Positively and Negatively Curved Space-time

[Silverious]

I was reading the lectures compiled in "Six not-so-easy pieces" by the great Feynman. He was trying to explain the curvature of space(and subsequently space-time) through varying temperatures on a hot plate. Much like the curve of a sphere. Anyways I'm going to assume most of your are familiar with this, since Feynman is pretty famous.

So he was saying, that if you reverse the temperatures on the hotplate(ie, cold in center, hot on out side, or vice versa) you would come up with negative curvature.

So, the way I understood it, mass curves space-time "positively". So if you were to gather negative mass(tachyons?) in a location, you could produce perhaps, anti-gravity, and maybe travel backwards in time due to the "negative" curvature of space.

I guess I need to say, that everytime I mention "anti-" or "negative" or "backwards" I mean relative to our ordinary FOR.

I really don't know much of what I'm talking about, but it's nice to discuss with people who DO know.

So how little do I understand this?

 

[Stingray]

Feynman was simplifying a lot for presentation. There is a curvature (single number) defined in geometry. This is sufficient to deal with things in one or two dimensions, but you need a lot more numbers to fully describe the real four dimensional world. There are actually 20 of them - most of which can be positive or negative. If you ignore all but the single curvature Feynman was talking about (not as ridiculous as it sounds actually), then it is always positive. A cosmological constant could change that though.

 

[turin]

I would think that a stress-energy tensor with a negative trace (i.e. negative mass) could also accomplish a sign reversal of the curvature scalar (i.e. curvature). According to Einstein's equation without the cosmological term:

R^μ_ν - (1/2)δ^μ_νR = κT^μ_ν

=>

R = -κT

If I remember correctly, T is "normally" positive, and therefore R is "normally" negative. This corresponds to (positive) mass inducing a negative value for R. If T could somehow be negative, then R would be positive.

 

[Stingray]

I would think that a stress-energy tensor with a negative trace (i.e. negative mass) could also accomplish a sign reversal of the curvature scalar (i.e. curvature). According to Einstein's equation without the cosmological term:

R^?_? - (1/2)?^?_?R = ?T^?_?

=>

R = -?T

If I remember correctly, T is "normally" positive, and therefore R is "normally" negative. This corresponds to (positive) mass inducing a negative value for R. If T could somehow be negative, then R would be positive.

T is usually negative actually. Think of a perfect fluid for example. Then T=-rho+3p<0. The argument can be made more general.

 

[turin]

T is usually negative actually. Think of a perfect fluid for example. Then T=-rho+3p<0. The argument can be made more general.

Isn't this a matter of convention? I have seen it both ways, but in most of my reading, I have seen the +--- signature.

 

[Stingray]

You're right. It can be either way. I used -+++, which is more common.

 

[turin]

Now I'm curious. Are you basing this primarily on recent articles, texts, converse with other physicists, or what? I will explicate my deference by suggesting that you probably know a bit more about GR than do I, but I have been reading about it for over a year now, and I can only think of two or three occassions that brought me to a source using a -+++ signature, whereas the rest used a +---. This is based primarily on text reading, some journal reading, and converse with my (former) major professor. I admit that this is a minor detail, but it may be an indication that I am not reading the most appropriate materials; that's why I'm asking.

 

[Stingray]

I'm basing it on all of those things. I actually can't think of any relativity text that uses +--- (there are some QFT ones that do). Wald, MTW, Hawking and Ellis, Synge, and Weinberg all use -+++. Which books have you been using?

As for papers, I think there are very few written in the past 30-40 years which use +---.

 

[turin]

I will have to look through my notes, but I probably didn't even write down the signature that was used. I happen to have one source from the library that I haven't gotten to yet (Synge, Relativity: The General Theory). I just opened it up and sure enough as you contend, it explicitly uses -+++ (I always put the time coordinate first, even though, in this case, Synge puts it last as +++-). I actually did make a note that Weinberg uses the -+++ signature; I don't know why I did that. I read some of MTW, but I don't remember anything about the signature. I think I read Ellis and Wald, but I don't remember. I know I didn't read any Hawking.

If I remember correctly, W. Rindler and Einstein (who also puts the time coordinate last) use the +---. I am absolutely positive that H. Ohanian uses the +---. My major professor uses the +---. I'm pretty sure I remember that Chandrasekhar uses the +---. I think J. Anderson uses +---. Does H. Weyl use +---, I can't remember? Doesn't Minkowski himself use +---? R.C. Tolman I believe uses +---. M. Mizushima uses +---. S. Donev uses +---. A.J. Accioly uses +--- (as well as &radic;(-g) which for some reason seems inconsistent to me). D.H. Kobe uses +---.

I think I remember G. Birkhoff using the -+++ signature, but I don't think he counts as his notation was so different than what I am familiar with on so many other issues (like using commas to denote the type of index and primes to denote parametric differentiation, which is almost the opposite of what I'm used to and certainly confusing). I just found that J.D. Romano uses -+++. K.C. Valanis uses -+++. F. Rohrlich uses -+++. From what I can tell M. Nowakowski uses -+++. L. Bel uses -+++.

It seems that many authors even use the 1,2,3,4 indices instead of the 0,1,2,3 indices, so that the signatures are actually +++- instead of -+++ (if that is any more or less significant). Sometimes, I even see definitions like: g_&mu;&nu; = &delta;_&mu;&nu;, which I must confess is a bit confusing to me.

Anyway, from this brief re-evaluation the unofficial (and probably quite unrepresentative) tally comes to:
+--- (12 users)
-+++ (10 users)

Well, it does seem that it was more of my imagination than anything else, probably fueled by the bias initiated by my major professor. I subconsciously convert the signature to +--- in my head even if it is -+++ (or any other variation). I will try to assume less about such issues in the future. This little tangent of ours has opened my eyes a bit.