4 ms·

On the subject of implicit assumptions (like linear trajectories transform to linear trajectories), it has always struck me that there is another assumption: if

On the subject of implicit assumptions (like linear trajectories transform to linear trajectories), it has always struck me that there is another assumption: if A sees B at speed v, then B sees A at the same speed v. It’s hard to imagine this symmetry not being true but who knows what’s possible in theory space?

Q-Q 3y ago
What are you guys on about >if A sees B at speed v, then B sees A at the same speed v is directly derivable from the Lorentz velocity addition formula which is the result of the usual two postulates. Why would you want that as an extra assumption... https://en.wikipedia.org/wiki/Velocity-addition_formula#Special_relativity https://en.wikipedia.org/wiki/Velocity-addition_formula#Spec... vAA = (vAB + vBA)/(1+(vAB*vBA/c^2) -> 0 = vAB + vBA so velocity of A relative to B and B relative to A both have magnitude v.
- superposeur 3y ago
  Of course, if you already assume special relativity then this is baked in: taking the inverse of the Lorentz transformation matrix effectively sends v to -v. And velocity addition for sure already assumes you have the Lorentz transformations. The point is that in the derivation of relativity from elementary postulates, the “reflectivity of velocities” assumption is used. For instance the linked paper uses it between equations (12) and (13). All such derivations use it at some stage. Intuitively, “reflectivity of velocities” is why time dilation leads to length contraction with same gamma factor. I don’t think it follows from absence of a preferred frame since one could imagine some complicated group structure relating the velocities of the various observers relative to each other in such a way that all are on equal footing and yet v_AB is not exactly -v_BA. As mentioned, this is certainly a reasonable assumption but who knows what whacky alternatives are out there? After all, in the different context of quantum mechanics, such “obvious” relationships as xp = px no longer hold true.
LudwigNagasena 3y ago
Another interesting velocity-related conundrum: It’s impossible to measure the one-way speed of light (from a source to a detector) because we have to synchronize the clocks somehow, which requires an algorithm based on the speed of light to account for time dilation. We can only measure the two-way speed of light (from a source to a reflector and back). So, from the standpoint of pure math, isotropy of the speed of light is purely a convention. The argument can only be made based on, so to speak, philosophical reasoning.
- superposeur 3y ago
  This is true — I take the upshot of this observation to be that only proper times are ever directly physically measurable. Coordinate times and concepts that rely on them (such as what you call one-way velocity of light) are always merely a bookkeeping device for organizing the results of different proper time measurements and will always entail some amount of arbitrariness. General relativity makes this more explicit: the arbitrariness is that of the choice of coordinates on spacetime, and one can choose any old coordinate system or none at all.
alok-g 3y ago
+1. When I was working out the maths myself, I had exactly the same thought.

Q-Q 3y ago

What are you guys on about >if A sees B at speed v, then B sees A at the same speed v is directly derivable from the Lorentz velocity addition formula which is the result of the usual two postulates. Why would you want that as an extra assumption... https://en.wikipedia.org/wiki/Velocity-addition_formula#Special_relativity https://en.wikipedia.org/wiki/Velocity-addition_formula#Spec... vAA = (vAB + vBA)/(1+(vAB*vBA/c^2) -> 0 = vAB + vBA so velocity of A relative to B and B relative to A both have magnitude v.

superposeur 3y ago
Of course, if you already assume special relativity then this is baked in: taking the inverse of the Lorentz transformation matrix effectively sends v to -v. And velocity addition for sure already assumes you have the Lorentz transformations. The point is that in the derivation of relativity from elementary postulates, the “reflectivity of velocities” assumption is used. For instance the linked paper uses it between equations (12) and (13). All such derivations use it at some stage. Intuitively, “reflectivity of velocities” is why time dilation leads to length contraction with same gamma factor. I don’t think it follows from absence of a preferred frame since one could imagine some complicated group structure relating the velocities of the various observers relative to each other in such a way that all are on equal footing and yet v_AB is not exactly -v_BA. As mentioned, this is certainly a reasonable assumption but who knows what whacky alternatives are out there? After all, in the different context of quantum mechanics, such “obvious” relationships as xp = px no longer hold true.

LudwigNagasena 3y ago

Another interesting velocity-related conundrum: It’s impossible to measure the one-way speed of light (from a source to a detector) because we have to synchronize the clocks somehow, which requires an algorithm based on the speed of light to account for time dilation. We can only measure the two-way speed of light (from a source to a reflector and back). So, from the standpoint of pure math, isotropy of the speed of light is purely a convention. The argument can only be made based on, so to speak, philosophical reasoning.

superposeur 3y ago
This is true — I take the upshot of this observation to be that only proper times are ever directly physically measurable. Coordinate times and concepts that rely on them (such as what you call one-way velocity of light) are always merely a bookkeeping device for organizing the results of different proper time measurements and will always entail some amount of arbitrariness. General relativity makes this more explicit: the arbitrariness is that of the choice of coordinates on spacetime, and one can choose any old coordinate system or none at all.

alok-g 3y ago

+1. When I was working out the maths myself, I had exactly the same thought.