10.1 The Schwarzschild Solution
This section will present a simple derivation for the Schwarzschild solution of general relativity and then derive perihelion procession.
In pretty much any general relativity text you will find a Schwarzschild solution derivation. Just to be different and do it my own way, this will be a Kerr-Schild coordinate derivation of the Schwarzschild solution that we will then transform to Schwarzschild coordinates resulting in:
Kerr-Schild coordinates are such that the coordinate speed of inward radial moving light is c. I am also looking for a spherically symmetric static vacuum solution. In other words I am looking for an exact vacuum solution to Einstein's field equations that will take the form:
Einstein's field equations are so nonlinear that when seeking an exact solution with a trial metric you'll want to start by finding the Einstein tensor for the simplest case of the trial first as you may end up with thousands of terms in the resulting Einstein tensor making it essentially impossible to solve, so for example we will actually first seek an exact vacuum solution in the form of
and should one not exist, then we would resort to finding the Einstein tensor for 10.1.2a and seek an exact vacuum solution from that.
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The Einstein tensor from 10.1.2b is
so all we have to do is find the solution for f + rdf/dr = 0 and we have an exact vacuum solution. The f that is the solution to this simple to solve first order linear seperable differential equation is f = r0/r where r0 is a constant so we have the Kerr-Schild coordinate expression for the Schwarzschild solution as:
Next one would calculate the Reimann tensor and make sure that it is not zero ensuring that the vacuum solution found isn't a mere frame transformation of the metric of special relativity, and doing so one finds there are many nonzero elements for the Riemann tensor for this vacuum solution. According to the Birkoff theorem any spherically symetric exact vacuum solution is nothing more than a frame transformation of the Schwarzschild solution, so we've found the solution. The transformation that takes this to Schwarzschild coordinates is
10.1 The Schwarzschild Solution 113
And defining a constant M by
yields 10.1.1 as the exact vacuum solution to Einstein's field equations called the Schwarzschild solution.
An exact calculation of the radial case of geodesic motion a test particle undergoes in terms of M for this spacetime yeilds
where is the proper time for the test particle, and so we see that this constant M in this exact vacuum solution for Einstein's field equations is what we think of as the active "gravitational mass".
For a black hole r0 is called the Schwarzschild radius. The Schwarzschild solution of general relativity is the solution for the case of a non-rotating uncharged black hole. The Schwartzschild radius describes a mathematical surface at which there is also a coordinate singularity for this kind of black hole.
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The coordinate singularity for general relativity is a place where infinities appear in our equations due to our choice of coordinates such as the
term has at r = r0 , but locally no physical quantities are infinite. In general relativity this surface is called an event horizon. Most generally an event horizon is any surface at which the coordinate speed of light vanishes.
Now lets derive perihelion procession.
The Schwarzschild solution is independent of both ct and yielding two Killing four-vectors
The first gives us the following conserved quantity for geodesic motion called the energy parameter for which we define by
The second killing four-vector results in another conserved quantity for geodesic motion called the angular momentum parameter,
Inserting these into the Schwarzschild solution itself and dividing through by results in
Orient the equatorial plane to match the orbit so .
With a little algebra we wind up with equations of motion that look very Newtonian
the difference being that the relativistic expression corresponding to conservation of energy here has a term that looks like a 1/r3 potential, or a 1/r4 force. Using the second in the first
Changing variables to u = 1/r
Differentiate with respect to u and chain rule
Its a strait forward back check to verify neglecting terms in the check above first order in (GM/c2) the following expression satisfies that differential equation
And so works as an approximate solution for a weak field.
The perihelions now occur at
If the period of the orbit is T, then the total change in angle for the location of perihelion over time is then
or in terms of the length of the semi-major axis a:
The complete exact equations of geodesic motion for a test mass in a Schwarzschild spacetime are
where is a constant of the motion and is equal to the of special relativity at , and is a constant of the motion and can be thought of as the angular momentum of the test particle in orbit divided by its mass.
Verify that far from the hole, the Schwarzschild solution obeys Eqn. 9.2.4 to order 1/r2.
Look up recent solar system data and use equation 10.1.20 to find for the mercury's procession after 100 of Earth's years. (Note - It is actually 575", but 534" are accounted for by the effect of other planets.)
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10.2 Hovering over a Schwarzschild Black Hole
In this section we derive the equation for the amount of force felt by an object held stationary over a Schwarzschild black hole for general relativity.
and then mention why we see from this that the assumption that we could hold it stationary underneath the event horizon was false.
Start four-acceleration Eqn. 5.3.1
Which simplifies to Eqn.10.2.1
This is the amount of force felt by the object hovering over the hole for general relativity. We should note that it
10.2 Hovering over a Schwarzschild Black Hole 117
becomes infinite at the event horizon and imaginary underneath and therefor the very assumption that an object underneath the event horizon of a Schwarzschild black hole could be held still was false.
