event at aqny point P. Let the two frames be parallel to each other i.e., X’-axis is parallel to X-axis, Y’-axis is parallel to Y-axis, axis. Let the co-ordinates of P be
and
-axis is parallel to Z-
relative to origins O and O’
respectively. The choice of the origins of two frames is such that their origins coincide at time Case I: Let the frame S’ have the velocity V only in X’-direction. Then O’ has velocity v only along X’- direction then O’ has velocity v only along X’-axis. The two systems can be combined to each other by the following equations
(1f) Case II: let the frame S’ have velocity v along any straight line in any direction such that . After a time t, the frame S’ separated from S by distances
along x, y,z axes
respectively. Then the two systems can be related by the following equations
(1g) 1.9 Lorentz Transformation [13] Only one space coordinate is considered. The thin solid lines crossing at right angles depict the time and distance coordinates of an observer at rest; the skewed solid straight lines depict the coordinate grid of a moving observer.
Lorentz transformation for frames in standard configuration
Fig-1.1.2 Standard configuration of coordinate systems for Lorentz transformations. Assume there are two observers O and Q, each using their own Cartesian coordinate system to measure space and time intervals. O uses
and Q uses
Assume
further that the coordinate systems are oriented so that the x-axis and the x' -axis are collinear, the y-axis is parallel to the y' -axis, as are the z-axis and the z' -axis. The relative velocity between the two observers is v along the common x-axis. Also assume that the origins of both coordinate systems are the same. If all these hold, then the coordinate systems are said to be in standard configuration. A symmetric presentation between the forward Lorentz Transformation and the inverse Lorentz Transformation can be achieved if coordinate systems are in symmetric configuration. The symmetric form highlights that all physical laws should be of such a kind that they remain unchanged under a Lorentz transformation. The Lorentz transformation for frames in standard configuration can be shown to be:
is called the Lorentz factor. 1.10 Matrix Form (Lorentz Transformation) [13] This Lorentz transformation is called a "boost" in the x-direction and is often expressed in matrix form as
Views of spacetime along the world line of a rapidly accelerating observer moving in a 1dimensional (straight line) "universe". The vertical direction indicates time, while the horizontal indicates distance; the dashed line is the spacetime trajectory ("world line") of the observer. The small dots are specific events in spacetime. If one imagines these events to be the flashing of a light, then the events that pass the two diagonal lines in the bottom half of the image (the past light cone of the observer in the origin) are the events visible to the observer. The slope of the world line (deviation from being vertical) gives the relative velocity to the observer. Note how the view of spacetime changes when the observer accelerates. For relative speeds much less than the speed of light, the Lorentz transformations reduce to the Galilean transformation in accordance with the correspondence principle. The correspondence limit is usually stated mathematically as,
so it is usually said that
nonrelativistic physics is a physics of "instant action at a distance". 1.11 Special Relativity Theory [10] Special relativity concerns itself with inertial frames of reference. Two observers who are in uniform, straight-line motion relative to one another are said to be in inertial frames of reference. Such observers are called inertial observers. If two observers are accelerating relative to one another, they are not inertial observers. Einstein developed Special Theory of Relativity first, by ignoring acceleration. Later on, he was able to generalize his theory to include acceleration and develop General Theory of Relativity. Special Theory of Relativity is based on two principles. First, the principle of relativity: the laws of physics are the same in all inertial frames of reference. This means that there is no absolute rest frame. Second, Einstein's light postulate: the speed of light in vacuum is constant, and has the same value in all inertial frames of reference. In order for the speed of light to be constant in different inertial frames which are in motion relative to one another, Einstein realized that space and time cannot be absolute. For example, if observer A is in motion relative to observer B, then, from B's point of view, A’s units of measuring space must be shorter than B’s (in A's direction of motion). Also, according to B, A’s units of measuring time must be longer (slower) than B’s. Relative to B, space for A is contracted in his direction of motion and time is lengthened or slowed down (dilated). (Two handy mnemonics are: "moving sticks are shortened" and "moving clocks run slow.") The sizes of these relative distortions of space and time are precisely what is
needed so that when A measures the speed of a passing light ray, he will always obtain the same value that B does. 1.12 General Relativity Theory [10] Special Theory of Relativity does not take into account acceleration. In 1907, the motion in an accelerating frame cannot be distinguished from motion in a uniform gravitational field. To put it another way, Einstein assumed that gravitational acceleration is equivalent to inertial acceleration (the acceleration due to inertia). Using this “principle of equivalence,” Einstein was able to extend the principle of relativity from inertial reference frames to all reference frames. He did this by showing that the same laws of physics that describe acceleration could also be used to describe motion due to gravity. General relativity is Einstein’s theory of gravity. It is founded on two core principles: (1) The principle of relativity: the laws of physics are the same in all frames of reference, (2) The principle of equivalence: accelerated motion and motion in a uniform gravitational field are equivalent. General Theory of Relativity is a geometric theory. Gravity is not treated as a force. Instead, according to General Theory of Relativity, gravitational acceleration is caused by the warping of space and time. The warping or curvature of spacetime is caused by the presence of matter, energy and pressure. CHAPTER TWO Crucial Tests in Relativity 2 Crucial Tests [3] For crucial tests in relativity, we know the followings (a) Advance of perihelion. (b) Gravitational deflection of light. (c) Shift in spectral lines. 2.1.1 Advance of Perihelion: To discuss the advance of the perihelion of a planet’s orbit around the sun, comparing the relativistic equations with those of classical mechanics. Proof: The differential equation of the path of a planet is
With
neglecting the small term
as a first approximation, we have
Where
and
are constants of integration giving eccentricity and longitude of the
perihelion. Putting this first approximation on the R.H.S of (2.1) we obtain
of the additional terms , the only one term which can produce and effect within the range of observation is the term
The particular integral of this term is
Here we are using the formula
As a second approximation, the complete
solution of (2.1) is
Taking
and observing that Since
is very small,
This is the required solution of (2.1). When a planet moves round the sun through one revolution, the perihelion of the planet advances a fraction of revolution equal to
[on using the well known formula of area
] From (2.4),
a
m
b l
Fig-2.1.1a
(a) Closed elliptic planetary orbit (predicted by Newtonian theory)
Fig-2.1.1b (b) Planetary orbit with a slow rotation of perihelion (predicted by Einstein’s Theory) being time period .Thus ,the relativistic theory leads to an advance of perihelion of a planetary orbit. That is to say, this theory leads to planetary orbit with a slow rotation of perihelion instead of to be perfectly closed elliptical orbits of the Newtonian theory. The advance of perihelion, given by (2.6), is very small for all the planets except Mercury, in which it is appreciable. For Mercury,
But
radian =
Hence, advance of perihelion in case of Mercury, is
Thus, the predicted value of advance of perihelion in case of Mercury is 43 seconds per century and the observed value is 43.5 seconds. Hence, the agreement is satisfactory. 2.1.2 Gravitational Deflection of Light Rays.(bending of light rays) Proof: treatment in general theory of relativity. We consider the deflection of a light ray in the gravitational field of the sun. According to the general theory of relativity, the track of a
light ray is given by geodesic equations with the added conditions
it means that the
differential equations of planetary orbit viz.
is also applicable to the path of light ray. If we put
in (2.8), then we get
putting
in (2.7), we get
Thus, the trace of a light ray is given by (2.9). Neglecting the small term
as a first approximation, we have
(2.10)
As a second approximation, the complete solution of (2.9) is
Changing this into Cartesian co-ordinates which may be assumed to be valid in the nearly Euclidean space surrounding the sun, we obtain
,
From (2.11) and (2.12), it is clear that the second term deviation from the path
(2.12)
in (2.11) shows a
Asymptotes to (2.11) are obtained by taking
compared to x so that asymptotes to (2.11) are
very large
let
be the angle between these asymptotes so that
Since
For
is very small and so
deflection
and hence neglected.
