Physical implications of not-smooth metric derivative matching

[FunkyDwarf]

Hey all,

My question pertains to interior metrics, for example the Schwarzschild interior metric given in post #5 of

https://www.physicsforums.com/showthread.php?t=323684

The radial derivative of the first term, the dt^2 coefficient, matches the radial derivative of the Schwarzschild exterior metric at the boundary, but the same cannot be said of the dr^2 coefficient. Would this not lead to a not-smooth effective potential at this point? What does this mean physically?

Thanks!
-G

 

[Sam Gralla]

Hi FunkyDwarf,

Matching metrics can be tricky business in GR, because you have to be careful to disentangle "coordinate effects" from "real effects". The conditions for a smooth matching were worked out by Israel a long time ago (60's?), but surprisingly aren't covered in very many references. The only treatment I know of is Poisson's fairly recent book "A Relativist's toolkit". (Poisson was Israel's student, by the way.) In any case the basic idea is to formulate matching conditions in a manner intrinsic to the matching hypersurface. The answer is that as long as the parts of the metric *tangential to the surface* agree in value and first derivative (technically, the "induced metric" and "extrinsic curvature" match), then the solution is said to be smooth across the hypersurface. With your metric and coordinate choices, "tangential" means just the $tt,\theta \theta,\phi \phi$ components, so the $rr$ component doesn't have to match (and your solution is perfectly smooth). Note that if there wre a mismatch (say) in $\partial_r g_{tt}$ then this wouldn't be a smooth solution. However, you can still interpret the singularity at the matching surface as being a thin shell of matter (i.e., delta-function in r stress-energy). One way to see that ignoring the $rr$ components makes sense is to compute what this delta-function should be in the usual way (derivative of theta-function = delta function). You'll find that $\partial_r g_{rr}$ cancels out of everything (equivalently that your metric has no delta function stress-energy, despite the discontinuity in the derivative of the non-tangential metric components).

I hope this helps! It's a confusing topic.

 

[FunkyDwarf]

Hey sgralla,

thanks for the reply! I guess that makes sense (at least the hypersurface matching bit i get). I recently realized that what i actually want to ask is what sort of conditions i can generate on the coefficients in the interior metric based on the matching to the exterior schwarzschild metric.

Say i have the metric ds = dt A(r,rs) + B(r,rs) dr +r^2 dOmega where i dropped the ^2 on the line element terms just...because :) and rs is the schwarzschild metric. I want this metric to describe some spherically symmetric interior which is matches to the vacuum solution, can i assume that when rs is the radius of the object R that A(r,R) is zero? Basically i have a collection of terms involving the A's and B's and their derivatives and i'd like to know which ones are likely to dominate (in general, if possible) in the limit that rs->R for instance.

Does that make sense?
-Z

 

[Sam Gralla]

I'm not sure I understand the question. What do you mean by "rs is the Schwarzschild metric", and what is this limit rs->R? One question I can answer is that if you have a metric

$$ds^2 = A(r) dt^2 + B(r) dr^2 + r^2 d \Omega^2$$

and you want to match to Schwarzschild (in Schwarzschild coordinates) at r=R>2M, then you need the metric components to match and the first r derivative of $g_{tt}$ to match. (The angular components already match in derivative, and the radial part doesn't matter). If the derivative of $g_{tt}$ doesn't match, you can still call your metric a solution, but the interpretation will be that it has a "surface stress-energy tensor", i.e., a thin shell of matter at the surface of the star. You'd want to calculate it and see if it satisfies energy conditions.

Note that you could also do the matching in "different coordinates on different sides", but the simplest choice is to use the same coordinates, in which case you demand continuity of all metric components.

 

[FunkyDwarf]

My apologies I mistyped, rs is the schwarzschild radius not metric.

I guess my question is given an interior metric as written in your post, are there any conditions i can place on A(r) and B(r) based on physical restrictions in the limit that R ->2m? Specifically, conditions over ALL r not just at the boundary?

Edit: for instance in the 3 interior metrics i have come across, including the Schwarzschild interior solution, the g_rr component is
$\left(1-\frac{r_s r^2}{R^3} \right)^{-1}$
where R is the radius of the body. Presumably this comes from a constant density requirement? Also in most cases it seems that A(r) when r_s = R is zero (previously mentioned time dilation issue), is this also a physical requirement of all metrics in this (black hole) limit or just a result that comes out? I would presume the former.

Thanks!
-Z

 

[atyy]

If I remember correctly, any stable constant density solution must have the boundary between matter and vacuum at greater than 9/8 of the Schwarzschild radius - try looking up Buchdal's theorem.