Advice on solving the Einstein Field Equations for the wormhole vacuum solutions

I'm doing a project for a class and we need to decide on a topic. I read something interesting that it's possible to solve the EFE for a vacuum solution and the result implies that wormholes exist.

My question is basically just asking for an overview of what such a project would consist of. I interpret this journey as "solve the differential equations under some assumption/constraints -> ??? -> wormholes are possible". Again, keep in mind that my understanding is pretty basic (I'm only an undergrad).

What I know so far: I know that the EFE need to have 4 assumptions, given here. One of my classmates also suggested that it might be a guess-and-check sort of endeavor, but I'm not worried because I think this solution already exists and the derivation is out there somewhere (although I have not yet found a derivation that also mentions wormholes - that Wikipedia link above does not have the word "wormhole" anywhere on the page via Ctrl+F).

• The link you have is to solving the Schwarzchild black hole problem, which is a warm up session as it incorporates a vacuum , and a very simplified meteric, and reduces the r.h.s of the EFE to 0. That's why the word wormhole does not appear, although vacuum does. Look at this page, which might help en.wikipedia.org/wiki/Wormhole – user108787 Nov 21 '16 at 22:54
• My point is that to learn about GR, I would suggest that you follow your first link, and see the simplications involved in the static, spherically symmetrical Schwarzchild solution, ( which is a toy black hole, as it does not rotate) and then go from there to the wormhole connected with that solution. – user108787 Nov 21 '16 at 23:02
• @whatwhatwhat See pg.398 in cmp.caltech.edu/refael/league/thorne-morris.pdf – udrv Nov 21 '16 at 23:39
• @CountTo10 is my approach correct? I think the order of tackling this problem is, in general, to show a solution to the einstein field equations for a vacuum, then show how this relates to wormholes. Am I on the right track? – whatwhatwhat Dec 2 '16 at 0:08
• It's not just solving Einstein's equation, and blammo! wormholes!. There are several layers of mathematical subetly to work through, and I don't think that doing this in any sort of quantitative way is recommeded for an undergraduate free-study project. – Jerry Schirmer Jan 24 '18 at 4:54

A quick answer, sorry if you know this, but I am just writing to remember for myself. Bear with me, and please check all this in any GR textbook.

1. Angular momentum is preserved, so when a star shrinks (after the core stops producing radiation), it's a.m. will still be present, so my point is that a non rotating Schwarzchild black hole would be very rare. But for teaching purposes, it's handy to make all the assumptions you can and be left with as few variables as possible.

2. You then need the place holding functions A, B and C (which can be reduced to just A and B). This will be of the form $$ds^2 = Adtdt - Bdrdr$$ ignoring the other angular terms, which are irrelevant as you have chosen them to be so.

3. You are left with the standard Schwarzchild metric anstanz, after assuming no angular dependence, no cross terms say $$drd\theta$$, and invariance under time reversal, and using the vacuum solution ($$T_{\mu\upsilon}=0$$). Also drop the cosmological constant.

4. Then you need to connect this place holding metric with $$R_{\mu\upsilon}$$, which you do by writing A and B as exponential terms, so they are always positive, and then, depending on your book, you either use forms or the $$\Gamma$$ symbols to get the variables that make up A and B.

5. Then, as you only have two variables to deal with, you can draw an embedding diagram illustrating the cross section of the wormhole.

Source: Wormhole embedding diagram

How you go from one step to another depends on the author you are following, but all this hoopla is just to show you the (no offence), baby steps in deriving a metric from the EFE and then using it to show how the curved wormhole is produced. Without choosing your physical situation to allow you to suppress variables, it would be very and (totally unnecessarily) difficult to get to the wormhole drawing stage.

What is the purpose of the holding function

We don't have the metric at the start, what we do have is this horror below, the Riemann tensor elements (all of them based on the metric):

$${\displaystyle R^{\rho }{}_{\sigma \mu \nu }=\partial _{\mu }\Gamma ^{\rho }{}_{\nu \sigma }-\partial _{\nu }\Gamma ^{\rho }{}_{\mu \sigma }+\Gamma ^{\rho }{}_{\mu \lambda }\Gamma ^{\lambda }{}_{\nu \sigma }-\Gamma ^{\rho }{}_{\nu \lambda }\Gamma ^{\lambda }{}_{\mu \sigma }}$$

The R. tensor is a set of second order, coupled, nonlinear partial differential equations, (as I am pretty sure you know already, sorry).

The holding functions are no value in themselves, they are literally just letters, they help remind us of the real functions we need to find (and we can manipulate them to reduce their number). In order to discover the metric and then use that to draw the wormhole, they are handy place holders, that's all. In flat space we don't need them, they are equal to one, but they are one of the tricks you need to use to find the curved space metric, as the functions are not equal to one in places where there is a gravitational mass.

• What is the purpose of the holding functions? – whatwhatwhat Dec 2 '16 at 1:28
• I'm only an undergrad physicist, so I'm new to all of this :) the GR, I mean. – whatwhatwhat Dec 2 '16 at 2:24
• I completely self study, so actually trying to explain it to someone else is good for me. I might have recommended this book before, but Relativity Demystified by David McMahon and it's the book for you, it's cheap on Kindle and he does more examples,in fact that's all he does, teach through examples and leaves the heavy stuff to Hartle, Carroll, Wald etc. Best of luck with it. – user108787 Dec 2 '16 at 2:33
• Thank you for your help! I am going through Thorne's derivation now so I wanted to get through that first and see how it compares with your answer. But regardless, you have been very helpful! Thanks!!! – whatwhatwhat Dec 2 '16 at 4:51
• My final comment, John knows GR at an expert level, I have absolutely no problem with you asking him to look over my answer, if you feel it will save you time. If fact, I highly recommend it. – user108787 Dec 2 '16 at 7:11