For a light ray grazing sun’s limb,
this prediction can be verified by observations at the time of
eclipse on the apparent positions of the stars. 2.1.3 Treatment in Newtonian Theory Let a light ray emitted from a star be moving parallel to y-axis and be passing through the mass m at a distance is given by
For a light ray moving parallel to y-axis,
We obtain
This is the equation of the path of a light ray according to Newtonian theory. The second term shows deviation from the path
. Asymptotes to (2.16) are obtain by
taking y very large compared with x so that asymptotes are
Let
be the angle between these asymptotes, then
Using the fact that
is very small and
This shown that the deflection on the path of a light ray due relativistic field is twice that predicted by Newtonian theory. 2.1.4 Gravitational Shift of Spectral Lines Proof: We consider the shift in spectral lines of light emitted by an atom situated in a gravitational field, when this light is observed on the surface of earth. Atoms of sodium vibrate with uniform frequency. Let one vibration and
be the interval between the beginning and the end of
the corresponding periodic time. Consider an observer who is moving
with sodium atoms . let the atom be momentarily at rest in co-ordinate system
, so
that , by Schwarzschild line element.z
We compare the periodic time of sodium atom at two places (1) On the surface of the sun. (2) On the surface of the earth. Let
and
be periodic times of a sodium atom on the surfaces of the sun and the earth
respectively. Then
is the ratio of the observed wave lengths of alight ray corresponding to a spectral line which originates on the surface of the sun, Using the fact that
remains invariant under
arbitrary co-ordinate transformation, we obtain
This is the required expression for the shift in spectral lines. If the spectral line originates on the surface of the sun, then
this prediction has been confirmed after the
experiment of Adams and St. John. Chapter Three Minkow ski Spacetime 3. Geometrical Interpretation of Lorentz Transformation Lorentz transformation is simply a rotation in four dimensional space. Assume
We know
The distance of a point and ordinates of
from the origin
through an angle be
is invariant under Lorentz transformation. is unchanged. Rotate the rectangular axes
so that the new rectangular axes are
and
. Let co-
relative to new axes. Then we have the relations. in in
-system -system (3a)
and or,
(3b)
We put values in
Putting the values in
The equations
and
represent Lorentz transformation. Thus, we have proved that
Lorentz transformations are equivalent to rotation of axes in four dimensional space or
through an hypothetical angle
3.1 Time Dilation [13] Time dilation is a phenomenon described by the theory of relativity. It can be illustrated by supposing that two observers are in motion relative to each other, and differently situated with regard to nearby gravitational masses. They each carry a clock of identically similar construction and function. Then, the point of view of each observer will generally be that the other observer's clock is in error (has changed its rate). Simple inference of time dilation due to relative velocity
Fig-3.1.3
Fig-3.1.4
Observer moving parallel relative to setup, sees longer path, time
, same speed
c.Time dilation can be inferred from the observed fact of the constancy of the speed of light in all reference frames. This constancy of the speed of light means, counter to intuition, that speeds of material objects and light are not additive. It is not possible to make the speed of light appear faster by approaching at speed towards the material source that is emitting light. It is not possible to make the speed of light appear slower by receding from the source at speed. From one point of view, it is the implications of this unexpected constancy that take away from constancies expected elsewhere. Consider a simple clock consisting of two mirrors A and B, between which a light pulse is bouncing. The separation of the mirrors is L, and the clock ticks once each time it hits a given mirror. In the frame where the clock is at rest (diagram at right), the light pulse traces out a path of length 2L and the period of the clock is 2L divided by the speed of light:
From the frame of reference of a moving observer traveling at the speed v (diagram at lower right), the light pulse traces out a longer, angled path. The total time for the light pulse to
trace its path is given by
The length of the half path can be calculated as a function of known quantities as
Substituting D from this equation into the previous, and solving for Δt' gives:
and thus, with the definition of
:
which expresses the fact that for the moving observer the period of the clock is longer than in the frame of the clock itself. 3.2 Length Contraction [13]
where is the proper length (the length of the object in its rest frame), is the length observed by an observer in relative motion with respect to the object, is the relative velocity between the observer and the moving object,
is the speed of light, and the Lorentz factor is defined as
3.3 Discuss and Derive Minkowski Spacetime [2] According to Minkowski the external world is not Euclidean space of three dimensions but it is composed of events whose coordinates are ( space coordinates and
) where the first three
involves time.The Minkowski space time
is the simplest
empty space-time in general relativity and in fact the space time of special relativity. In terms of natural coordinates
the metric
can be expressed in the form (3.1.1)
If we use spherical coordinates
where, then,
Now
are
(3.1.2) [
]
Now we switch it to null coordinate
Where, The metric in these Coordinates is given by,
Then,
Therefore,
(3.1.3)
Now we want to change the coordinates in which “Infinity� takes the finite coordinate value. Let, The coordinate range are now,
Now,
Similarly,
Now,
Therefore the Minkowski metric in these coordinates is,
Again,
Therefore (3.1.5) becomes,
We can transform it back to a time like coordinate , Where,
and a space like coordinate
via,
And
Similarly,
Now (3.1.6) becomes,
3.4 Minkowski space. (Four Dimensional Continuum) [2] According to Minkowski, the external world is net Euclidean space of three dimensions, i.e., it is not composed of points whose co-ordinates are (x, y, z) where
are real numbers;
but it is composed of events whose co-ordinates are (x1, x2,x3 x4) where the first three (x1, x2, x3) are space coordinates and the fourth one x4 involves time. If an event occurs in the space, then the position of the point where it occurs and the instant when it occurs both are represented by the location of the event in the four dimensional continuum. All the four directions are not equivalent. For, an axis which measures the distance in -direction, can be rotated to measure the distance in
and
directions; but the same axis cannot be rotated to
measure the tune interval due to the fact that a meter stick cannot be converted into a clock. The direction of time interval is not unique.This distinction is expressed by saying that the space time continuum is
dimensional rather than as four dimensional.The interval
between two events whose co-ordinates are and
) is given by (3.1.7)
where the co-ordinate
involves
This interval must be independent of transformation from
one system to another system.We have seen that the expression is Lorentz-invariant. Consequently the invariant interval between two neighboring points must be of the form (3.1.8) Comparing
and
we get
where
Hence, the four co-
ordinates of an event in the space are
In Newtonian Physics, an event may be identified by four members (x, y, z, t) where (x, y, z) are the rectangular cartesian co-ordinates of the place where it occurs and the time at which it occurs. Hence, it is clear that an event needs four numbers to identify it and for this reason we say that in Newtonion Physics to totality of all possible events form a four dimensional continuum. This continuum is called space-time; and we are not in a position to remove the hyphen and speak of space and time respectively. Thus, the co-ordinates of an event may also be taken at
where
.
