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This section will present a simple derivation for the Schwarzschild solution.

(10.1.1) 

In spherical coordinates, but otherwise flat space-time the invariant interval took the form:

ds² = dct² - dr² - r²dq ² - r²sin²qdf²

Let's examine the slightly more general possibility that a vacuum field solution exists in the form:

ds² = A(r)dct² - A^-1(r)dr² - r²dq ² - r²sin²qdf²

(10.1.2)

Then:

g₀₀ = A, g₁₁ = -A^-1, g₂₂ = -r², g₃₃ = -r²sin²q and all other g_mn = 0.

Recalling Eqn. 4.3.3

g^rng_ns = d^r_s

its obvious that:

g⁰⁰ = A^-1, g¹¹ = -A, g²² = -r ^-2, g³³ = -r ^-2sin ^-2q and all other g_{m n} = 0.

 111

112 Chapter 10 The Schwarzschild Black Hole

From the equation for the affine connection Eqn. 4.4.3

G^l_mn = (1/2)g^ls(g_sm ,_n + g_sn ,_m - g_mn ,_s)

the following can be calculated:

G ⁰₀₀ = G ⁰₁₁ = G ¹₀₁ = G ¹₁₀ = 0.

G ¹₂₂ = -rA

G ²₁₂ = G ²₂₁ = G ³₁₃ = G ³₃₁ = 1/r

G ²₃₃ = -sinq cosq

G ³₂₃ = G ³₃₂ = cotq

G ¹₃₃ = -rAsin²q

Plugging these into the Ricci tensor R_mn, Eqn.6.3.1 from Eqn.6.3.3 outcome in the following:

and all other R_mn = 0.

10.1 The Schwarzschild Solution 113

For the case that space is vacuum T_mn = 0 except for the possibility of mass at the origin, and if we take

l » 0, then according to Eqn. 6.3.21 we have

R_mn = 0.

This is satisfied with the expressions we arrived at for the Ricci tensor components if

Let:

Then:

Separating variables and integration results in:

where r₀ is a constant of integration.

So:

Separating variables and integrating again results in:

where b is another integration constant.

114 Chapter 10 The Schwarzschild Black Hole

So:

(10.1.3)

In the limit as r goes to infinity space-time goes flat and the solution should therefor become that for special relativity, Therefor b = 1.

So:

So ds can be written:

(10.1.4)

In a weak field limit we should have g₀₀ = 1 + 2f /c² where f is the Newtonian gravitational potential. For a point mass at the origin this should be f = - GM/r What we have is g₀₀ = r₀/r. Relating the two results in

(10.1.5)

Therefor the integration constant r₀ which is a parameter describing this spacetime geometry is what we think of as mass.

For a black hole r₀ is called the Schwarzschild radius. The Schwarzschild solution 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.

10.1 The Schwarzschild Solution 115

The coordinate singularity 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. This surface is called an event horizon. Most generally an event horizon is any surface at which the coordinate speed of light vanishes.

Problem 10.1.1

Verify that far from the hole, the Schwarzschild solution obeys Eqn. 9.2.4 to order 1/r². Why needn't all f _ii obey the Poisson equation or the inhomogenous wave equation Eqn. 9.2.3 for a delta function source?

Problem 10.1.2

Other constant values for b in Eqn. 10.1.3 besides 1 also result in vacuum field solutions. Why should they not correspond to "exterior" solutions?

Problem 10.1.3

There are two concentric thin spherical mass shells. One is radius R₁ and mass M₁ and the other is radius R₂ and mass M₂. Use b - r₀/r = 1 + 2f /c² where f is the Newtonian potential, to model the spacetime by Eqn 10.1.3 for each vacuum region. What should b and r₀ be in the region r less than R₁, r between R₁ and R₁, and r greater than R₁? (It turns out this is a bad model for thick shells because g₁₁ takes on an odd behavior. For a fluid sphere g₁₁ goes back to -1 at the origin.)

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116 Chapter 10 The 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.

