3 ms·

I've always intuitively understood this to be the reason why it would take infinite force to achieve light speed for a massive object. When we apply a physical

I've always intuitively understood this to be the reason why it would take infinite force to achieve light speed for a massive object. When we apply a physical force, it is applied in the spatial axes, so it is always perpendicular to the time axis. Acceleration is just rotating some magnitude of your fixed velocity vector out of the time axis and into the spatial axes. When your spatial velocity is apparently zero, then the component of force that is perpendicular to your velocity is large, so you achieve a large deflection. But as you rotate velocity out of time and into space, it becomes more perpendicular to time, so any force applied perpendicularly to time is now more parallel to your velocity, having a smaller component perpendicular to one's velocity. You can't rotate a vector with a parallel force. This is also why you can't travel backwards in time through just acceleration. There is no way to impart a force perpendicular to your velocity vector when it is already perpendicular to time, giving you no way to rotate the vector to have a component that points backward in time. So I've always wondered, whether general relativity allows for forces parallel to time, and we just don't know of any mechanism to actually do so, or if it does not cover such cases because we have no mechanism, or if it disallows it entirely.

Sharlin 6y ago
This is a useful way to think about it, but you have to keep in mind that (even flat) spacetime is not Euclidean but Minkowskian: in the distance metric, time has an opposite sign to the spatial dimensions. So when you "rotate" a four-vector, it actually follows the surface of a hyperboloid rather than a sphere, which means that rotation has a discontinuity at Θ = π/4 (in normalized units) and the vector escapes to infinity! Only massless objects can travel "at the speed of π/4", everything else can only approach but never reach that speed.

Sharlin 6y ago

This is a useful way to think about it, but you have to keep in mind that (even flat) spacetime is not Euclidean but Minkowskian: in the distance metric, time has an opposite sign to the spatial dimensions. So when you "rotate" a four-vector, it actually follows the surface of a hyperboloid rather than a sphere, which means that rotation has a discontinuity at Θ = π/4 (in normalized units) and the vector escapes to infinity! Only massless objects can travel "at the speed of π/4", everything else can only approach but never reach that speed.