38E (General Relativity)

\frac{\partial x ˉ ^{\overset{α}{ˉ}}}{\partial x ^{μ}} = γ - γ v 00 - γ v γ 00 00100001 .

It is a boost along the

x

-axis from the frame in which the black hole moves to its rest frame. Direct matrix multiplication shows that replacing

v

- v

gives its inverse.

b

Put

B^{2} = \frac{( 1 - A ) ^{2}}{( 1 + A ) ^{2}}, C = (1 + A)^{4} .

The rest-frame metric can be written as

C d \overset{s}{ˉ}_{Minkowski}^{2} + (C - B^{2}) d \overset{ˉ}{t}^{2}

. The Minkowski term is invariant under the Lorentz transformation, while

d \overset{ˉ}{t} = γ (d t - v d x)

. Therefore

f (A) = (1 + A)^{4}, g (A) = (1 + A)^{4} - \frac{( 1 - A ) ^{2}}{( 1 + A ) ^{2}},

and

A = \frac{m}{2 γ ^{2} ( x - v t ) ^{2} + y ^{2} + z ^{2}} .

These substitutions give precisely

d s^{2} = f (A) (- d t^{2} + d x^{2} + d y^{2} + d z^{2}) + γ^{2} g (A) (d t - v d x)^{2} .

Hold

p = mγ

fixed. At a point with

x \neq = t

A = \frac{p}{2 γ γ ^{2} ( x - v t ) ^{2} + y ^{2} + z ^{2}} ⟶ 0,

but

γ^{2} A ⟶ \frac{p}{2∣ x - t ∣} .

The small-

A

expansions are

f (A) = 1 + 4 A + O (A^{2}), g (A) = 8 A + O (A^{2}) .

Consequently, with

H = 4 p /∣ x - t ∣

d s^{2} = - d t^{2} + d x^{2} + d y^{2} + d z^{2} + H (d t - d x)^{2}, x \neq = t .

Equivalently,

g_{tt} = - 1 + H, g_{t x} = g_{x t} = - H, g_{xx} = 1 + H, g_{yy} = g_{zz} = 1,

with all other components zero.

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