I am looking for good references (pdf papers) that discuss and justifies the use of the "modified" dot product: \[ {\langle}A,B{\rangle}=-A^0B^0+A^1B^1+A^2B^2+A^3B^3 \] in the 4-dimensional space time metric (Lorentz, Minkowski) by way of basic principles using just ideal clocks and light signals. I tracked down A.A. Robb (1936) but that was not useful. I am looking for papers written at the advanced undergraduate level or at the beginning graduate level. Any such references would be appreciated.
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$\begingroup$ This is elementary text book material, not the stuff of papers. I cover it in troubador.co.uk/bookshop/computing-science-education/… $\endgroup$– Charles FrancisCommented May 24, 2020 at 20:55
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$\begingroup$ How do you "justify" the use of a particular metric? Is it not just used because it works? $\endgroup$– CharlieCommented May 24, 2020 at 21:09
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$\begingroup$ Charlie, Thanks for your response. I would think that it's possible to have a number of different non-equivalent metrics that "work" and so to choose the most appropriate metric one would choose the metric that most closely agrees with the way clocks and light signals behave (according to the axioms of Special Relativity). Basically, I'm looking for a sanity check. $\endgroup$– user54738Commented May 24, 2020 at 23:46
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$\begingroup$ Thanks safesphere. I am familiar with the approach you mentioned. I am looking for something that I could explain to a non-specialist and I'm thinking that Lorentz transformations my be just too much for such an audience. I'm thinking Bondi's method (k-calculus) mentioned below by robphy might be worth looking into. $\endgroup$– user54738Commented May 24, 2020 at 23:49
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3 Answers
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The Bondi k-calculus uses the radar method [using light signals and clocks (wristwatches)] to express the square-interval operationally as a product of time-intervals involved in the radar measurements (resulting in a variant of the expressions used by Robb).
[$k$ refers to the Doppler factor, an eigenvalue of the Lorentz transformation. Bondi is essentially (i.e. secretly) working in light-cone coordinates.
The standard form of the Lorentz Transformation and the $\gamma$-factor are
delayed, and are not explicitly needed to obtain the results Bondi obtains.]
When the square-interval is expressed in rectangular coordinates, you obtain the Minkowski signature.
I think the best readable reference for this method is Bondi’s Relativity and Common Sense. https://archive.org/details/RelativityCommonSense
You can see a summary of Bondi’s method here: https://physics.stackexchange.com/a/508251/148184 https://www.physicsforums.com/insights/relativity-using-bondi-k-calculus/
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This paper seems to fit what you are looking for precisely: it starts with a derivation of the Euclidean distance formula with respect to two dimensional creatures who live in a three-dimensional world and can measure the travel they go through in the third dimension using altimeters. It then naturally extends this argument to our case, that of three dimensional creatures living in a four-dimensional world, who can nevertheless measure our travel through the fourth dimension using clocks. From this the 'spacetime distance formula' is derived, which is the same thing as the Minkowski metric.
http://skepticaleducator.org/wp-content/uploads/2015/01/DerivationLorentzTransformation.pdf
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You can make any theory formally covariant but the point of the principle of relativity is that you have to make your theory invariant among all inertial frames, i.e., the laws in one inertial frame should not refer to its velocity with respect to another inertial frame. This can be achieved if and only if the metric is invariant. The only metric that is invariant among all inertial frames while ensuring that the speed of light remain invariant and that the principle of inertia remain true is the Minkowskian metric. See, Chapter $2$ of Gravitation and Cosmology by Weinberg.