Confused on the types of solutions to Einstein field equations in General Relativity

Question

Context

While reading about the types of solutions to The Einstein Field Equations in General Relativity, I came across the following article.

Where they explain that Karl Schwarzschild provided the first Exact solution to the Einstein field equations in General Relativity.

I have 2 questions. They are both very closely related so I thought it would be appropriate to include both in this 1 post.

2 questions

1. What is the difference between a non-exact and a exact solution to the Einstein Field Equations in General Relativity?

2. Are the Schwarzschild metric and Schwarzschild solution the same thing?

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What I think currently of Q1

When a solution is "exact" it is not "simplified" or "compacted" - that is; it perhaps can be "applied" right away. (If that makes sense, if at all)

What I think currently of Q2

To me they seem like it's the same thing, the metrics and solutions phrasings that is. (in this context I should add).

Resources I have tried

• Searched for both questions on this stack*site, Wikipedia, Google, and some other sites. However I didn't find any question that I think is answering this question.

While I don't know if the following is helpful or not, I tried to provide another example of a solution just for the context.

• Kurt Gödel
• Closed time-like curve

adding this too as it's about closed timelike curves as well, as well* as to not only use Wikipedia

• Does negative energy density (i.e. weak energy condition violation) create closed timelike curves?

Note I am aware that many Wikipedia links is being used.

and please if any of this is nonsense/wrong, point it out directly!

My background in Physics currently is not so top notch; as also described in my previous question

I do not yet study physics in any serious way. That is, not university yet. I do read books like

• spacetime physics by Taylor/Wheeler
• Wikipedia articles(that doesn't have warnings, or I read with extreme caution on those articles anyway)
• and some Documentaries.

Answer 1

The Schwarzschild metric is indeed the first (non-trivial) exact solution to the Einstein field equation (the trivial one being Minkowski space).

The term of "exact solution" used here is to contrast it with Einstein's previous attempts at finding relativistic gravitational solutions, such as his approximated solution of a gravitational field for the analysis of the perihelion of Mercury, where he simply takes the generic form of a spherically symmetric metric and simply takes a first order approximation of it. You'll notice that nothing in this paper makes use of the Einstein field equations (this was slightly before those equations were finalized), this is simply an Ansatz that seem to give the appropriate results in the classical limit.

By contrast, the Schwarzschild solution is the exact solution of a vacuum spherically symmetric metric for the Einstein field equations.

For the second part of your question, there is indeed generally some ambiguity of use between "solution", "metric" and "spacetime". I don't think there is really much of a consensus as to what those terms mean, but here are some things to keep in mind :

First, in general relativity, every metric tensor $g$ is also a solution to the Einstein field equation, as it is a differential equation in the metric :

\begin{equation} R_{\mu\nu}[g] - \frac{1}{2} g_{\mu\nu}R[g] = T[g] \end{equation}

For any metric $g$, there exists a $T$ such that $g$ is a solution of the Einstein field equation. If you have a specific form of $T$ in mind, that is a constraint on the set of solutions. In the case of the Schwarzschild metric, it is a solution to the vacuum equation, $T = 0$.

Furthermore, generally physicists tend to keep the distinction between the metric tensor $g$ (which is independent of any coordinate system) and the components of the metric tensor $g_{\mu\nu}$ in a given coordinate system $x$, such that $g = g_{\mu\nu} dx^\mu dx^\nu$, a bit ambiguous. The metric tensor itself does not change, but what people typically call the Schwarzschild metric is the one given in the original Schwarzschild coordinates,

\begin{equation} ds^2 = - (1 - \frac{r_s}{r}) dt^2 + (1 - \frac{r_s}{r})^{-1} dr^2 + d\Omega^2 \end{equation}

But there are many more coordinates you could express this metric in. I don't think there is much standard terminology as to what people refer to when they use such words, so this is something to watch out for.

Answer 2

Exact solution means exact mathematical expression that solves the equation. For example exact solution to the equation $$x^2+1=10000,$$ would be $x=\sqrt{9999}$.

You can also have numerical solution, which is solution produced by a computer. In our example this would be $x = 99.99499987499375$ that was produced by my windows calculator.

There is also an approximate solution that you can get by noting that $1 \ll 1000 $, neglecting 1 and solving simplified equation $x^2=10000$ exactly with $x\approx 100.$

Of course Einstein field equations (EFE) are differential equations, so the solution is bunch of functions (metric to be more precise) instead of one number. But the principle is the same.

Are the Schwarzschild metric and Schwarzschild solution the same thing?

The solution to EFE is a metric, so they should be the same. But in the quote

Schwarzschild provided the first exact solution to the Einstein field equations of general relativity, (...) The Schwarzschild solution, which makes use of Schwarzschild coordinates and the Schwarzschild metric, leads to a derivation of the Schwarzschild radius, which is the size of the event horizon of a non-rotating black hole.

it looks to me like they call Schwarzschild solution any solution for spherically symmetric vacuum spacetime, while by Schwarzschild metric they might mean metric written in Schwarzschild coordinates. Hard to say... I don't think the terminology in the quote is rigidly defined - people just use these words as they see fit, hoping the meaning should be clear from the context.