I am working through "General Relativity" by Wald, and am currently going through the brief section on Special Relativity. The spacetime metric is defined as $\eta_{ab} = \sum\limits_{\mu, \nu=0}^3 \eta_{\mu,\nu} (dx^\mu)_a(dx^\nu)_b$ where $\eta_{\mu, \nu} = \mathrm{diag}(-1,1,1,1)$. My question is, what is the physical motivation for this? The last three terms in the summation I understand - they give the physical distance between two events. But what about the negative sign infront of the "time term"?

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    $\begingroup$ Nothing stops you from making the temporal coefficient positive and the spatial coefficients negative. I think MTW has a table enumerating which textbooks that came before it use which notation. If you don't understand why there have to be two different signs, then you may want to go back one step and study special relativity some more. $\endgroup$
    – CuriousOne
    May 20 '16 at 2:41

The metric is usually defined this way because of the speed limit (c).

Imagine that: If you are centered at the origin of a coordinate frame (x,y,z) a emit a signal, this signal would never travel a distance bigger than the distance the light would travel. So,

$$ {|r|} ^{2} = {x}^{2} + {y}^{2} + {z}^{2}$$

And set it equal to the distance the light would travel (c²t²):

$$ {x}^{2} + {y}^{2} + {z}^{2} = {c}^{2}{t}^{2}$$

Now you can see that:

$${x}^{2} + {y}^{2} + {z}^{2} - {c}^{2}{t}^{2} = 0 $$

Work theese expressions for infinetesimal intervals to see it straight. ${ds}^{2}$ must have this signature...

  • $\begingroup$ Not trying to be anal, but relativity doesn't say that the speed of light is a speed "limit", it says that the speed of light is constant in all inertial systems (which I believe is what you are trying to say here). $\endgroup$
    – CuriousOne
    May 20 '16 at 3:36
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    $\begingroup$ @CuriousOne you are right, I'm trying to say that C is constant. But in S.R. c is the speed limit which information can travel through space. Only massless particles could travel at c. $\endgroup$
    – IamZack
    May 20 '16 at 3:46
  • $\begingroup$ There is a not so subtle difference between the two statements. You could have a speed limit in the universe (see e.g. optical phonon dispersion) without the speed of light being exactly the same in all inertial systems. The latter is a much stronger statement and, at the same time, it actually defeats the speed limit. The eigentime difference between any two points in the universe for the connecting traveler can be made as small as you wish, i.e. there is no real speed limit. What is limited is the possibility to visit another space and then return into one's original timeline. $\endgroup$
    – CuriousOne
    May 20 '16 at 4:50
  • $\begingroup$ @CuriousOne , I got it. But do you agree that c is constant and we can't have information traveling through space-time at a speed greater than c, ok? $\endgroup$
    – IamZack
    May 20 '16 at 13:19
  • $\begingroup$ @CuriousOne I understood what you tried to say, it's just my wrong way of explaining things. Your correction is welcome. $\endgroup$
    – IamZack
    May 20 '16 at 13:26

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