Motivation for the usage of 4-vectors in special relativity

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Given that, in this limiting case we require that ds2=0=ds′2ds2=0=ds′2ds^{2}=0=ds'^{2}, then in general, we must have that
ds′2=F(t,x,y,z)ds2ds′2=F(t,x,y,z)ds2​
ds'^{2}=F(t,x,y,z)ds^{2}
I like this line of reasoning very much, except that I am not sure about this step. I don't understand why this would follow.
 
  • #27
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Yes, I think that's a very good motivation (if not a proof). I like particularly the discussion about the overall factor F(t,x,y,z)F(t,x,y,z).
Thanks :)

I like this line of reasoning very much, except that I am not sure about this step. I don't understand why this would follow.
I have to be honest, I was a bit "hand-wavy" in this part, but I think it follows because the two line elements must agree in the vanishing case, i.e. if ##ds^{2}=0## then we must also have that ##ds'^{2}=0## (as I showed before generalising). This can only be true if the line element in the primed frame is related to the unprimed frame by ##ds'^{2}=F(t,x,y,x)ds^{2}## in general, since otherwise ##ds^{2}## could vanish and ##ds'^{2}## would not necessarily vanish.

I may be wrong in my reasoning though and would be interested in a more formal proof of the statement that ##ds'^{2}=F(t,x,y,x)ds^{2}## in general if anyone can help out?!
 
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  • #28
PeterDonis
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the Galilean spacetime square interval between any two events is just the square of the time-difference between those events. This is Galilean invariant.
I agree that the "time distance" between two events in Galilean spacetime is Galilean invariant. I'm not sure I agree with calling this the "Galilean spacetime square interval", because the whole point is that "time distance" and "space distance" are different things and are never mixed together, so there is no "spacetime distance". But that's a matter of terminology.
 
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vanhees71
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Well, the mathematical structure of the Galilei-Newton spacetime is more like a fibre bundle than a pseudo-scalar space, i.e., at each point of the time line (oriented ##\mathbb{R}^1##) you have a Euclidean ##\mathbb{R}^3## as space. I don't think that the definition of some spacetime bilinear form makes any physical sense, while it is very natural in SRT to introduce a pseudo-scalar product of signature (1,3) (and further understood as "local" in the sense of the tangent spaces on a pseudo-Riemannian manifold also in ART).
 
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robphy
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I agree that the "time distance" between two events in Galilean spacetime is Galilean invariant. I'm not sure I agree with calling this the "Galilean spacetime square interval", because the whole point is that "time distance" and "space distance" are different things and are never mixed together, so there is no "spacetime distance". But that's a matter of terminology.
Well, the mathematical structure of the Galilei-Newton spacetime is more like a fibre bundle than a pseudo-scalar space, i.e., at each point of the time line (oriented ##\mathbb{R}^1##) you have a Euclidean ##\mathbb{R}^3## as space. I don't think that the definition of some spacetime bilinear form makes any physical sense, while it is very natural in SRT to introduce a pseudo-scalar product of signature (1,3) (and further understood as "local" in the sense of the tangent spaces on a pseudo-Riemannian manifold also in ART).
Admittedly, all of these Galilean [and Newtonian-Gravitational] spacetime constructions are in hindsight following Special Relativity and Minkowski Spacetime [and General Relativity]. Taken in isolation, it doesn't seem to have much mathematical or physical motivation.

I didn't make up these notions myself [they are in the literature, as I posted here in Motivation for the introduction of spacetime and Generalisation of Time Dilation]. It seems the main mathematical-physical motivation was to explore the limiting case (i.e. the classical nonrelativistic limit) of the transformation group and of the differential-geometric structures--the correspondence principle. In addition, it could be argued that "regarding Galilean physics as a limiting case" constrains the possible interpretations one may have had on a physical concept [like momentum]. A similar method of reasoning may help constrain how one might generalize Special Relativity to General Relativity, or General Relativity to Quantum Gravity.

I am, however, actively working out aspects of this idea (one of my "new ideas for teaching relativity") , as applied to various physical situations treated in introductory and intermediate physics.
It's motivated by the alternative-to-the-historical storyline suggested by Jammer and Stachel's "If Maxwell had worked between Ampère and Faraday: An historical fable with a pedagogical moral" [ http://scitation.aip.org/content/aapt/journal/ajp/48/1/10.1119/1.12239 ]... that maybe one could have discovered Galilean relativity and aspects of its structures first [like a Galilean-invariant electrodynamics explaining a subset of experimental results], then be confronted with a new phenomenon [Faraday's Law] that needed to be explained, resulting in Special Relativity [a Lorentz-invariant electrodynamics]. (reference to old post in Looking for complete explanation of logic behind relativity)
 
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In Newtonian physics, yes.
Would it be correct to say that in Newtonian physics, since time (and simultaneity) is (are) absolute the approach of considering a 3 dimensional space at each instant in time is well-defined, since it is observer independent, hence we can consider 3 dimensional space parametrised by time, which governs how the 3D space evolves. In special relativity, both time and simultaneity are relative and so there is no well-defined way of considering 3-dimensional space at each instant in time, since this can not be achieved in an observer independent manner. Since there is no observer independent way to separate time and space coordinates we must consider them as coordinates of a single 4-dimensional spacetime.
 
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Would it be correct to say that in Newtonian physics, since time (and simultaneity) is (are) absolute the approach of considering a 3 dimensional space at each instant in time is well-defined, since it is observer independent, hence we can consider 3 dimensional space parametrised by time, which governs how the 3D space evolves. In special relativity, both time and simultaneity are relative and so there is no well-defined way of considering 3-dimensional space at each instant in time, since this can not be achieved in an observer independent manner. Since there is no observer independent way to separate time and space coordinates we must consider them as coordinates of a single 4-dimensional spacetime.
Hi Frank. 3D space being relative is what Einstein referred to in his quotes I posted in that other thread.
 

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