# Help in derivation of Birkhoff's theorem

• user1139
In summary: Delta^{2}(\mathrm{d}\theta,\mathrm{d}\phi)=-A(r,t)\left(\frac{\partial f}{\partial r}\right)^{-2}\left(\frac{\partial f}{\partial t}\right)^{2}-A(r,t)\left(\frac{\partial f}{\partial r}\right)^2\left(\frac{\partial f}{\partial t}\right)^{-2}\mathrm{d
user1139
Homework Statement
The most general spherically symmetric line element is of the form ##\mathrm{d}s^2=-A(r,t)\mathrm{d}t^2+2C(r,t)\mathrm{d}t\mathrm{d}r+B(r,t)\mathrm{d}r^2+r^2(\mathrm{d}\theta^2+\sin^2{\theta}\mathrm{d}\phi^2)##.
Show that with a transformation ##t\rightarrow t+f(r,t)## for some ##f(r,t)##, the line element can be written as ##\mathrm{d}s^2=-A(r,t)\mathrm{d}t^2+B(r,t)\mathrm{d}r^2+r^2(\mathrm{d}\theta^2+\sin^2{\theta}\mathrm{d}\phi^2)##.
Relevant Equations
Using the transformation for ##t##, I obtained

$$\mathrm{d}t'=\left(1+\frac{\partial f}{\partial t}\right)\mathrm{d}t+\frac{\partial f}{\partial r}\mathrm{d}r$$.

Using this equation, I substituted it into the general line element to obtain

\begin{align*}
\mathrm{d}s^2=&-A(r,t)\left(1+\frac{\partial f}{\partial t}\right)^{-2}\mathrm{d}t'^{2}-A(r,t)\left(\frac{\partial f}{\partial r}\right)^2\left(1+\frac{\partial f}{\partial t}\right)^{-2}\mathrm{d}r^2+2A(r,t)\left(1+\frac{\partial f}{\partial t}\right)^{-2}\left(\frac{\partial f}{\partial r}\right)\mathrm{d}t'\mathrm{d}r\nonumber\\
&+2C(r,t)\left(1+\frac{\partial f}{\partial t}\right)^{-1}\mathrm{d}t'\mathrm{d}r-2C(r,t)\left(\frac{\partial f}{\partial r}\right)\left(1+\frac{\partial f}{\partial t}\right)^{-1}\mathrm{d}r^2+B(r,t)\mathrm{d}r^2+r^2(\mathrm{d}\theta^2+\sin^2{\theta}\mathrm{d}\phi^2).
\end{align*}

Is the equation that is immediately above correct? How do I proceed to show that the line element can be written as ##\mathrm{d}s^2=-A(r,t)\mathrm{d}t^2+B(r,t)\mathrm{d}r^2+r^2(\mathrm{d}\theta^2+\sin^2{\theta}\mathrm{d}\phi^2)##?

Last edited:

## 1. What is Birkhoff's theorem?

Birkhoff's theorem is a mathematical theorem in general relativity that states that any spherically symmetric vacuum solution of Einstein's field equations must be static and asymptotically flat.

## 2. How is Birkhoff's theorem derived?

Birkhoff's theorem is derived using mathematical techniques from differential geometry and tensor calculus, specifically the use of the Einstein field equations and the Schwarzschild solution.

## 3. What are the implications of Birkhoff's theorem?

Birkhoff's theorem has important implications for the understanding of black holes and the behavior of matter in their vicinity. It also provides a powerful tool for solving problems in general relativity.

## 4. Are there any exceptions to Birkhoff's theorem?

Yes, Birkhoff's theorem only applies to vacuum solutions, meaning that there is no matter or energy present. In the presence of matter or energy, the theorem does not hold true.

## 5. How does Birkhoff's theorem contribute to our understanding of the universe?

Birkhoff's theorem is an important result in general relativity that helps us understand the behavior of space and time in the presence of massive objects. It also plays a crucial role in the study of black holes and their properties.

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