F'¹_felt = (1 - 2GM/rc²)^-1/2GMm/r²

(10.2.1)

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

A^l = dU^l/dt + G^l_mnU^mUⁿ

The only nonzero U^m is U^ct resulting in

A^l = dU^l/dt + G^l_ctctU^ctU^ct

We then insert U^ct = dct/dt = c(1 - 2GM/Rc²)^-1/2

A^l = dU^l/dt + G^l_ctctc²(1 - 2GM/Rc²)^-1

The only nonzero A^l is A^r

A^r = G^r_ctctc²(1 - 2GM/Rc²)^-1

We then insert the affine connection

A^r = (1/2)g^rr(-g_ctct,r)c²(1 - 2GM/Rc²)^-1

resulting in

A^r = GM/r²

We then insert this into equation Eqn.5.2.2

|F| = m(-g_mnA^mAⁿ)^1/2

resulting in

|F| = m[(1-2GM/rc²)^-1(GM/r²)( GM/r²)]^1/2

Which simplifies to Eqn.10.2.1

|F| = (1-2GM/rc²)^-1/2GMm/r²

This is the amount of force felt by the object hovering over the hole. 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.

Problem 10.2.1

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

Problem 10.2.2

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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118 Chapter 10 The Schwarzschild Black Hole

In this section we will look at the behavior of a test object thrown up or down in a Schwarzschild spacetime from the perspective of a hovering observer. If the hovering observer and the object are far from the hole then the result for the coordinate acceleration will be shown to be

a¹ = -(GM/r²)[1 - (v²/c²)]

(10.3.1)

Note that at high radial speeds where v²/c² can't be neglected, the object acts as if its gravitational mass shifts down from its inertial mass. In fact, with their definitions as made below they both shift up from m which is an invariant, but M_i as defined below shifts up more, making is appear as though gravity has less hold at higher radial speeds.

To start the process we find the Schwarzschild interval in the hovering observer's coordinates. There are several coordinate transformations one might choose from to represent a transformation to a hovering observer's frame. Any will work that reduce the interval to the interval for special relativity at the location of the hovering ship. One might choose the transformation for its own simplicity. However, this increases the complexity of the global form of the invariant interval. Here we choose a transformation to a hovering ship frame which keeps the interval in simple form. The transformation is

t = t'(1 - 2GM/Rc²)^-1/2

ò r' [(1 - 2GM/Rc²)/(1 - 2GM/r'c²)]^1/2dr' = ò r (1 - 2GM/rc²)^-1/2dr

(10.3.2)

The integration constant is chosen so that r = r' at R.

This transformation globally transforms the form of the Schwarzschild solution into

(Note the lack of primes on the first and last r.) (10.3.3)

The metric components in the following expressions are then taken from this. We then look at the expression for coordinate acceleration Eqn. 5.3.8 in the absence of four-forces.

a^l = - G^l_mnu^muⁿ + (u^l/c)G ⁰_mn u^muⁿ

10.3 "Apparently" Lighter With Speed 119

The only nonzero u^l are

u⁰ = c

u¹ = v

And we are only interested in a¹ so it becomes

a¹ = - G¹₁₁v² - 2G¹₀₁cv - G ¹₀₀c² + (v/c)(G⁰₁₁v² + 2G⁰₀₁cv + G⁰₀₀c²)

Of these the only nonzero affine connections are

G¹₁₁ = (1/2) g¹¹(g₁₁,₁)

G¹₀₀ = - (1/2)g¹¹(g₀₀,₁)

G⁰₀₁ = (1/2) g⁰⁰(g₀₀,₁)

These reduce it to

a¹ = - (1/2)g¹¹(g₁₁,₁)v² + (1/2) g¹¹(g₀₀,₁)c² + g⁰⁰(g₀₀,₁)v²

Simplified

a¹ = (1/2){(1/g₁₁)(g₀₀,₁)c² + v²[2(1/g₀₀)g₀₀,₁ - (1/g¹¹)g₁₁,₁]}

We work out the metric derivatives to be

g₀₀,₁ = (dg₀₀/dr)(dr/dr') = (2GM/r²c²)(1 - 2GM/rc²)^1/2/[(1 - 2GM/Rc²)(1 - 2GM/r'c²)]^1/2

g₁₁,₁ = - (1 - 2GM/Rc²)(2GM/r'²c²)/(1 - 2GM/r'c²)²

After insertion of these and g₀₀ and g₁₁ and simplification, this results in

a¹ = -(GM/r²){( 1 - 2GM/r'c²)^1/2(1 - 2GM/rc²)^1/2/(1 - 2GM/Rc²)^3/2 - (v²/c²)[2(1 - 2GM/Rc²)^1/2/[(1 - 2GM/rc²)^1/2(1 - 2GM/r'c²)^1/2] - (r/r')²/(1 - 2GM/r'c²)]}

(10.3.4)

This is the exact solution.

Next we might look at approximations.