Chapter Four Schwarzschild Solutions
4 Schwarzchild Solution [2] Solution: We have the line element
In matrix form
Now the inverse of
is
=
The geodesic general form is
All the other values are zero, now
Now in empty spaces
Multiplying (1.1) by
, we have
(4.1.6)
Integrating we have,
, where
is constant
Putting the value of B and B’ in (1.2) we get ,
, where
is constant
Now putting the value of A and B in
Which is the Schwarzchild Solution for the empty space time outside a spherical body of mass M. 4.1 Singularity Removing Process Let
and
then the equation becomes
We observe in equation (4.1.7) that Schwartzchild metric has a singularity at
. It can
be removed by the some co-ordinate transformation. Kruskel transformation is such a transformation which is defined by
Now (4.1.8) Again differentiating with respect to r we have the equation (4.1.8)
From (4.1.9)
Now we put this value in (4.1.7) we get
This has no singularity at So that the singularity at
is removed.
4.2 Solution of Einstein Gravitational Equation for Material World [6] Let us consider a static distribution of matter which exhibits spherical symmetry. The line element having this property in spherical polar co-ordinates is expressed as
Where
and
are function of r only, ie.,
Einstein’s field equations in the presence of matter are
In the mixed tensor form the above equation may be written as
We can also write
Yields
Where
And the scalar curvature
Hence for the static symmetric line element, we get
From equation (4.2.3) and (4.2.7), different component of
are written as
To proceed further let us assume that the matter is spherically symmetric and contains incompressible perfect fluid of proper density momentum tensor
and proper pressure
, so that the energy
is expressed as
On lowering the index, we get
Since the distribution of mass is static, all velocity components of fluid matter is zero,
So equation (1) yields
Now the tensor
given by equation, (10) may be written as
Adding equation (4.2.13a)And (4.2.13c), we get
Differentiation of (4.2.13a) yields
Now equating (4.2.13a) & (4.2.13b), we get
Multiplying the above equation by 2/r and rearranging, we get
With the help of equation (4.2.14) and (4.2.15), the above equation yields
4.3 Schwarzchild’s Exterior and Interior Solutions [6] We now want to determine the line element due to this static and spherically symmetric distribution of the perfect fluid in the following two cases : (i)
Outside the sphere is empty space.
(ii)
Inside the sphere
(iii)
The solution (i.e. the line element) in the first case is called the Schwarzschild exterior solution and in the second case is called Schwarzschild interior solution
The line element in the both cases is given by the equation (1) But the function cases in turn
and
are different in the both cases. We shall now consider both the
4.3(a) Schwarzchild’s Exterior Solution In the empty space outside the static sphere of perfect fluid all the component of energy momentum tensor is zero. So we have from the equation (4.2.8).
Adding equation (4.2.18a) and (4.2.18b) we get
Integrating, we get
Being constant of integration, Above equation
may be written as
Substituting the value of
Which is equivalent to
Integrating we get
Integrating again we get
in equation (4.2.18b) we get
From the above equation it is obvious that it is obvious that the constants are A , B, kWhich are integrating constant for . It will be shown that these entire three constant are not arbitrary. The solution (4.2.22) can also be obtained from the field equation
Since this equation holds in this case too. Using equation (4.2.20) and (4.2.21), we get
Comparing this two equations (4.2.22) and (4.2.24), we get Hence from equation (4.2.23) we get
From equation (4.2.21)
For convenience we may choose
So that
Hence schwarzchild’s exterior solution is
4.3(b) Schwarzchild’s Interior Solution We shall now determine line element inside the sphere of matter. It is natural that such a solution must depend on the properties of fluid of which the sphere is composed. Schwarchild solve this problem by assuming that the sphere is composed of an incompressible perfect fluid of proper density ρ₀ and proper pressure p₀. The Equations governing the line element inside the sphere of incompressible fluid are given by equations (4.2.13) & (4.2.17). The solution of this equation must satisfy the following boundary conditions: (i)
The pressure is zero at the boundary of the sphere,
(ii)
The density ρ₀ is uniform throughout the sphere. From equation (13c), we have
Integration of the above equation yields
being integrating constant
Since at r we have
,
, therefore to remove singularity at r=0, we put
,Thus
Now we have to find the solution for v. From equation (4.2.17), we have
As
is constant, integration of the above equation yields
Where
is constant of integration, i.e., ( or,
Using equation (4.2.14) and putting
we get
Equation (4.2.27) yields on differentiation with respect to r so that equation (4.2.28)may be written as
Now substitute the value of
from equation (4.2.27), we get
To solve the above equation we substitute
So that Hence equation (4.2.29) yields
=-
Where constant Since the solution of So that
Therefore the solution of equation (31) is
Hence finally we have
Hence Schwarzchild’s interior solution is given by
Now we have simply determine the constants A and B .Differentiating (33ii) we get
Using equation (4.2.33) and (4.2.35), Equation (4.2.13a) gives
To fix values A and B let us impose the following important requirements: (i)
The pressure is zero at the boundary of the sphere i.e.,
at
, where
is the radius of the sphere. (ii)
The cosmological constant A may be neglected for
.
(iii)
The exterior and interior solutions become identical at the boundary
of the
sphere. Applying (i) & (ii) requirements to the equation (4.2.36), we get
Applying (iii) requirement [i.e., equating exterior and interior solutions (4.2.26) and (4.2.34) at
] and using equation (4.2.37)we get
The equation represents the required values of A and B in interior solution (4.2.34) The interior solution will be real only if
This equation provides as upper limit on the possible size of a sphere of a given density and on the mass of a sphere of given radius.
Kurscal Metric and Schwarzchild Black Hole 5 Einstein Field Equation [3] Proof: Field equations in classical mechanics are given by (5.1.1) Where
stand for gravitational potential, Newtonian constant of gravitation and density
of material distribution respectively. While discussing the principle of equivalence, we noted that
can be interpreted either as potential function or metric tensor
analogue of the equation (5.1.1) in general theory of relativity, metric tensor
. That is to say, we consider
. In order to get an
must be replaced by the
to be gravitational potential. It follows
from (5.1.1) that the field equations in general theory of relativity are expressible in terms of second order derivatives of
. The most appropriate tensor which contains second order
derivatives is the Ricci tensor
. Hence, L.H.S. of (5.1.1) will be either or its linear . combination. While describing the relativistic field equations we must keep in mind that the field equations must be invariant under the tensor law of transformation. It means that both the sides of (5.1.1) must be expressed in terms of tensor. Hence,
in (5.1.1) must be replaced
by second rank tensor. This tensor is commonly known as energy momentum tensor. All the above facts are met in the equation. (5.1.2) where
stands for Einstein’s constant of gravitation and
For Science divergence of energy tensor is zero. Now (5.1.2) becomes
Taking cosmological constant
into account, we obtain (5.1.3)
This is the required field equation in general theory of relativity, For
,
(5.1.3)gives (5.1.4)
Multiplying (5.1.4) by
we get
Substituting the value
in (5.1.4),
Or,
(5.1.5)
This is an alternative form of the field equation (5.1.4). For empty space, Then (5.1.5) gives
(5.1.6)
Thus, the field equation in empty space is given by (5.1.6). These equations are in number. All of them are not independent to each other. Some of them satisfy the four Bianchi identities. As a result of which these are reduce to 6 in number. We are to determine
unknown components of
from these equations. Hence, we cannot
determine uniquely. Therefore, in any gravitational situation all the ten
cannot be
uniquely determined; (5.1.6) gives a solution. (5.1.6) The existence of such solution goes against Mach’s principle of inertia according to which a particle is expected to have inertia in presence of matter; and in absence of matter, it is expected to have no inertia. In fact inertia of a particle is due to matter. In order to avoid such contradiction Einstein and De Sitter took the equation (5.1.7) as the field equation in place of (5.1.4) Here
is constant. It is easy to verify that the divergence of L.H.S of (5.1.7) is Zero.