Consider a cable lowering an observer of m down to the event horizon of a black hole. The work required to do this turns out to be mc2 and so the reaction force that the hovering cable wench feels must always be finite, right down to the event horizon. However, the tension in the bottom part of the cable attached to the lowered observer diverges and so it always snaps just prior to reaching it. Show that the tension in the bottom end of the cable near the event horizon a height h above it is given by
Use Eqn. 10.2.1 to find the weight felt by a 75 kg observer standing 100m over the event horizon of a 3 solar mass black hole.
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10.3 Exact Spacetime Inside Static Spherically Symmetric Matter
By a static spherically symmetric matter distribution I will mean matter for which coordinates can be found such that the line element for the interior solution can be written as
where f, g, and g00 are functions of r. In derivations of the Schwarzschild solution it is common to choose Schwarzschild coordinates such that g is absorbed into the definition of r leaving only the functions f and g00 to find. It is more convenient in this case to instead choose isotropic coordinates. Since the f in front of the dr2 term and the g in front of the terms are functions of the coordinate r, one is free to choose a radial coordinate such that the line element becomes
where hn is some function of the new radial coordinate r. Exactly working out the Einstein tensor for this line element and referring to Einstein's field equations one finds that the time-time element of the stress-energy tensor T 00 that exactly corresponds to this line element which merely required static spherical symmetry expressed in isotropic coordinates is
One might immediately recognize the (dh/dr)2 term as associated with an electric field's energy density and h as proportional to its electric potential. Lets consider however, that we are interested in uncharged matter first. The choice of n=4 will yield a zero energy density wherever the Laplacian of h is zero which means that choosing n=4 would correspond to h having an interpretation of being proportional to the Newtonian gravitational potential yielding vacuum where the Laplacian of h or the Laplacian of the proportional gravitational potential is zero. So for the case of uncharged matter it is natural to choose n to be 4. In that case the line element becomes
And the time-time element of the stress-energy tensor that corresponds exactly to that is
At this point we can define a new function in terms of h that will correspond to the Newtonian gravitational potential as
In doing so the line element is written
and the time-time element of the stress-energy tensor exactly corresponding to this is
we see that the time-time element of the stress-energy tensor can be written
In places of weak gravitation where we see now that the energy density corresponds to the Newtonian energy density at that same place where the spatial elements of the line element correspond to Euclidean coordinates. An observer for which these coordinates are appropriate, for which they are a mere spherical coordinate transformation of that of special relativity nearby him finds that nearby him the energy density observed corresponds to that of Newton where is the matter's Newtonian gravitational potential.
Next I will define a function in terms of g00 and as
In doing this, the line element is written
and the full stress-energy tensor exactly corresponding to this is
Outside the spherically symmetric matter where you choose and , all of the stress-energy tensor terms vanish and the line element becomes the isotropic coordinate expression of the Schwarzschild solution
If the matter is a fluid or gass, then inside the matter you have
but for a liquid the relation between the two potentials from this equation will break at the surface where there is surface tension.
So let us look at an example of how to apply this in modeling the interior spacetime of a nonrotating sphere of uncharged matter. The energy density term of the stress energy tensor is related soley to the behavior of . As such, choosing the behavior of , corresponds to choosing the behavior for the density of the matter. We may then use the Newtonian potential for the interior of a uniform sphere to model a solid or liquid as an approximation for the interior, through it wouldn't model the interior of a gass well where the density of the gass would greatly increase with depth, so let us start this model as a liquid or solid choosing the density so that the potential is well enough described by that of Newton yielding for the interior
If the matter that we are modeling is a liquid, then we must apply equation 10.3.15 in order to find the appropriate corresponding potential . The solution for this is
So our model for the interior solution for a liquid is
If the surface tension is negligible then A and B go to 1 which is an application of boundary conditions that g00 and dg00/dr are continuous at the boundary with the exterior solution being the isotropic coordinate expression of the Schwarzschild solution.Exercises
a. Show that outside of matter where, , that in order for the pressure terms to correspond to vacuum, you must have
b. Show that k other than zero corresponds to a mere time scaling coordinate transformation.
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10.4 Behavior of Light in a Schwarzschild Spacetime
For this section on general relativity, first lets derive the equation for the gravitational red shift for a Schwarzschild space-time. Then we will compare deflection of light according to a local observer to deflection over large trajectories, then derive the photon sphere and finally gravitational slowing of the remote coordinate speed of light known as the Shapiro effect.