First we might look at the approximation where the ball is instantaneously at the location of the hovering ship. (r = r' = R) At this instant, it becomes

a¹ = -(GM/R²){1/(1 - 2GM/Rc²)^1/2 - (v²/c²)[2/(1 - 2GM/Rc²)^1/2 - 1/(1 - 2GM/Rc²)]}

(10.3.5)

120 Chapter 10 The Schwarzschild Black Hole

There are two different approximations of interest to make from here.

First we might let the speed of the ball instantaneously be at v = 0 at which moment it becomes:

a¹ = -(GM/R²)/(1 - 2GM/Rc²)^1/2

(10.3.6)

This form is expected as the observer is held still considering Eqn. 10.2.1

|F| = (1 - 2GM/rc²)^-1/2GMm/r²

And we should expect a¹ = |F|/m. For v = 0.

Second, if we go back to Eqn .10.3.5 and make an approximation where both the observer and the ball are far from the hole, then it becomes Eqn. 10.3.1

a¹ = -(GM/r²)[1 - (v²/c²)]

At high radial speeds where v²/c² can't be neglected, the object acts as if its gravitational mass shifts down from its inertial mass.

Merely to demonstrate this oddity, consider the following. Recall from special relativity that when an ordinary force is applied in the direction of motion we have Eqn. 3.2.11

f = g³ma

The Newtonian gravitational mass M_g is given by Eqn. 4.1.1

f = - GMM_g/r²

This results in

- GMM_g/r² = g³ma

Now insert the acceleration from Eqn. 10.3.1 and it becomes

- GMM_g/r² = - g³m(GM/r²)[1 - (v²/c²)]

This results in

M_g = gm

(10.3.7)

If we think of inertial mass as the proportionality between ordinary force and ordinary acceleration (which we don't typically do), then the inertial mass is given by

10.3 "Apparently" Lighter With Speed 121

M_i = g³m

(10.3.8)

At high radial speeds then, gravitational and inertial mass are not equivalent. They are related by

M_g = (1 - v²/c²)M_i

(10.3.9)

Here we see that the object does behave as if its gravitational mass shifts below its inertial mass at high radial speed.

Problem 10.3.1

What would Eqn.10.3.8 become for motion perpendicular to the ordinary force applied. If 10.3.7 were to remain the same, what would the relationship between the new M_i and M_g be?

Problem 10.3.2

According to Eqn. 10.3.1, how fast would something have to move radialy in order to have half the accleration that it would have staying stationary?

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122 Chapter 10 The Schwarzschild Black Hole

For this section, first lets derive the equation for the gravitational red shift for a Schwarzschild space-time.

Starting with the interval for the metric for the Schwarzschild solution 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

dct² = (1 - 2GM/rc²)dct²

or

dt = dt(1 - 2GM/rc²)^-1/2

(10.4.1)

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 we get

T₁(1 - 2GM/r₁c²)^-1/2 = T₂(1 - 2GM/r₂c²)^-1/2

(10.4.2)

Using that frequency is the inverse of the period we arrive at an expression relating the frequencies observed

(f₁)^-1(1 - 2GM/r₁c²)^-1/2 = (f₂)^-1(1 - 2GM/r₂c²)^-1/2

or

f₁(1 - 2GM/r₁c²)^1/2 = f₂(1 - 2GM/r₂c²)^1/2

(10.4.3)

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

f₁(1 - GM/r₁c²) » f₂(1 - GM/r₂c²)

(10.4.4)

This phenomenon is called gravitational red shift.

10.4 Behavior of Light in a Schwarzschild Spacetime 123

Next lets look at the remote observer frame coordinate speed of light. Refer again to Eqn. 10.1.1

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 coordinate speed of light.

0 = (1 - 2GM/rc²)c² - v_r²/(1 - 2GM/rc²) - v_W²

or more simply

v_r² + (1 - 2GM/rc²)v_W² = (1 - 2GM/rc²)²c²

(10.4.5)

From this we see that the 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

v_r = ±(1 - 2GM/rc²)c

(10.4.6)

For instantaneously moving perpendicularly to the radial vector to the hole the speed is

v_W = ±(1 - 2GM/rc²)^1/2c

(10.4.7)

Lets now take a closer look at the coordinate speed for radial moving light.Eqn.10.4.6

v_r = ± (1 - 2GM/rc²)c

As we will discuss in the section on wormholes there are actually two interior regions of a Schwarzschild spacetime. 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.

v_r = (1 - 2GM/rc²)c

124 Chapter 10 The Schwarzschild Black Hole

From this equation we find that the coordinate speed of light exceeds c close to the center of the hole. In fact we find that for

r < GM/c²

(10.4.8)

The coordinate speed of radialy moving light is greater than c.