From (5.1.7),
Or, Or, For empty space, Consequently
(5.1.8) so that , by (5.1.8)
With these values (5.1.7) becomes
Or,
Thus, Einstein took (5.1.9) As the field equation in empty space. He felt that his new field equation (5.1.9) would not provide any solution in empty space. But a little latter it was investigated that his new field equation (5.1.9) gave a solution as given below:
Thus, the introduction of
would not serve the purpose for which it was introduced.
5.1 From the Field Equations to the Kruskal Metric [12] The usual way of arriving at the Kruskal metric for a spherically symmetrical field in general relativity is to first derive the Schwarzschild metric and then make a coordinate transformation that eliminates the singularity in the Schwarzschild time coordinate at r = 2m by “analytically continuing” the geodesics. However, the introduction of the singular time coordinate, followed by a transformation that is itself singular at r = 2m, sometimes leaves people uneasy – which is the main reason the actual characteristics of the manifold at r = 2m remained unclear for so many years. The singular behavior of the Schwarzschild time coordinate at and inside the 2m radius can be confusing, and the legitimacy of the transformation procedure can only be established by some fairly sophisticated reasoning. These complications can be avoided (to some extent) by proceeding directly from the Einstein field equations to the Kruskal metric, without introducing the problematic Schwarzschild coordinates. Our objective is to determine a spherically symmetrical metric that satisfies the vacuum field equations R = 0. Furthermore, to clearly exhibit the causal structure, we would like null paths (which represent the paths of light) to always be at 45 degrees from vertical in a drawing of the coordinates. These conditions immediately imply a diagonal metric of the form
where the parameter r represents the radial position, and A(r) is some function of r, which in turn is a function of the coordinates u and v. The partial derivatives of the metric coefficients are
In these terms the components of the Ricci tensor are
Equating each of these components to zero, we have five conditions, but the condition for R is redundant to the condition for R, so we really have just four distinct conditions. It’s convenient to express the conditions imposed by Rvv and Ruu in terms of their sum and difference, so we can express the conditions as
Now, if r is an explicit function of u and v, we could proceed to determine the functions A(r) and r(u,v) from these conditions, but to allow for the possibility that r is an implicit function of u and v, let us consider two functions f and g such that
The partial derivatives of r with respect to u and v are then given by
Substituting for the partial derivatives of r in terms of g and f, and simplifying, we get
This differential equation is satisfied by any arbitrary function of
, and also by any
arbitrary function of u2 – v2. We will find it very convenient later to choose a function h such that
= 0, which leads to the requirement
= 0. The simplest quadratic function
satisfying these conditions is simply u2 – v2. Thus we assure the equivalence of conditions (5.2) and (5.4) by setting.
which has the partial derivatives
Substituting these expressions into equation (5.2) gives
Multiplying through by f, and making use of the fact that
, this can be
written in the form
Differentiating both sides, we get
Where
Dividing the left and right sides of this equation by the left and right sides of (5.10) respectively, the quotient on the left side is
where we’ve made use of relation (5.9). Subtracting this from the quotient on the right side, we get the differential equation
If we divide through equation (5.10) by f, and take the derivative of A, and require that the ratio of this derivative to A satisfies (5.9), we would arrive at this same differential equation for f. Thus if we take a solution of (11) for our f function, and use this compute our A function by equation (5.10), we will satisfy conditions (5.2), (5.3), and (5.4). To show that these functions also satisfy condition (5.1), note that if we substitute from the equations (5.8) into condition (5.1) and simplify, we get
Multiplying through by the denominator of the right hand side, this gives
Again making use of (5.9), this can be written as
This would be equivalent to (5.11) if
which is automatically satisfied given that (5.9) is satisfied, as can be seen by taking the derivative of the relation
Differentiating this expression gives
Dividing through by the derivatives of A and f, we get
which is the same as (5.12). Therefore, with h(u,v) given by (5.7), and with f(r) being a function that satisfies (5.11), and with A(r) computed from (5.10), all of the components of the Ricci tensor vanish, so we have a solution of the vacuum field equations. To characterize the solutions of (5.11) in a systematic way, we can express f(r) as a Laurent series expansion, but after substituting this into (5.11) we find that all the coefficients of negative powers of r must be zero, which implies that f(r) must be analytic, i.e., it must have a power series representation of the form Substituting this into (5.11) and setting the coefficients of each power of r to zero, we have the following conditions on the coefficients.
and so on. If f1 is not zero, then the first condition shows that f0 and f2 must also be non-zero. In this case we can freely choose arbitrary (non-zero) values f0 = and f1 = , and the remaining coefficients are then fully determined, having the values
and so on. Therefore, in this case (i.e., with f1 ≠ 0) the function f(r) is of the form
for arbitrary constants and = /. The corresponding function A(r), given by equation (10), is
Recalling that the value of r in this expression is a function of u and v given by the relation f(r) = u2 – v2, we can write r and A explicitly as functions of u and v.
Thus we have the line element
For this system of coordinates the radial position r = 0 corresponds to the hyperbolic locus of points with u2 – v2 = , and all positive values of r are in the right-hand quadrant as shown in the figure below.
