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Maybe my question seems simple, I am reading about general gravity but the first obstacle faced me is the definition of the distance in a four dimension space. Why does the infinitesimal distance in space-time contain a minus sign before the dt squared?

Please someone help me to jump this problem.

Qmechanic
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Malek_Physics
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    Maybe you should start reading about special relativity first (in particular, the notion of invariant interval). – Yvan Velenik Jul 06 '20 at 10:40
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    Related, possible duplicate: https://physics.stackexchange.com/q/304799/ – PM 2Ring Jul 06 '20 at 10:54
  • Here’s a physically motivated development of the minus sign in the metric signature using operational radar measurements and the Bondi’s k-calculus: https://physics.stackexchange.com/a/508251/148184 – robphy Jul 06 '20 at 13:27

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Spacetime unifies space and time. So a spacetime interval is a “distance” in space and time. However, we need to distinguish intervals measured with clocks and intervals measured with rulers. The minus sign allows that. If the interval squared is negative then it is measured with a clock, and if it is positive then it is measured with a ruler.

Dale
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  • Subsequently, the minus sign has a physical argumentation and not a mathematical one. Is this true? – Malek_Physics Jul 06 '20 at 11:03
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    @Malek_Physics I don’t think I would go so far as to say it does not have a mathematical explanation. But it certainly does have a physical one. – Dale Jul 06 '20 at 11:19
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It is not just a matter of definition. It is an incredible physically observable fact about spacetime which is is not so easy to observe directly but was predicted by Einstein Minkowski etc based on somewhat indirect evidence but nowadays we have lots of more direct evidence such as lifetimes of fast moving particles. So basically experiments tell us that is how space and time are, with a strange connection. It couldn’t be a plus sign as we would not then notice any difference between space or time as expanded in other comments.

blanci
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