Next lets look at a case of deflection of light as observed from a hovering ship. 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 that the light is deflected so that it hits below target as long as the ship hovers still by demonstrating that

a^r = (c²/r) - GM/r².

(10.4.9)

We start with the equation for coordinate acceleration Eqn. 5.3.8

a^l = (dt/dt)²[F^l - (u^l/c)F⁰]/m - G^l_mnu^muⁿ + (u^l/c)G⁰_mnu^muⁿ

In this case there is no four-force, as none act on the photons, it is

a^l = - G^l_mnu^muⁿ + (u^l/c)G⁰_mnu^muⁿ

The component we need is

a^r = - G ^r_mnu^muⁿ + (u ^r/c)G⁰_mnu^muⁿ

Then we make an approximation what for the entire length of the beam u^r << c.

a^r = - G ^r_mnu^muⁿ

Inserting the expression for the affine connection we have

a^r = -(1/2)g^rr(g_{m r},_n + g_{n r},_m - g_{m n},_r)u^muⁿ

(no sum on r as its not a variable index)

For Simplicity we'll let the beam be oriented in the f direction at q = p /2. Doing the sums we have

a^r = (1/2)g^rr(g_ff,_r)u^f u^f - (1/2)g^rr(g_rr,_r)u^ru^r + (1/2)g^rr(g_ctct,_r)u^ctu^ct

Using g_rr = -1/(1 - 2GM/rc²) and g_ctct = 1 - 2GM/rc² we have

a^r = (1/2)g^rr(g_ff,_r)u^fu^f - g^rr[(GM/r²c²)/(1 - 2GM/rc²)²]u^ru^r + g^rr(GM/rc²)u^ctu^ct

10.4 Behavior of Light in a Schwarzschild Spacetime 125

Now we make the approximations u^r << c and u^f = c/r for the entire length of the beam and using g_ff = -r²sin²q = -r² and r >> 2GM/c² we have

a^r = -(1/2)(-2r)(c²/r²) - (GM/r²c²)c²

Or more simply Eqn. 10.4.9

a^r = (c²/r) - GM/r²

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.

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.

r_ps = 3GM/c².

(10.4.10)

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

a^l = (dt/dt)²[F^l - (u^l/c)F⁰]/m - G^l_mnu^muⁿ + (u^l/c)G⁰_mnu^muⁿ

No four forces will act on the photon leaving us with

a^l = - G^l_mnu^muⁿ + (u^l/c)G⁰_mnu^muⁿ

For simplicity we will consider the circular orbit, u^r = 0, that is at the equator q = p/2 leaving us with

a^r = 0 = - G^r_ctctu^ctu^ct - 2G^r_ctfu^ctu^f - G^r_ffu^fu^f

(10.4.11)

We have already seen that the coordinate speed will be given by

v_W = (1 - 2GM/rc²)^1/2c = ru^f

0 = - G^r_ctctc² - 2G^r_ctf (1 - 2GM/rc²)^1/2c²/r - G^r_ff(1 - 2GM/rc²)c²/r²

126 Chapter 10 The Schwarzschild Black Hole

0 = - G^r_ctctc² - 2G^r_ctf (1 - 2GM/rc²)^1/2c²/r - G^r_ff(1 - 2GM/rc²)c²/r²

(10.4.12)

Now we consider the affine connections

G^r_ctct = -(1/2)g^rrg_ctct,_r = (1 - 2GM/rc²)GM/c²r²

G^r_ff = -(1/2)g^rrg_ff,_r = -(1 - 2GM/rc²)r

G^r_ctf = 0

(10.4.13)

Inserting into Eqn. 10.4.12 these we have

0 = - c²(1 - 2GM/rc²)GM/c²r² + (1 - 2GM/rc²)²c²/r

Now factor

0 = [- GM/r² + (1 - 2GM/rc²)c²/r](1 - 2GM/rc²)

simplified

0 = (1 - 3GM/rc²)(1 - 2GM/rc²)

(10.4.14)

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.

Problem 10.4.1

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.

Problem 10.4.2

Where is the coordinate speed of radial moving light (1/2)c? Where does this occur for tangential moving light?