Fig-5.1a Notice that a light ray from the point r = 0 and v = 0 can propagate freely to all larger values of r, which might seem contrary to our expectation for a gravitational field. But notice also that no source mass appears in this metric. It is simply a non-inertial coordinate system for flat Minkowski spacetime. To confirm this, we can evaluate the components of the full Riemann curvature tensor, and show that they are all zero for this metric. It’s particularly useful to evaluate the component R in terms of r, and , because these three parameters have the same physical significance for all of our metrics as well as for the Schwarzschild metric, permitting a direct comparison. For the general family of metrics described above, it can be shown that
Inserting the function f(r) = er and the corresponding expression for A(r) into this equation, we find R = 0, whereas for the Schwarzschild metric we have
Thus the above metric corresponds to the case m = 0. From this we conclude that no line element of the desired form can represent a gravitational field with m ≠ 0 if the coefficient f1 in the power series expression for f(r) is non-zero. Therefore we return to the conditions for the coefficients of the power series, but this time we set f1 = 0. The result is that the first three conditions are automatically satisfied, without imposing any restriction on the values of f0, f2, and f3. The remaining conditions are
and so on. In this case we can freely choose f0 = , f2 = , and f3 = , and then recursively compute all the remaining coefficients from these conditions. This gives the coefficients
and so on. Notice that the odd coefficients are all multiples of , so if = 0 the function f(r) is an even function of r. In that case the non-zero coefficients are
and so on. Therefore, in this case we have
where = /. Since A(r) is zero, the metric is singular for the u and v coordinates. Moreover the reference curvature component, given by (5.13), is
which is infinite for any non-zero value or r. If we take r = 0 the metric vanishes completely, so this “Gaussian” expression for f(r) doesn’t lead to a realistic field. Therefore we require f3 ≠ 0. In a sense, the three non-zero parameters f0, f2, and f3 represent three degrees of freedom for the solutions of (5.11), set aside the flat solution and the singular “Gaussian” solution. For any specified values of these three coefficients, equation (5.11) uniquely determines all the remaining coefficients. The only solutions in finite polynomials are of the form
for arbitrary values of a, , and n. Furthermore, these constitute all the solutions of (5.11) (aside from the flat and Gaussian), because the series expansion of this function is
so for any given values of f0, f2, and f3 we can compute
Since (5.11) gives unique values for all the remaining coefficients, and since (5.16) represented solutions with all possible values of these initial coefficients, it follows that all the solutions of (5.11) (aside from the flat and Gaussian) are of the form (5.16). The corresponding metric coefficients and reference curvature are
For any value of n other than 1, the metric coefficient A either vanishes or becomes infinite at r = n/. Therefore, to give well-behaved metric coefficients for all positive values of r, we must set n = 1, and for convenience we also set a = -1, leading to the functions
Thus we have the line element
which is finite and non-singular at every positive radial location. We can already see that the value of must be 1/(2m) in order for the curvature component R to equal the corresponding component given by (5.14) for the Schwarzschild metric for a gravitational field of mass m. However, this would be making essential use of the Schwarzschild solution, because the value of R for Newtonian gravity is zero. To complete the derivation without invoking the Schwarzschild solution at all, we need to determine the value of more directly. This amounts to determining the value of the gravitational constant, which we’ve taken as unity by choosing suitable units. In practice this is based on empirical observations of the inverse square relation for the radial acceleration of gravity from rest. To evaluate this relation in terms of , we must first determine the geodesic equations for the metric. We only need to consider the radial and time components, because we will work with fixed angular values of and . Evaluating the Christoffel symbols and inserting them into the expressions for the geodesic equations for the two-dimensional space with coordinates u and v, we get
We note in passing that for null geodesic paths (i.e., light pulses) we have dv/d = ±du/d, in which cases these equations reduce to
and therefore such paths continue in the “45 degree” direction with dv/d = ±du/d.Now, we wish to determine the acceleration experience by a test particle at rest at a radial position r from a mass m, assigning to r and m the approximate physical meanings they have in Newtonian gravity, and evaluating the acceleration in terms of the proper time of the particle. Thus we wish to find d2r/d2 based on the metric (5.17) for a test particle that is at rest (meaning dr/d = 0) on the u axis at the radial position r corresponding to some initial u0. Differentiating the relation
which shows that the condition dr/d = 0 at v = 0 and u = u0 requires du/d = 0. From this and the metric (5.17) it follows that, at the initial condition, we have
Furthermore, differentiating (5.19) again, we get
Inserting the values for our condition and solving for d2r/d2, this reduces to
We can get the second derivative of u from the second geodesic equation for this condition
Substituting this into the previous equation, we get
where we’ve made use of (5.18) and (5.20) for our specified conditions. Inserting the functions A and f, this reduces to
It follows that we must have = 1/(2m) in order to match the Newtonian inverse-square relation d2r/d2 = m/r2. Therefore the line element (5.17) must be
where r is given as a function of u and v implicitly by the relation
This completes the derivation of the Kruskal metric directly from the field equations, without transforming from, or making any use of, the Schwarzschild coordinates.
Fig- 5.1b it possible to cover the interior and the exterior region of the Schwarzschild black hole with one coordinate patch Consider the coordinate transformation,
The metric becomes
And it is regular at 5.2 Schwarzschild Radius [15] One of the remarkable predictions of Schwarzschild's geometry was that if a mass M were compressed inside a critical radius
, now a days called the Schwarzschild radius, then its
gravity would become so strong that not even light could escape. The Schwarzschild radius of a mass M is given by
where G is Newton's gravitational constant, and c is the speed of light. For a 30 solar mass object, like the black hole in the fictional star system here, the Schwarzschild radius is about 100 kilometers. Schwarzschild radius had already been derived (with the correct result, but an incorrect theory) by John Michell in 1783 (this reference is from Erk's Relativity Pages) in the context of Newtonian gravity and the corpuscular theory of light. Michel derived the critical radius by setting the gravitational escape velocity v equal to the speed of light c in the Newtonian formula
for the escape velocity v from the surface of a star of mass M and radius 5.3 Horizon [15] The Schwarzschild surface, the sphere at 1 Schwarzschild radius, is also called the horizon of a black hole, since an outside observer, even one just outside the Schwarzschild surface, can see nothing beyond the horizon. 5.4 Gravitational Slowing of Time [15] In general relativity, clocks at rest run slower inside a gravitational potential than outside In the case of the Schwarzschild metric, the proper time, the actual time measured by an observer at rest at radius r, during an interval
of universal time is
dt, which is
less than the universal time interval dt. Thus a distant observer at rest will observe the clock of an observer at rest at radius r to run more slowly than the distant observer's own clock, by a factor
This time dilation factor tends to zero as r approaches the Schwarzschild radius rs, which means that someone at the Schwarzschild radius will appear to freeze to a stop, as seen by anyone outside the Schwarzschild radius. 5.5 Gravitational Redshift [15] The gravitational slowing of time produces a gravitational redshift of photons. That is, outside observer will observe photons emitted from within a gravitational potential to redshifted to lower frequencies, or equivalently to longer wavelengths. Conversely, observer at rest in a gravitational potential will observe photons from outside to blueshifted to higher frequencies, shorter wavelengths.