Starting with the interval for the metric for the Schwarzschild solution of general relativity we have Eqn.10.1.1
Consider a stationary observer so that the spatial displacements are zero and the interval is the proper time for this observer's world line. This reduces the expression to
Now if we were to relate the observed periods of light that two different stationary observers at two different altitudes find coming from a common source for general relativity we get
Using that frequency is the inverse of the period we arrive at an expression relating the frequencies observed
Sometimes a Taylor expansion is done at this point where we keep only terms to first order in 1/r for an approximation that is a simpler expression and we then have
This phenomenon in general relativity is called gravitational red shift.
10.4 Behavior of Light in a Schwarzschild Spacetime 123
Next lets look at a case of deflection of light as observed locally from inside a high hovering ship. We will find that this amount which is what is referred to as the amount of deflection due to equivalence is only half the value of deflection over large distances. That's the curvature of space effect. Imagine that a ship hovers above a planet, star, or Schwarzschild black hole at a distance to the center of "r" which we will take to be large compared to the dimensions of the ship. We orient a laser so that it points at the center of a target in a direction perpendicular to the direction of the massive body. We will show for general relativity that the light is deflected so that it hits below target as long as the ship hovers still by demonstrating that
We start with the equation for coordinate acceleration Eqn. 5.3.8
In this case there is no four-force, as none act on the photons, it is
The component we need is
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Then we make an approximation what for the entire length of the beam .
Inserting the expression for the affine connection we have
(no sum on r as its not a variable index)
For Simplicity we'll let the beam be oriented in the direction at . Doing the sums we have
Using grr = -1/(1 - 2GM/rc2) and g00 = 1 - 2GM/rc2 we have
10.4 Behavior of Light in a Schwarzschild Spacetime 125
Now we make the approximations and for the entire length of the beam and using and we have
Or more simply Eqn. 10.4.9
The first term c2/r arises solely due to using spherical coordinates. If there were no mass and we chose to use spherical coordinates to describe the rectilinear motion of an object moving in that direction we would still have a v2/r term. The other term -GM/r2 we note is identical to the Newtonian expression for gravitation. This tells us that if the ship itself were to have been in free fall, the beam would have hit dead center. But that is what we would also expect for a ship in an inertial frame in the absence of massive bodies.
We found that light deflects over short distance in a way consistent with Newtonian gravitation over a short distance above,
however the deflection of light over large distances we will show here to differ from Newtonian gravitation by a factor of 2. It is this general relativity prediction that agrees with astronomical observation. That is experimental proof for the curvature of the space.
From equation 10.1.12 considering a nonzero angular momentum there will be a minimum radius the light approaches the gravitating mass at which the equation yields
From this the angular momentum parameter can be solved for in terms of the energy parameter
Inserting this into equation 10.1.15 results in
Light speed particles are described by this as the limit as resulting in
It is a strait forward back check to verify that neglecting higher order terms than GM/rmin2c2 the following satisfies that differential equation
At the angles at which the light comes in from and goes out to infinity u then goes to zero. This yields a quadratic equation in resulting in
Since must be less than 1 in magnitude it is the top sign that is the correct solution. This can then be written
And for a small angle of deflection this yields
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The incoming deflection angle is equal to the outgoing deflection angle so the sum of the two giving the total deflection is twice that value:
It is the value of this prediction that matches the deflection of light from stars passing our sun on the way to us and results in the correct gravitational lenzing observed in nature.
Finally lets derive the photon sphere. The photon sphere of a Schwarzschild black hole is a sphere given by a radius at which a photon can make a complete closed orbit around the black hole.
rps = 3GM/c2.
In this section we will prove this.
When we use the coordinate variables for the indices no Einstein summation will be implied.
We begin with the expression for coordinate acceleration Eqn. 5.3.8
No four forces will act on the photon leaving us with
For simplicity we will consider the circular orbit, ur = 0, that is at the equator leaving us with
The coordinate speed will be given by equation 10.4.24
Now we consider the affine connections for general relativity
Inserting into Eqn. 10.4.12 these we have
We have two possible solutions.
One is Eqn.10.1.5
r = 2GM/c2
and the other is Eqn 10.4.10
r = 3GM/c2
The first is the Schwarzschild radius. The second is the correct solution for the photon sphere.
Next lets look at the remote observer frame coordinate gravitational slowing of the speed of light which, unlike special relativity which uses globally rectilinear Lorentz frames, varies remotely. Refer again to Eqn. 10.1.1
For general relativity for a lightlike path ds = 0 and so dividing through this equation by dt2 we have an expression for the radial and angular components of the remote coordinate speed of light.
or more simply
From this we see that the remote coordinate speed of light near a black hole depends on the direction in which it travels. For light moving radialy away from the hole this reduces to
For instantaneously moving perpendicularly to the radial vector to the hole the speed is
Lets now take a closer look at the remote coordinate speed for radial moving light.Eqn.10.4.23
As we will discuss in the section on wormholes there are actually two interior regions of a Schwarzschild spacetime of general relativity. The sign we choose below is chosen to indicate an inward direction of travel for light in the black hole interior region.
