- #1
MrRobotoToo
- 109
- 50
- TL;DR Summary
- I provide an argument for why the conventionality of the one-way speed of light doesn't undermine special relativity
I recently watched this video by Veritasium on the Einstein synchronization convention and its implication for the one-way speed of light and it got me wondering if it in any way undermines the results of special relativity. I think the following argument shows that it doesn't, but any constructive criticism is welcome.
Let’s begin by considering two inertial observers O and O′ that may or may not be in relative motion. They both carry clocks with them and are able to send and receive light signals. If O sends out a light pulse at τ₁ and it bounces off an object and comes back to her at τ₂, she’ll label the ‘bounce event’ with the coordinates t and x, where
t = ɑ(τ₂ + τ₁)
x = β(τ₂ - τ₁)
She’s free to choose ɑ and β–for example, if she believes that the speed of light for each leg of the round trip has the same constant value, then ɑ = β = ½…but any other values will do. Now, with respect to the object, at the exact moment that the O pulse bounces off, another pulse also bounces off that had initially been sent by O′ at τ′₁ (as measured by his clock), and which he’ll receive at τ′₂. He labels the bounce with his coordinates t′ and x′:
t′ = ɑ′(τ′₂ + τ′₁)
x′ = β′(τ′₂ - τ′₁)
Again, O′ is free to choose ɑ′ and β′.
If O and O′ are at rest with respect to each other, they can synchronize their clocks in the usual way: O sends out a light pulse and it bounces off O′ and returns to O. Their clocks are synchronized so long as t = t′ for the bounce. For O′, this event will clearly have τ′₂ = τ′₁. Keeping this in mind and setting the equations for the time coordinates equal to each other will yield (after a bit of algebra):
τ′₁ = ½(ɑ/ɑ′)(τ₂ + τ₁)
The Einstein synchronization convention results when ɑ = ɑ′.
Now that we have the two coordinate systems (t, x) and (t′, x′), we can relate them to each other using Pelissetto and Testa’s postulates, for instance. In the case where the two systems are related by the Lorentz transformations, the constant speed that appears in the equations isn’t 𝘯𝘦𝘤𝘦𝘴𝘴𝘢𝘳𝘪𝘭𝘺 the speed of light. Unusual choices of the alphas and betas above will imply that the speed of light is anisotropic, which would be the case if light is a wave in some material medium and the two observers are moving with respect to it. This doesn’t conflict with SR since it merely implies that light is demoted to an unremarkable phenomenon akin to sound traveling through air. Their postulates also allow for the systems to be related by Galilean transformations, which simply the limiting case when the constant speed that appears in the equations goes to infinity.
Let’s begin by considering two inertial observers O and O′ that may or may not be in relative motion. They both carry clocks with them and are able to send and receive light signals. If O sends out a light pulse at τ₁ and it bounces off an object and comes back to her at τ₂, she’ll label the ‘bounce event’ with the coordinates t and x, where
t = ɑ(τ₂ + τ₁)
x = β(τ₂ - τ₁)
She’s free to choose ɑ and β–for example, if she believes that the speed of light for each leg of the round trip has the same constant value, then ɑ = β = ½…but any other values will do. Now, with respect to the object, at the exact moment that the O pulse bounces off, another pulse also bounces off that had initially been sent by O′ at τ′₁ (as measured by his clock), and which he’ll receive at τ′₂. He labels the bounce with his coordinates t′ and x′:
t′ = ɑ′(τ′₂ + τ′₁)
x′ = β′(τ′₂ - τ′₁)
Again, O′ is free to choose ɑ′ and β′.
If O and O′ are at rest with respect to each other, they can synchronize their clocks in the usual way: O sends out a light pulse and it bounces off O′ and returns to O. Their clocks are synchronized so long as t = t′ for the bounce. For O′, this event will clearly have τ′₂ = τ′₁. Keeping this in mind and setting the equations for the time coordinates equal to each other will yield (after a bit of algebra):
τ′₁ = ½(ɑ/ɑ′)(τ₂ + τ₁)
The Einstein synchronization convention results when ɑ = ɑ′.
Now that we have the two coordinate systems (t, x) and (t′, x′), we can relate them to each other using Pelissetto and Testa’s postulates, for instance. In the case where the two systems are related by the Lorentz transformations, the constant speed that appears in the equations isn’t 𝘯𝘦𝘤𝘦𝘴𝘴𝘢𝘳𝘪𝘭𝘺 the speed of light. Unusual choices of the alphas and betas above will imply that the speed of light is anisotropic, which would be the case if light is a wave in some material medium and the two observers are moving with respect to it. This doesn’t conflict with SR since it merely implies that light is demoted to an unremarkable phenomenon akin to sound traveling through air. Their postulates also allow for the systems to be related by Galilean transformations, which simply the limiting case when the constant speed that appears in the equations goes to infinity.