an be an be
The gravitational slowing of time produces a gravitational redshift of photons. That is, an outside observer will observe photons emitted from within a gravitational potential to be redshifted to lower frequencies, or equivalently to longer wavelengths. Conversely, an observer at rest in a gravitational potential will observe photons from outside to be blueshifted to higher frequencies, shorter wavelengths. In the case of the Schwarzschild metric, a distant observer at rest will observe photons emitted by a source at rest at radius r to be redshifted so that the observed wavelength is larger by a factor
than the emitted wavelength. The redshift factor tends to infinity as r approaches the Schwarzschild radius , which means that someone at the Schwarzschild radius will appear infinitely redshifted, as seen by anyone outside the Schwarzschild radius. That the redshift factor is the same as the time dilation factor (well, so one's the reciprocal of the other, but that's just because the redshift factor is, conventionally, a ratio of wavelengths rather than a ratio of frequencies) is no coincidence. Photons are good clocks. When a photon
is redshifted, its frequency, the rate at which it ticks, slows down. In the illustration shown, a source at rest at 1.18 Schwarzschild radii emits light rays with the same initial wavelength in 6 equally spaced directions. The light ray going out is redshifted, while the rays falling in become blueshifted, from the point of view of observers at rest in the Schwarzschild geometry. Five of the 6 rays end up falling into the black hole No stationary frames inside the Schwarzschild radius. According to the Schwarzschild metric, at the Schwarzschild radius , proper radial distance intervals become infinite, and proper time passes infinitely slowly. Inside the Schwarzschild radius, proper radial distances and proper times appear to become imaginary (that is, the square root of a negative number). The problem with the Schwarzschild metric is that it describes the geometry as measured by observers at rest. It is now realized that once inside the Schwarzschild radius, there can be no observers at rest: everything plunges inevitably to the central singularity. In effect, the very fabric of spacetime falls to the singularity, carrying everything with it. No pressure can withstand the inexorable collapse. To paraphrase Misner, Thorne & Wheeler (1973, ``Gravitation'', p. 823), that same unseen power of the world which impels everyone from age 20 to 40, and from 40 to 80, impels objects inside the horizon irresistably towards the singularity. The Schwarzschild metric remains valid inside the Schwarzschild radius. It is fine to perform mathematical calculations using the Schwarzschild metric. Inside the Schwarzschild radius, if you transform to frames of reference which fall inward (or outward, for a white hole) faster than the speed of light, then the geometry becomes `normal' again. 5.6 Schwarzschild Spacetime Diagram [15]
Fig-5.1c This spacetime diagram illustrates the temporal geometry of the Schwarzschild metric, at the expense of suppressing information about the spatial geometry. By comparison, the embedding diagram at the top of the page illustrated the spatial geometry, while suppressing information about the temporal geometry. The horizontal axis represents radial distance, while the vertical axis represents time. The cyan vertical line is the central singularity, at zero radius, Each point at radius r in the spacetime diagram represents a 3-dimensional spatial sphere of circumference 2 pi r,The Schwarzschild spacetime geometry appears ill-behaved at
the horizon, the Schwarzschild radius (vertical red line). However, the pathology is an artifact of the Schwarzschild coordinate system. Spacetime itself is well-behaved at the Schwarzschild radius, as can be ascertained by computing the components of the Riemann curvature tensor, all of whose components remain finite at the Schwarzschild radius. The curious change in the character of the Schwarzschild geometry inside versus outside the horizon can be seen in the spacetime diagram. Whereas outside the horizon infalling and outgoing light rays move generally upward, in the direction of increasing Schwarzschild time, inside the horizon infalling and outgoing light rays move generally leftward, toward the singularity. General Relativity permits an arbitrary relabelling of coordinates. Some coordinate systems which behave better at the Schwarzschild radius are illustrated below. 5.7 The Schwarzschild Black Hole [10] Is it possible to cover the interior and the exterior region of the Schwarzschild black hole with one coordinate. The metric has two singularities r = 0,
= Schwarzschild
radius and it is regular at
Fig-5.1d the end of a star's life occurs when the star has exhausted all of its nuclear fuel. In its death, the star could collapse to form a black hole if it were massive enough. The spacetime of a black hole is curved in such a way as to cause the future light cones to tip inward. At a specific distance from the black hole, the light cones are so tipped-over that the "outgoing edge" of each light cone is vertical in the diagram below. These "edges" form a surface (drawn as a cylinder in the diagram). This surface (called the event horizon) is the characteristic feature of a black hole.
Fig-5.1e The event horizon is a boundary that divides this spacetime into an "inside" and an "outside". Once inside, particles and light-rays can never escape outside. In fact, since all of the light cones point to the singularity (a really bad place), their worldlines will end. (Realize that the light cones restrict the fates of worldlines that encounter them.) [The following diagram shows an observer's worldline in the outside region. This observer is periodically sending out light-pulses. The light-pulses could be detected by, say, another observer who ventures into the black hole.] 5.8 Inside the Black Hole [12] Now let’s consider the Schwarzschild solution for
(inside a black hole). A
small but very important change must be made to the metric for this case. When the coefficient negative.
is positive. However, for
this coefficient is
In order to work with positive coefficients for this case, we use The metric then becomes
Notice how the minus sign has moved from the t coordinate to the r coordinate. This means that inside the event horizon, r is the timelike coordinate, not t. In General Theory of Relativity, the paths of material particles are restricted to timelike world lines. It is the coordinate with the minus sign that determines the meaning of “timelike.� According to General Theory of Relativity, inside a black hole, time is defined by the r coordinate, not the t coordinate. It follows that the inevitability of moving forward in time becomes, inside the
black hole, the inevitability of moving toward occurs at
Thus,
This swapping of space and time
marks a boundary, the point where space and time
change roles. For the observer inside this boundary, the inevitability of moving forward in time means that he must always move inward toward the center of the black hole at All timelike and lightlike world lines inside at
must move toward decreasing r and end
(the end of time!) Because it is not possible for any particle or photon inside to take a path where r remains constant or increases, the boundary
is
called the event horizon of the black hole. No observer inside the event horizon can communicate with any observer outside the event horizon. the outside observer can never see an infalling observer reach or cross the event horizon, because any light radiating from the infalling observer slows down and redshifts, with the redshift approaching infinity as the infalling observer nears the event horizon. The infalling observer does in fact cross the event horizon. Remember that the singularity at
(the
event horizon) was shown to be a coordinate singularity, not a real, physical singularity. Using transformed coordinates, it can be shown that the infalling observer passes from to
in a finite amount of time (his proper time, or the interval along his world
line). Furthermore, it can be shown that the maximum amount of time from
to
for an observer who has fallen through the event horizon, even if he has at his disposal a rocket of unlimited power, is given by meter Where M is the geometrized mass used in the Schwarzschild metric. M is related to the Newtonian mass m by
where G is the gravitational constant in S.I. (standard international) units. Let’s look at a reallife example. Astronomers believe that there is a supermassive black hole at the center of our galaxy, with an estimated mass of about 3.7 million solar masses. The tidal force near the event horizon of such a large black hole is weak. (The tidal force, or tidal acceleration gradient, is the difference in the gravitational acceleration between two points in a nonuniform gravitational field. The smaller the black hole, the larger this gradient is near the event horizon, because the curvature of spacetime is greater. An astronaut approaching a stellar black hole of a few solar masses would be torn apart by the tidal force before reaching the event horizon.) Thus, it is possible that an astronaut, if well protected from radiation, could survive to cross the event horizon of this supermassive black hole and continue
inward. Let’s calculate the maximum time this astronaut could avoid reaching the center of the black hole. (For simplicity, we assume the black hole is not rotating, so that the above formula can be used.)
Our intrepid astronaut has less than a minute to explore the black hole. The Schwarzschild radius of this black hole is
5.9 Physical Interpretation of the Event Horizon [14] We found earlier that the Schwarzschild metric has a coordinate singularity at the event horizon, where the coordinate time becomes infinite. However, a calculation using transformed coordinates shows that the infalling observer falls right through the event horizon in a finite amount of time (the infalling observer's proper time). Although we can never actually see someone fall through the event horizon (due to the infinite redshift), he really does. As the free-falling observer passes across the event horizon, any inward directed photons emitted by him continue inward toward the center of the black hole. Any outward directed photons emitted by him at the instant he passes across the event horizon are forever frozen there. So, the outside observer cannot detect any of these photons, whether directed inward or outward. Consider two observers far from the black hole. Suppose they synchronize their watches, then one of them remains far from the black hole while the other descends slowly (at first) toward the event horizon. Then the time on the watch of the descending observer as he reaches and falls through the event horizon will be approximately equal to the time on the watch of the far away observer as she sees her companion disappear very near the event horizon.
