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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"?

Qmechanic
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Jonathan Gafar
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1 Answers1

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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...

IamZack
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