Deriving Schwarzschild Solution: Easier Strategies?

[davidge]

Is there a less boring way of deriving the Schwarzschild solution? The derivation itself is easy to going with; what I don't like is computing all the Christoffel symbols and Ricci tensor components --there are so many possible combinations of indices. I know that by using some constraint conditions one don't need to consider some of the Christoffel symbols, but still one end up with many equations.

(I wonder how many time Schwarzschild himself took to derive that solution for the first time...)
BTW, I've read that he was at a hospital at that time; good place to do calculus if you can.

 

[Vanadium 50]

"Less boring" seems awfully subjective.

 

[romsofia]

GR is one of the more computational fields of physics. Unless you have some physical intuition, this is what you'll be spending your time doing (or just use an algebraic package to do it for you...)

 

[PeterDonis]

Is there a less boring way of deriving the Schwarzschild solution?

By "less boring" I assume, from your further statements about having to compute so many possible combinations of indices, that you mean something like "requires less computational drudgery". The best way I know of to do that is to first derive Schwarzschild coordinates, i.e., to demonstrate that any spherically symmetric spacetime can be described by coordinates $(t, r, \theta, \phi)$ in which the metric looks like this:

$$
ds^2 =j(t, r) dt^2 + k(t, r) dr^2 + r^2 \left( d\theta^2 + \sin^2 \theta d\phi^2 \right)
$$

Notice that we have already considerably reduced the possible combinations of indices, since the angular part of the metric is just the standard metric on a 2-sphere with radius $r$, and as far as computing the two unknown functions $j$ and $k$ goes, you don't even have to worry about the angular part at all; in other words, the problem is effectively reduced to only two coordinates instead of four. Also, since there is no $dt dr$ cross term in the metric, the possible index combinations are further reduced. MTW takes this approach in their derivation; I think Carroll's notes also go into it.

Once you have the metric as above, computing its Einstein tensor (you only need to compute the $tt$, $tr$, and $rr$ components, as noted above) and setting each component equal to zero (because you want a vacuum solution, with vanishing stress-energy tensor), enables you to find the two unknown functions, and to show that they turn out to be functions of $r$ only, not $t$ (the latter requires rescaling the $t$ coordinate). I wrote up this part in an Insights article some time ago (you can search on "A Short Proof of Birkhoff's Theorem" in the Insights section), but I think it's worth working through it yourself.

 

[vanhees71]

Well, as Leibniz has put it, it's insulting the human mind to do "boring" purely technical calculations. Nowadays it's easy to get good computer-algebra programs (even for free) that do the job for you. I use Mathematica, but deriving the Christoffel symbols and the Einstein tensor from the usual ansatz (I prefer @PeterDonis 's in #4, because it's then also shown that a spherically symmetric pseudometric implies that it's static) involves only taking derivatives and sums and thus should be possible with any CA, and as I said, there are free ones (e.g., maxima).

 

[martinbn]

If you want to see a different way, although you didn't say which way you found boring, you can look at Geroch lecture notes.

If you want to compute curvature, then it is better to do the computations the Cartan way.

 

[davidge]

vanhees71 good way of avoiding calculus at hand. By the way, I have tried to compute the Ricci tensor on Mathematica (for the Schwarzschild metric) and I got zero all times. Is it a coincidence for this particular metric that that tensor vanishes? It's not obvious at first that it has to be zero for the Einstein's equations to be satisfied, because there seems to be possible to get zero from the combination $R_{\mu \nu} - \frac{1}{2}g_{\mu \nu}R$ other than every component $R_{\mu \nu}$ (and consequently $R$) being equal to zero.

PeterDonis I think this is the most beautiful way of learning something, because as you pointed out, it's doing the calculations that one can discover more and more things that otherwise would be hidden in a software.

If you want to see a different way, although you didn't say which way you found boring, you can look at Geroch lecture notes.

If you want to compute curvature, then it is better to do the computations the Cartan way.

I will try to check out these two. Thanks.

 

[PeterDonis]

there seems to be possible to get zero from the combination $R_{\mu \nu} - \frac{1}{2}g_{\mu \nu}R$ other than every component $R_{\mu \nu}$ (and consequently $R$) being equal to zero

It isn't. The easiest way to see that is to take the trace of the equation.

 

[davidge]

It isn't. The easiest way to see that is to take the trace of the equation.

So that means that whenever $T_{\mu \nu} = 0$, $R_{\mu \nu} = 0 = R$ as well?

 

[PeterDonis]

So that means that whenever $T_{\mu \nu} = 0$, $R_{\mu \nu} = 0 = R$ as well?

Yes; again, you can see this for yourself by taking the trace of the vacuum field equation.

 

[George Jones]

And then use the trace to find another useful form of the Einstein field equation,
$$R_{\mu\nu} = 8\pi \left( T_{\mu \nu} - \frac{1}{2}T^\alpha_\alpha g_{\mu\nu} \right).$$
The equation above shows that $T_{\mu \nu} = 0$ implies that $R_{\mu \nu} = 0$, and the usual form of the field equation shows $R_{\mu \nu} = 0$ implies that $T_{\mu \nu} = 0$. i.e., a vacuum if and only if $R_{\mu \nu} = 0$.

 

[davidge]

Can you show how to obtain the trace? I'm not getting it.

 

[George Jones]

Can you show how to obtain the trace? I'm not getting it.

What do you get when you multiply both sides of
$$R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} = 8\pi T_{\mu\nu}$$
by $g^{\mu \nu}$?