Chapter Six Static and Non-Static Cosmological Models
6 Cosmological Models [6] The speculation about the nature of the universe is as old as man himself. It has been known for a long time that Newton’s gravitational theory meets with serious difficulties when applied to the universe as a whole. The three crucial tests of general theory of relativity to indicate that it has provided some significant modification over the Newtonian theory and has furnished an acceptable solution of the problem of the field of star in the empty space surrounding it at least to the distance of the order of the dimensions of the solar system. It then seems to be great interest to extend the application of the general theory of relativity to the universe as whole. The question was first taken up by Einstein shortly after the development of the general theory of relativity. Since then it has been the object of many investigators. This is the very interesting program because certain large scale properties of the
universe are experimentally known and capable of comparison with such a model of the universe. The following are the most important of these properties. (a) On the average, matter is distributed in a fairly uniform manner in the universe, with
the density of approximately (b) The universe appears to be fairly isotropic from the solar system. (c) Light reaching us from the distant nebulae is shifted towards the red in proportional to
the distance traversed, according to the law (d) According to the measurement of the radioactive remains, some rocks in the crust of the earth are at least 3.5 to 4 billion years. Einstein’s modified field equations are :
The constant
is such that its effect is negligible for phenomenon in the solar system or even
in our own galaxy; but becomes important when the universe a whole considered. By combining various value of
with different various possibilities of
different models of
the universe may be constructed. The static solution of equation (6.1.1) represent “static cosmological models” Here w shall first consider the static homogeneous and isentropic model of universe originally proposed by Einstein’s de-Sitter’s cosmological models are based on the following assumptions: (1) The Universe is static, i.e., in a proper co-ordinate system matter is at rest and the
proper pressure
and proper density
are the same everywhere.
(2) The universe is isotropic, i.e., all special direction is equivalent. (3) The universe is homogeneous, i.e., no part of the universe can be distinguished from the other. (4) For small values of r the line element should reduce to special relativity from for flat-space time since local gravitational fields can be neglected in small space time regions. The line element of the static, homogeneous, isentropic universe has the familiar from
Where
are spherical polar co-ordinates,
radial distance r to be determined, i.e.,
and
are some unknown functions of the
For the universe containing perfect fluids, the pressure
and density
determined by the
field equations (6.1.1) are given by
Where the primes denote derivatives with respect to r, i.e., assumption (i)
etc. According to
, therefore the equation. (6.1.3c) gives
This equation is satisfied with any of the three possibilities (i) (ii)
(iii)
These three equations respectively lead to the Einstein de-Sitter and the special theory and the special theory of relativity line elements of the universe and contain in themselves all the possibilities of a static isentropic and homogeneous universe. 6.1 The Einstein Line Element (I)
Einstein line element arises from the possibility
(II)
This leads on integration Appling condition (iv) to the above equation i.e.
at
We get Thus we have Substituting the value of
0 in equation (6.1.5a), we get
So the line element (6.1.2) becomes
This is the line element is called the Einstein line element for static, isotropic and homogeneous universe. 6.2 Geometry of the Einstein Universe In order to understand the geometry of space-time characterized by Einstein line element; it is convenient to rewrite the Einstein line element by the transformation of co ordinates. (a) Consider the transformation
So the Einstein line element transforms to
(b) Further substituting
Equation (7.1.6) leads to Einstein line element in the form
(c) Again by putting
In equation (7.4); Einstein line element take the
form
(d)Finally considering the transformation
We see that Einstein line element takes the form
Einstein line element in this form suggest that the special geometry of Einatein universe may be regarded as the immersion of the whole three dimension spherical surface in a four dimensional Euclidian space, viz.,
This form also represents the isentropic and homogenous character of Einstein universe. It contradicts with actual universe where, according to Hubble and Humason, a definite red shift is observed in the light from the nebulae which red shift increases at least very closely in linear propagation with distance. 6.3 Spherical space. Consider the space a spherical the total special volume of Einstein universe, from equation, (7.1.8) is (7.1.11) Although the volume of the spherical space is finite
yet it is unbounded. Neither there
is any boundary nor any centre of the spherical surface, every point in this space stand in the
same relation to the rest of the space as any other point. The total distance around the universe (major circle) is (7.1.12) 6.4 Elliptic Space The Einstein line element is given by (7.1.4) defines the elliptical space. The element given by (7.1.4) is real only when
the special expansion of the physical space in Einstein
universe is defined by the expression
The total volume of the Einstein universe is then
And the total proper distance is
Thus we see that the total volume and the total proper distance for elliptical space are just half of the corresponding quantities for spherical space. 6.5 Density and Pressure of Matter in Einstein Universe For general line element
For Einstein universe, we have
So that equation (7.1.16a) and (7.1.16b) yield
This equation represents required expression for the pressure and density are known, the two unknown quantities and
and
and density
if the pressure
can be expressed in terms of
from equation (7.1.18a) and (7.1.18b) as
6.6 Incoherent Matter in the Universe Let us assume that universe is filled with incoherent matter ( )Exerting no pressure. Then we have from equation (7.1.18a) and (7.1.18b)
Hence the total mass of the spherical universe is
And the total mass of the elliptic universe is
6.7 Radiation in the Universe
Let us assume that the matter fluid filling the universe is radiation only. For this case . so that the parameter
and
are given by
Hence the total mass of the spherical universe
The total mass of the elliptic universe
6.8 Empty Universe Let us assume that the universe in entirely empty. Then So that the equation (7.1.19a) and (7.1.19b) yield
Thus in this case, Einstein line element takes in the form of special relativity line element for the flat space time, i.e., Einstein universe degenerates to the flat space time of the special relativity. It is obvious that the general theory of relativity provides a quantitative description of the universe based on some simplifying assumptions.
6.9 Motion of a Test Particle in Einstein Universe Einstein line element is
The motion of a test particle in a gravitational field corresponding to the line element given by equation (7.1.26) would be described by geodesic equations.
