The Paradox of Oppositely Traveling Objects in Spacetime

[dennis83704]

It is my understanding that all things travel through spacetime at the same speed.
Additionally , I am told we are moving through space at some velocity.
My question is this: If an object were made to travel in a direction opposite to our direction of motion , would that object not then be traveling through space at a slower speed? Wouldn't time pass faster for that object as it would be moving slower through space?

 

[TRodrigues]

Let's elaborate on that point a bit more. All things travel through spacetime at the same speed, or more technically, everything's "four-velocity" -- that is, the velocity at which things move in the four-dimensional spacetime -- has the same magnitude.

Then, you say that you are told we are moving through space at some velocity.

Velocity compared to what?

 

[dennis83704]

Wikipedia's entry on CMB states we are moving at 627 km/sec towards Virgo with respect to the cosmic rest frame.

 

[bcrowell]

Wouldn't time pass faster for that object as it would be moving slower through space?

Relativistic time dilation depends on the velocity of one observer relative to another observer, not the velocity of an object relative to space.

Wikipedia's entry on CMB states we are moving at 627 km/sec towards Virgo with respect to the cosmic rest frame.

The cosmic rest frame is simply the frame in which the CMB is not moving.

 

[JesseM]

It is my understanding that all things travel through spacetime at the same speed.

I think you are thinking of Brian Greene's explanation of relativity, I posted the math behind it in [post=430613]this post[/post]--I've never seen any textbook that explains relativity this way though, and it has potential to create conceptual confusion.

Additionally , I am told we are moving through space at some velocity.

Not in any objective sense--if you pick an inertial reference frame we have some speed relative to that frame, but we have a different speed relative to other frames, and every inertial reference frame is considered equally valid. In any frame, if you take the square of your speed through space [tex]d\vec{x}/dt[/tex] and add it to the square of the rate that your clock is ticking relative to the time coordinate in that frame ([tex]d\tau / dt[/tex], what Greene calls your "speed through time"), multiplied by the speed of light squared, the sum of these terms is always equal to the speed of light squared, i.e. [tex](d\vec{x}/dt)^2 + c^2(d\tau/dt)^2 = c^2[/tex]. Greene uses the shorthand "your speed through spacetime is always c" to describe this fact, though like I said I think this has the potential to create some conceptual confusion.

My question is this: If an object were made to travel in a direction opposite to our direction of motion , would that object not then be traveling through space at a slower speed? Wouldn't time pass faster for that object as it would be moving slower through space?

If we are using an inertial frame where your speed is nonzero, then an object moving at a lower speed in this frame (or at rest in this frame) will have its clock tick more quickly than your clock, relative to the time coordinate of this frame. But again it's all completely relative, you could analyze things from the point of view of the inertial frame where you are at rest and then it would be your clock ticking faster than the other clock.

 

[dennis83704]

Thank you all for taking the time to comment , it was very helpful.