A question about movement in 4D

[curiouschemist]

Okay, I understand that if you dedicate some motion to one dimension at a constant rate you consequently pull motion from another dimension. Like driving at an angle rather than a straight line traveling a longer distance to a said stopping line (traveling in two dimensions rather than one). I understand how this translates to time dilation at fast acceleration, the more you move through space the less you move through time. This has only been given to me in one spatial dimension and the time dimension. What happens if a rotor spins at close to the speed of light? It is, in turn (no pun intended), spinning in two spatial dimensions, would you only be able to spin the rotor at 1/2 the speed of light since you are distributing motion to two spatial dimensions from one time dimension? In this respect, what about the third spatial dimension? Like, say, a collapsing massive star. Would it only have to accelerate to 1/3 the speed of light? Anyway... I hope you see my question, I hope you (anyone) can give me some clarity. Thanks!

 

[JesseM]

Okay, I understand that if you dedicate some motion to one dimension at a constant rate you consequently pull motion from another dimension. Like driving at an angle rather than a straight line traveling a longer distance to a said stopping line (traveling in two dimensions rather than one). I understand how this translates to time dilation at fast acceleration, the more you move through space the less you move through time. This has only been given to me in one spatial dimension and the time dimension. What happens if a rotor spins at close to the speed of light? It is, in turn (no pun intended), spinning in two spatial dimensions, would you only be able to spin the rotor at 1/2 the speed of light since you are distributing motion to two spatial dimensions from one time dimension?

No--it's not as if an individual atom in a solid object "knows" whether other atoms in the same object are moving in different directions (as with a rotor) or if they are all at rest with respect to each other (as with a solid object moving in a linear way).

I don't think the idea of explaining relativity in terms of "movement in space vs. movement in time" should be taken too literally, it's really just a sort of mathematical trick. Brian Greene is the only author I have seen who describes relativity this way, and he explains the justification in an endnote in The Elegant Universe (p. 392):

For the mathematically inclined reader, we note that from the spacetime position 4-vector $$x = (ct, x_1, x_2, x_3) = (ct, \vec{x})$$ we can produce the velocity 4-vector $$u$$, $$dx/d\tau$$, where $$\tau$$ is the proper time defined by $$d\tau^2 = dt^2 - c^{-2}(dx_1^2 + dx_2^2 + dx_3^2)$$. Then, the "speed through spacetime" is the magnitude of the 4-vector $$u$$, $$\sqrt{((c^2dt^2 - d\vec{x}^2)/(dt^2 - c^{-2}d\vec{x}^2))}$$, which is identically the speed of light, $$c$$. Now, we can rearrange the equation $$c^2(dt/d\tau)^2 - (d\vec{x}/d\tau)^2 = c^2$$, to be $$c^2(d\tau/dt)^2 + (d\vec{x}/dt)^2 = c^2$$. This shows that an increase in the object's speed through space, $$\sqrt{(d\vec{x}/dt)^2}$$, must be accompanied by a decrease in $$d\tau/dt$$, the latter being the object's speed through time (the rate at which time elapses on its own clock, $$d\tau$$, as compared with that on our stationary clock, $$dt$$).

I don't really like the idea of labelling $$d\tau/dt$$ the "speed through time", since it's really just the speed that a clock is ticking as seen in a different reference frame, and labelling the magnitude of the 4-vector $$u$$ above (usually known as the '4-velocity') as the "speed through spacetime" makes even less sense to me.

 

[R0man]

Your question is a very interesting one and it touches on some important concepts in physics. To answer your question, we need to first understand the idea of spacetime and how we measure motion in 4D.

In 4D, we have three spatial dimensions (length, width, and height) and one time dimension. These dimensions are all interconnected and cannot be separated. This means that any motion in one dimension will affect the other dimensions as well. This is why, in your example of driving at an angle, you are traveling a longer distance in two dimensions rather than just one.

Now, when we talk about motion in 4D, we use the concept of spacetime intervals. This is a way of measuring how far an object travels in 4D. The faster an object moves through space, the slower it moves through time. This is what we call time dilation. So, if an object is moving at the speed of light, it is not moving through time at all. This is because all of its motion is dedicated to the three spatial dimensions.

In the case of a rotor spinning close to the speed of light, the same principle applies. The rotor is spinning in two spatial dimensions and therefore, it can only spin at a fraction of the speed of light. This is because some of its motion is dedicated to the time dimension.

As for your question about a collapsing massive star, the same concept applies. The star would have to accelerate to a fraction of the speed of light in order to collapse in the third spatial dimension. This is because its motion is distributed among all four dimensions.

I hope this helps to clarify your question. Keep exploring and asking questions, as these concepts can be complex and fascinating to learn about.