4 ms·

You have to be careful when talking about 'speed' when you make time a dimension in your geometry. Traditionally, speed is a measure of how much of your space-t

You have to be careful when talking about 'speed' when you make time a dimension in your geometry. Traditionally, speed is a measure of how much of your space-time curve is in along the time axis. For any space-time path, you can consider the coordinate system as an observer traveling that path would see it, in which that observer would see itself traveling through time at a rate of one second per second. Geometrically, if you were to draw where on the path the observer's clock ticks, the distance between ticks as measured along that path is constant regardless of the path. The speed of light limit says something different. It limits what paths a physical observer can take. Condsider a 1+1 dimensional universe (or our 3+1 universe with a test particle moving along a single spatial dimension). Pick a non accelerating observer to construct the 'stationary' coordinate system. Plot spatial coordinates along the horizontal axis, and the time coordinate as the vertical axis. Pick units such that the speed of light is 1. A particle moving at a constant velocity will follow a straight line. If the line is vertical the particle is stationary. If the line is at an angle, the speed of the particle is the inverse of the slope of the line. The speed of light limitation says that this line cannot be shallower than 45 degrees. In more analytic terms, the distance metric for our 2 dimensional spacetime is given by ds^2 = dt^2 - dx^2. The speed of light limitation says that ds^2 cannot be negative for any path a particle actually takes. In other words all particles must must have at least half of their travel be along the time dimension.

moron4hire 5y ago
To me, this was why the statement that "B would see the objects growing further apart while A would see them getting closer together", if FTL was allowed, was not a very satisfying answer. So given what you've described, that means that forces applied to bodies need to have a time component equal in magnitude to the spatial components. Forces must always exist along that 45° line. The limit of the force required to continue to rotate that spacetime velocity out of the time component and into the spatial components goes to infinity as the vector approaches that 45° line. The fact that forces are unidirectional is the unexplained part. If they weren't, then we could rotate that vector further, and start traveling backwards in time. Then wouldn't B expect to see the objects moving apart, while A sees them moving closer together? To me, the impossibility of the disparity in observations is a consequence of, dependant on, no-FTL, not an explanation thereof. Addendum: I don't understand why you said 45° instead of 90°. I thought objects traveling at the speed of light would experience infinite time dilation, and thus be observed as having 0 passage of time.
- formerly_proven 5y ago
  > Addendum: I don't understand why you said 45° instead of 90°. I thought objects traveling at the speed of light would experience infinite time dilation, and thus be observed as having 0 passage of time. Think about what the diagram shows: Every (s[pace], t[ime]) coordinate pair on the spacetime diagram shows an observation of a particle. So in natural units, a photon's wordline is given by s=t or s=-t (traveling in one or the other direction). If you draw that, it's a 45° line. It also gives you the light cone of the observer at (0, 0). A horizontal wordline would be something moving at infinite speed, not the speed of light, as it is observed at every place at the same time.

moron4hire 5y ago

To me, this was why the statement that "B would see the objects growing further apart while A would see them getting closer together", if FTL was allowed, was not a very satisfying answer. So given what you've described, that means that forces applied to bodies need to have a time component equal in magnitude to the spatial components. Forces must always exist along that 45° line. The limit of the force required to continue to rotate that spacetime velocity out of the time component and into the spatial components goes to infinity as the vector approaches that 45° line. The fact that forces are unidirectional is the unexplained part. If they weren't, then we could rotate that vector further, and start traveling backwards in time. Then wouldn't B expect to see the objects moving apart, while A sees them moving closer together? To me, the impossibility of the disparity in observations is a consequence of, dependant on, no-FTL, not an explanation thereof. Addendum: I don't understand why you said 45° instead of 90°. I thought objects traveling at the speed of light would experience infinite time dilation, and thus be observed as having 0 passage of time.

formerly_proven 5y ago
> Addendum: I don't understand why you said 45° instead of 90°. I thought objects traveling at the speed of light would experience infinite time dilation, and thus be observed as having 0 passage of time. Think about what the diagram shows: Every (s[pace], t[ime]) coordinate pair on the spacetime diagram shows an observation of a particle. So in natural units, a photon's wordline is given by s=t or s=-t (traveling in one or the other direction). If you draw that, it's a 45° line. It also gives you the light cone of the observer at (0, 0). A horizontal wordline would be something moving at infinite speed, not the speed of light, as it is observed at every place at the same time.