Eir the sake of simplicity lat the test particle be infinitely at rest, so that the component of the special velocity of the test particle are zero i.e.,
Then equation (7.1.27) will become
Hence equation (7.1.29) yields
i.e., the particle has zero acceleration. Hence in Einstein universe a rest particle would remain permanently at rest. We may also interpret that the matter Einstein universe is without motion. 7.1 The de-Sitter Universe Line Element The de-Sitter universe line element arises from the possibility
This on integration gives Where c is integration constant. Appling the boundary condition that should apporch the special relativity from, i.e., We get
Hence
, the line element
Equation (6.1.3b) may be written as
, we get
So that line element (6.1.2) becomes
This line element called de-Sitter line element for static, isotropic and homogeneous universe. 7.2 Geometry of de-Sitter Universe In order to understand the geometry of space time characterized by de-Sitter line element, it is convenient to re-write de-Sitter line element by transformation of co-ordinates. The de-Sitter line element is
(a) Considering the transformation
the de-Sitter line element transforms to
(b) For further simplification of de-Sitter line element, let us substitute
So that
Hence de-Sitter line element reduce to
(c) Further substituting in
This equation determine that four dimensional surface in the five dimensional manifold which corresponds to the space-time and we may regard the geometry of de-Sitter universe as that holding on the surface of a sphere embedded in four dimensional Euclidian space. 7.3 Lemaitre-Robertson Transformation The de-Sitter line element I another useful form may be obtained by considering the transformation
Using this transformation de-Sitter line element 7 becomes
Substituting
and dropping primes we get
This transformation was introduced by Lemaitre and Robertson independently. By the help of this transformation it is possible to change the static line element into a non static one. 7.4 Pressure and Density Matter in de-Sitter Universe(absence of matter and radiation)
The de-Sitter line element is obtained by the condition Since the proper density
can either be zero or positive, ie.,
, hence
above equation will be satisfied only if
This means that de-Sitter universe is completely empty. It contains neither matter nor radiation. 7.5 Motion of a Test Particle in de-Sitter Universe The de-Sitter line element is With
(7.5.17)
The motion of a test particle is governed by the geodesic equations
The non vanishing Christoffel’s symbols of second kind corresponding to line element (17) are
geodesic equations (7.5.18) are written as
Using equation (7.5.18a) and 7.5.19 above equation give
Let us choose the initial motion to be in the plane
then
So that equation (7.5.20b) gives
This equation implies that the particle will continue to move in the plane Using equation (7.5.21), (7.5.20a) and (7.5.20d)
Equation (7.5.24) and (7.5.25) may be written as
The integration of the above equation yield
Where h and k are constant of integration. The constant h is measure of angular momentum of the motion. Further instead of working with the equation. (7.5.22) due to its troublesome integration, we use the line element (7.5.17), which by use of (7.5.21) yields
Substituting the value of
Putting
and
from(7.5.25) and (7.5.26), we get
we get
Differentiating with respect to
and simplifying we get
Now using (4) and (16) we get
This equation represent the orbit of the particle in the de-Sitter universe and corresponds in Newtonian mechanics to the motion of a particle under a central repulsive force proportional to the distance r where 7.6 Similarity and Difference between Einstein and de-Sitter Line Element Topics
Einstein line element
de-Sitter line element
(1)Geometry
(1)The physical space of Einstein universe may be embedded in a Euclidean space of higher dimension. This also suggests that the geometry of the Einstein universe is one which holds on the surface of a sphere embedded in a Euclidean space
(1)The physical space in deSitter line universe can be embedded in a Euclidean space of higher dimensions. The geometry of this universe is one which holds on the surface of sphere embedded in a Euclidean space of five dimensions.
of four dimensions.
(2)Density and pressure
(2) Density and pressure of the matter (2) The de-Sitter line element in Einstein universe is as follows is based on the assumption
(a)
Since
and there fore
we have (b)
From (a) and (b) we have This is the unique solution of . The equation
implies that
the de-Sitter universe is completely empty. It contains neither matter nor radiation.
(3) Motion of a test particle
(3) In the Einstein universe the motion of a test particle has zero acceleration. It means that in Einstein universe a particle at rest remains at rest.
(3) In the de-Sitter universe the motion of a test particle has zero acceleration. It means that in de-Sitter universe a particle at rest at origin with
remains at
rest. (4) Shift in spectral lines
(4) From the Einstein universe we
(4) From the de-Sitter
have
universe we have
meaning three
no
red shift. It means that Doppler Effect is observed. In other words, the nebulaes do not seen to sun away. It contradicts the matter is receding in actual universe and the universe is expanding at every moment.
this
shown that red shift is proportional to the distance measured from the origin if we take
It also supports
Weyl’s theory according to which nebulaes are receding with velocity proportional to the distances from us. We see that de-Sitter is completely empty yet predicts the recession of nebulae.
8(a) Robertson Walker Model [1] To us the universe appears to be homogeneous and isotropic on a sufficiently large scale. It is unlikely that we are in a special position of the universe. This which states roughly speaking that that the universe looks the same from all positions in space at a point are equivalent. To define a moment of time. We introduced a series of non-intersecting space like hyper-space. Let t be denoted the synchronized properties of all galaxies and
be denoted the
co-ordinates, which are constant for all galaxies. Then space-time metric is defined by , The spatial distance ) and
;
of any two nearly galaxies on the same hyper-space
constant at (
is ,
Let us consider the triangle formed by three nearly galaxies at same particular time and also triangle formed by the same galaxies at same lower time. By the cosmological principle yhe
second triangle must be similar to the first triangle and magnification factor must be independent of position of the triangle of through a common factor
.This means that time can enter
that the ratio of small distance may be same of all times. thus ,
The
only
where
is homogeneous, isotropic and independent on time. This
must be a space of constant curvature also. The of such space can be expressed as a interms of properties of curvature
curvature alone. By the definition of symmetry
can be written as where
is a constant
Let it can be verified that the three dimensional curvature tensor can be represented as the following metric.
With the transformation
Again by transformation
, this becomes
of the space
This is the Robertson-walker metric. The spatial metric of the Robertson-walker space-time metric is given by
We study this equation for For
this has a positive curvature
with the transformation
, then the metric
becomes
(8.2)
Such an space can be embedded. For four dimensional Euclidean space with
This corresponds to the hypersurface In the Euclidean space
Clearly all the points and directions on this We get
are equivalent. Putting . It has surface areas
hyper-surface is sweept by the co-ordinate range . The total volume is For
the universe has zero spatial curvature. In this case
This is the ordinary Euclidean
with range of
is .
constant. . The entire
Whose spatial volume is infinite. For
the universe has negative spatial curvature. In
this case the spatial part of the Robert Walker space-time in to the hyper-surface
With trans formation
which corresponding
in the Minkowski space.
Then This is the ordinary
is,
with range of
. Whose spatial volume is infinite.
8(b) Friedmann Model [5] We Robertson walker line element is (8.3) Here we have from the above metric, ,
We know the Christoffel symbols of second kind is If we label our co-ordinate according to
Then we define only the non-vanishing Christoffel symbols are:
We know that,
Which gives,
And With
, we know,
and then (8.5)
In our moving co-ordinate system, So that Hence Therefore from (8.6)
Extracting
from the line element of (6.11) we see that
Now from Einstein field equation
But from Robertson-walker line element, we obtain
(8.6)
by (6.14) Applying (6.15) into (6.16) we obtain, +
Putting
in right hand side we get
This is the general form of Robert walker model. Here k is a scalar and
is gravitational
constant. We shall refer to this equation as Friedmann equation. Note that the pressure has completely cancelled out of this equation. We know that
yields the continuity
equation and equation of motion of the fluid particles with
those becomes,
The continuity equation (8.7) may be written as
And with
, this reduces to
This contains the pressure. As for the standard friedmann model. We put equation and it becomes
Integrating we have
Which is friedmann model of the universe, where
,
in the above
This leads to possible models each of which has
at some point in time and it is
natural to take this point as the originof t, so that
and t is then takenthe age of the
universe. Let
be the present age of the universe and
values of
and
are present day
and .
We write equation Therefore the friedmann models (6.17) becomes,
The three friedmann models arise from integrating equation (6.23) for the three possible of k.
(i)
If
(field model) then (8.11) implies
This model is known as the Einstein de-Sitter model. Therefore , therefore
(ii)
If
Substituting
(closed model): then from (8.10) we get
gives
,where
as
If
These two equations give
via the parameter. The graph of
the figure.
And (iii)
(open model), If
then
is cycloid and shown in
These two equations give Comments: we see that
via a parameter and
its graph is also shown in the above figure. gives modelwhich continually expand,
gives a model which expand to a maximum value of only spatial. But also temporally closed.
and then contracts, so the latter is not