 

[davidge]

What do you get when you multiply both sides of
$$R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} = 8\pi T_{\mu\nu}$$
by $g^{\mu \nu}$?

$\frac{1}{2}R = 8 \pi T$ ??

 

[George Jones]

$\frac{1}{2}R = 8 \pi T$ ??

This is not what I get.

What does $g_{\mu \nu} g^{\mu \nu}$ equal?

 

[davidge]

This is not what I get.

What does $g_{\mu \nu} g^{\mu \nu}$ equal?

For a diagonal metric in $n$ dimensional space, would it be $g_{\mu \nu} \ g^{\mu \nu} = n$?

Maybe then $$R = - 8 \pi T \ (1) \\ R_{\mu \nu} = 8 \pi\bigg(T_{\mu \nu} - \frac{1}{2}T^\alpha{}_\alpha \ g_{\mu \nu}\bigg) \ (2)$$ for then if we operate on $(2)$ with $g^{\mu \nu}$ we get $(1)$ again.

 

[George Jones]

For a diagonal metric in $n$ dimensional space, would it be $g_{\mu \nu} \ g^{\mu \nu} = n$?

For the coordinate components of any (semi)Riemannian metric (not just for coordinate systems in which the components are diagonal),
$$g_{\mu \nu} g^{\alpha \nu} = \delta_\mu^\alpha .$$
Taking $\alpha = \mu$ gives
$$g_{\mu \nu} g^{\mu \nu} = \delta_\mu^\mu = 4$$ (for spacetime.)

Maybe then $$R = - 8 \pi T \ (1) \\ R_{\mu \nu} = 8 \pi\bigg(T_{\mu \nu} - \frac{1}{2}T^\alpha{}_\alpha \ g_{\mu \nu}\bigg) \ (2)$$ for then if we operate on $(2)$ with $g^{\mu \nu}$ we get $(1)$ again.

Yes,
$$\begin{align}
\left( R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} \right) g^{\mu \nu} &= 8\pi T_{\mu\nu} g^{\mu \nu}\\
R_{\mu\nu} g^{\mu \nu} - \frac{1}{2}Rg_{\mu\nu} g^{\mu \nu} &= 8\pi T\\
R - \frac{1}{2}4R &= 8\pi T\\
-R &= 8\pi T.
\end{align}$$

Substituting this for the scalar $R$ in
$$R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} = 8\pi T_{\mu\nu}$$
gives the result.

 

[davidge]

Thanks. The form of the equation $-R = 8 \pi \ T$ allow us to see that when $T_{\mu \nu} = 0$ and consequently $T = 0$, $R = 0$ and consequently $R_{\mu \nu} = 0$.

 

[George Jones]

The form of the equation $-R = 8 \pi \ T$ allow us to see that when $T_{\mu \nu} = 0$ and consequently $T = 0$, $R = 0$ and consequently $R_{\mu \nu} = 0$.

I am having a little difficulty understanding what you mean here, particularly the last part

$R = 0$ and consequently $R_{\mu \nu} = 0$.

If all the components $T_{\mu \nu}$ are zero, then the right side of
$$R_{\mu\nu} = 8\pi \left( T_{\mu \nu} - \frac{1}{2}T^\alpha_\alpha g_{\mu\nu} \right)$$
is zero, and consequently the the left side, $R_{\mu \nu}$, must also be zero.

Similarly, if all the components $R_{\mu \nu}$ are zero, then the left side of
$$R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} = 8\pi T_{\mu\nu}$$
is zero, and consequently the the right side, $T_{\mu \nu}$, must also be zero.

 

[davidge]

That's what I was trying to say.

Now how do one shows that $T - 2 T^{\alpha}{}_{\alpha}$ is the same as $T - 2T$? That is, why are we free to clear out the indices $\alpha$ appearing in $T$?

 

[George Jones]

Now how do one shows that $T - 2 T^{\alpha}{}_{\alpha}$ is the same as $T - 2T$? That is, why are we free to clear out the indices $\alpha$ appearing in $T$?

What is the definition of $T$?

 

[davidge]

What is the definition of $T$?

$g^{\mu \nu}\ T_{\mu \nu}$, which maybe can be seen as $T^{\nu}{}_{\nu}$. In this case that equality makes sense.

 

[George Jones]

$g^{\mu \nu}\ T_{\mu \nu}$, which maybe can be seen as $T^{\nu}{}_{\nu}$.

Yes, these quantities are the same. There are various ways to see this; here are three (not independent):

1) consider $g^{\mu \alpha} T_{\mu \nu} = T^\alpha_\nu$ and let $\alpha = \nu$;

2) the sum over $\mu$ in $g^{\mu \nu}T_{\mu \nu}$ means "raise the $\mu$ index for $T$ and repace it by the other index for $g$, i.e., raise $\mu$ and replace it by $\nu$, i.e. $T^\nu_\nu$;

3) explicitly doing the sum over the repeated $\mu$ in $g^{\mu \nu}T_{\mu \nu}$ (for brevity, pretend that spacetime is 2-dimensional) gives $g^{0 \nu}T_{0 \nu} + g^{1 \nu}T_{1 \nu}$, the repeated $\nu$ in the first term raises $\nu$ and changes it into $0$, the repeated $\nu$ in the second term raises $\nu$ and changes it into $1$, so $T^0_0 + T^1_1$, which, from the summation convention, is $T^\nu_\nu$.

 

[davidge]

Thanks