Timeline for What's the physical content in the invariance of spacetime interval in GR?
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 17, 2022 at 13:57 | history | edited | Dale | CC BY-SA 4.0 |
added 19 characters in body
|
| Apr 17, 2022 at 13:55 | comment | added | Dale | @DivjDC it is a little tricky for spacelike intervals. Suppose that you have lengths marked on an ideal string which you have pulled taut. The string forms a worldsheet which measures $g_{\mu\nu}$ along certain geodesics lying in the worldsheet. Those geodesics are found by taking any event on the worldsheet and going “straight” in the direction perpendicular to the string’s congruence and parallel to the string. It is definitely not trivial which length is being measured, but it is indeed a measurement of length. | |
| Apr 17, 2022 at 13:47 | comment | added | Myridium | So the measurements of distance and time (the coordinates, tangent vectors) change, but the metric changes covariantly, so that the spacetime interval is invariant. Or, you may say that the physics change, so that all process proceed at half the speed (accounting for a scaling of how time is measured). | |
| Apr 17, 2022 at 13:43 | comment | added | Dale | @Myridium I disagree. A clock measures the same amount of time between two events on its worldline regardless of the coordinates used. In fact, you can even use coordinates that have no timelike coordinates or multiple timelike coordinates. Regardless of the coordinates, the clock measures the same | |
| Apr 17, 2022 at 13:29 | comment | added | Myridium | Measurements of distance and time do depend on the choice of coordinates; that's what the choice is! We have so-called inertial frames as those special subset of coordinate choices which reproduce the same physics in a 'form-invariant' way (e.g. a choice of Lagrangian, or path integral in QFT, 'look' the same when written in two different coordinate systems-- this is a symmetry). More generally, other choices of coordinates may be symmetries of some theories, e.g. so-called 'scale invariance' in QED with a massless fermion. | |
| Apr 17, 2022 at 6:50 | comment | added | user87745 | Either a nit or I might be wrong: $g_{\mu\nu}dx^\mu dx^\nu$ is an actual measurement only if the interval is timelike/lightlike, right? For the spacelike interval, any actual measurement of length would be a "two-way" measurement and it won't be simply $g_{\mu\nu}dx^\mu dx^\nu$. For example, Zee, Chapter V.3, Eq (9). I realize that one can define a proper length along a spacelike curve and it will be the integral of $\sqrt{g_{\mu\nu} dx^\mu dx^\nu}$ but I'm not sure if it's something one can measure. | |
| Apr 17, 2022 at 1:18 | history | answered | Dale | CC BY-SA 4.0 |