Why is there a Minus in Spacetime Interval Formula?

[nitsuj]

The kind of causality you are thinking of always holds locally in GR but globally you can have all kinds of weird things happen with regards to causality; DimReg gave the example of closed time-like curves.

definitely weird stuff to me. I lean heavily on the precedence of SR to GR, so suppose I was speaking only locally .

 

[DimReg]

@ WannabeNewton: SOME of the uchicago lecturers are amazing, others are actually quite terrible. Geroch and Wald are both quite good.

@nitsuj: Don't worry, not all of us can be experts in everything! Actually, if everyone were experts than this conversation would have been useless.

 

[nitsuj]

@nitsuj: Don't worry, not all of us can be experts in everything! Actually, if everyone were experts than this conversation would have been useless.

Am unsure of the emphasis put on being an "expert", but I reiterate the topic "Why is there a negative sign in the spacetime interval" and the numerous ways an "explanation" can be presented. I choose the length time relationship.

No worries on not being an SR expert, I refer to myself as a Fan of SR. In fact I am so far from being an expert, someone who actually studies SR or performs calculations would have a more "expert" understanding of SR then me...that's for sure!

I did miss this though;

These curves essentially are the signature of time travel: you can meet up with an earlier version of yourself.

That's well into the realm of "physically possible according to theory", or leaning heavily on the theory. In what sense could I meet up with an earlier version of myself? Everything I know about SR tells me that isn't even remotely possible. Are there "real world" examples of this, or is it strictly hypothetical?

The closest I can envision is traveling to the same instant, but not to a "previous" "version" or "earlier time" of myself, there is of course, only one of me; no matter which metric you choose/

EDIT: Reading about the Gödel metric the wiki seems to say it is possible to "See" an earlier version of yourself. I see that as a world of difference...womp womp womp...I can "see" old light that I shouldn't be able to "see". That said I still don't know about CTCs or what they are.

Einstein said this in retort to the metric "It will be interesting to weigh whether these are not to be excluded on physical grounds."

 

[robphy]

Although it may not be the intention of the OP, questions like the OP's question
(in my opinion) sometimes places the burden on Relativity to explain its peculiarities.

One way to respond to such questions is show that similar peculiarities occur elsewhere...although often not noticed. I think the following example is compatible with some of the early comments in this thread...

Inspired by the SR-formulation of spacetime, one can go back and consider the situation for the Galilean case (i.e. the ordinary position vs. time graph in PHY 101). Depending on conventions and definitions, the analogous question is
"Why in the Galilean spacetime-interval [in my conventions... it gives the square of the elapsed proper time in a Galilean spacetime]
$ds^2=dt^2 -\left(\frac{c_{signal}}{c_{light}}\right)^2 dy^2 = dt^2 - 0 dy^2=dt^2.$
is there a "zero" instead of a "plus 1"?" (c_{signal}=\infty in the Galilean case.)

So, one answer is that the position-vs-time-graph (whether in SR or ordinary PHY 101) does have not a Euclidean geometry [where the Pythagorean theorem holds]. For PHY 101 (as well as SR), this means that two line-segments on a pos-vs-time graph (say one for an object at rest and one moving with a nonzero velocity) with the same Euclidean length do not correspond to the same elapsed time.

So this can be start of a line of reasoning in which it is realized that not every geometry that arises in physics is necessary Euclidean... e.g. spherical geometry, phase space, PV-diagrams, etc... One can then go on and try to give some physical or mathematical intuition [likely based on what concepts are viewed as fundamental] as to why it is so... as others here have done.


The answer I gave above based on the Galilean spacetime is an extension of Minkowski's 1907 formulation, which is based on the Cayley-Klein projective geometries.

To me, causality (in the sense that not all events can be totally-ordered by causal relations) is at the root of the answer. (Along these lines, one can start at AA Robb's 1914 formulation http://archive.org/details/theoryoftimespac00robbrich... then somehow make the appropriate assumptions [e.g. continuity, homogeneity, etc...] to arrive at the Minkowski metric (for example.. something like http://www.mcps.umn.edu/assets/pdf/8.7_Winnie.pdf ). From this point of view, it is research problem to start with causality at a microscopic level to recover [i.e. explain] the geometrical structures of continuum general relativity.

 

[Vanadium 50]

The OP has not posted a second time in this thread, everyone.

 

[DimReg]

That's well into the realm of "physically possible according to theory", or leaning heavily on the theory. In what sense could I meet up with an earlier version of myself? Everything I know about SR tells me that isn't even remotely possible. Are there "real world" examples of this, or is it strictly hypothetical?

EDIT: Reading about the Gödel metric the wiki seems to say it is possible to "See" an earlier version of yourself. I see that as a world of difference...womp womp womp...I can "see" old light that I shouldn't be able to "see". That said I still don't know about CTCs or what they are.

Einstein said this in retort to the metric "It will be interesting to weigh whether these are not to be excluded on physical grounds."

A CTC is a physically accessible path (for massive particles) that forms a complete loop. So essentially you end up at the same point of time and space, after some finite proper time. I don't really want to get into the details of causal structure, it's a huge topic and my ability to replicate it from memory is not exceptional. However, CTC's are typically considered unphysical, and there are some reasons to believe there are no CTC's outside of just "they are confusing" (in order for there to be an initial value formulation, space-time has to be globally hyperbolic, which has no CTC's).

My point with CTC's was just to note that the relative minus sign in the minkowski metric is not by itself enough to ensure causality. That said, there aren't any CTC's in minkowski space, but it's just one of many examples of a metric without CTC's. So while the minkowski metric ensures causality, causality does not ensure the minkowski metric.

 

[nitsuj]

My point with CTC's was just to note that the relative minus sign in the minkowski metric is not by itself enough to ensure causality. That said, there aren't any CTC's in minkowski space, but it's just one of many examples of a metric without CTC's. So while the minkowski metric ensures causality, causality does not ensure the minkowski metric.

Ah, thank you, your point went over my head. It'll take some time to "see" why that is; it doesn't "jump out at me".

Perhaps I feel too strongly that the simple logic of causality has a physical significance & that properly defined measurements do too.