This is also our exterior region's solution representing outgoing light.
From this equation we find that the remote coordinate speed of light exceeds c close to the center of the hole. In fact we find that for
The remote coordinate speed of radialy moving light is greater than c.
In order to experimentally observe this Shapiro effect sending signals between far distant space probes we have to derive the equation of motion for light for experimental comparison to the theory for a more arbitrary geodesic trajectory because we can not set them hovering over the sun at constant radial position. So first go back to equations 10.1.6, 10.4.7, and 10.1.10
Inserting the first two into 10.1.10 results in
Light is described by the limit as . So this reduces to
This first order differential equation is separable so the time it takes according to a remote observer for light to go from the high position r=a to the lowest position r=rmin corresponding to the minus sign if we switch the limits to drop the sign is given by
Doing some expansion and keeping only terms to first order in GM/c2
Its a strait forward differentiation to back check that the following is the solution to that integral
Going from 10.4.36 to 10.4.37 we will neglect terms on the order (rmin2/a2) so the first term of 10.4.33 becomes what the flat spacetime prediction would become, so the Shapiro delay for this part of the trip is
If we then find the Shapiro delay for the light traveling up to some higher place r=b on the other side, that would be
giving us a one way trip delay from a to b of
Now if we do the experiment between two planets near superior conjunction, rmin will be small compared to both a and b reducing this to
Now if the signal goes for a round trip during which the planets don't significantly change their relative positions with respect to each other the total Shapiro delay will be
In some big texts like "General Relativity" by Wald, the Shapiro delay is defined in terms of Schwarzschild time as above. However some like "Gravitation" by Misner Thorne and Wheeler defines it in terms of time according to the lab. If you're not going to convert your labs time values from experiments to Schwarzschild time then you have to adjust this result with systematic error correction parameters to account for things such as gravity time dilation of earth near the sun, time dilation moving around the sun, earth's gravity time dilation on the lab, time dilation of lab moving as earth spins or even to account for things like signal lag through atmosphere. With systematic error correction parameters in place the equation in terms of lab time can be written (see problem 10.4.6) :
Notice that since this is proportional to GM/c2, this highest order delay term is completely relativistic. As such Newtonian gravitation not only can not account for it, but since a Newtonian acceleration field would cause everything to speed up going by the sun by ,
Treating the photon as obeying Newtonian gravitation particle dynamics you would get:
Which comparing with no sun present in approximation results in a round trip difference of
Newtonian gravity predicts the wrong sign completely expecting an early time of arrival for the light instead of a delay. Experiments such as the Earth-Venus time delay measurement that first experimentally confirmed the effect in the early '70s really is the clincher for the validity of general relativity as compared to Newtonian gravitation. The theories make opposite predictions here and it was general relativity that turned out to be right.
Use small number approximation to find the change in frequency a locally 655nm wavelength source undergoes rising 15.5m above the surface of the earth.
Where is the coordinate speed of radial moving light (1/2)c? Where does this occur for tangential moving light?
Show that changing radial coordinate to r' where
r = r'(1 + GM/2r'c2)2
puts the Schwarzschild metric in to the isotropic coordinate expression
In isotropic spherical coordinates a good weak field approximation for a general static matter distribution is
where is the Newtonian gravitational potential for the matter distribution.
a. Show that this reduces to the Schwarzschild metric equation of problem 10.4.3 given and h = 0.
b. Write this metric for pseudo Cartesian coordinates x,y,z.
c. In these pseudo Cartesian Coordinates, calculate the Einstein tensor with your choice of software and then find the cooresponding stress energy tensor keeping only terms to first order in and h(x,y,z) and their derivatives because products will be lower in order of magnitude for a weak field approximation. Show that
and to show what px , py, and pz are in terms of partial derivatives of h.
a. Use the timelike Killing vector cooresponding to the time isometry in the metric of problem 10.4.4 to show that there is a conserved energy parameter of geodesic motion and use it with the metric of problem 10.4.4 to calculate for general geodesic motion in the presence of a static matter distribution.
b. Show that if the Potential is indeed weak that the resulting geodesic motion calculation for indeed matches the calculation for Newtonian gravitation.
Go back to equation 10.4.33 and follow thereafter showing that doing a time correction accounting for the lab being at the earth-sun distance leads to the sign difference on the first term in the brackets between equations 10.4.38 and 10.4.39. If all other corrections are small and are about 1.
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