# Question about relativity -- speed through space plus speed through time

1. Sep 30, 2015

### Starlover

Hello Everyone!

I have a question for better minds than mine. :)

This pertains to speed through space plus speed through time always needing to equal the speed of light.

I believe I've read that in order to add speed through space and speed through time, we would convert the two numbers to common units, by multiplying time by the speed of light. This sounds simple enough, but I can't figure out how to do it. (For example, which units would I use: Light years, years, miles per second, etc.?)

I'm about as far from Sheldon Cooper as a person can get -- can't follow physics equations, symbols, etc. -- so I'm hoping someone would be kind enough to provide an example of the conversion process in the form of ordinary sentences.

By the way, am I even correct about converting the two numbers to common units by multiplying time by the speed of light? And if so, would such a conversion be part of the Lorentz equations I've read about?

2. Sep 30, 2015

### andrewkirk

I can't think of anything that would give meaning to the phrase 'speed through time', so I'll skip that one. 'Speed through space' is interpreted easily enough once one has chosen a reference frame. Then it is just the speed of the object relative to that frame.
Yes, that is a respected technique. For instance Schutz uses it in his text 'A first course in General Relativity'. The approach is that time is measured in metres, so that the unit of time is the time it takes for light to travel 1 metre. Then the factor c becomes 1 so it disappears out of all the equations and it makes the calcs much cleaner.

He does a similar trick later on for gravity, where he measures mass in metres too!

The conversion doesn't remove the need for the Lorentz adjustments, it just changes how they're written. eg $\sqrt{1-\frac{v^2}{c^2}}$ becomes $\sqrt{1-v^2}$.

3. Sep 30, 2015

### Starlover

Is there any chance you could give me an example -- in sentence form -- with numbers? Something like this:

"Jack's ship was moving through space at XX (SPEED). This means that he was moving through time at XX (SPEED), because
______ (movement through space) + __________ (movement through time) = _____________."

4. Sep 30, 2015

### andrewkirk

Sorry, I'm afraid I can't write a meaningful sentence of that form because the words 'he was moving through time at xx speed' have no meaning, irrespective of what we put in place of the xx.

To measure a speed, we need a time dimension, so we can't use time to measure movement through itself.

A velocity 4-vector has, in any given coordinate system, a spatial and a time component, but I think the concepts involved in that are too technical for what you're looking for.

5. Sep 30, 2015

### bcrowell

Staff Emeritus
This seems to be a variation on the following FAQ, which is in turn based on a misconception.

FAQ: Why do massive particles move through spacetime at the speed of light?

They don't. This idea seems to be something that the popularizer Brian Greene has perpetrated on the world. Objects don't move through spacetime. Objects move through space. If you depict an object in spacetime, you have a world-line. The world-line doesn't move through spacetime, it simply extends across spacetime.

Greene's portrayal of this seems to come from his feeling that because the magnitude of a massive particle's velocity four-vector is traditionally normalized to have magnitude c, it makes sense to describe the particle, to a nonmathematical audience, as "moving through spacetime" at c. This is simply inaccurate. A good way to see that it's inaccurate is to note that a ray of light doesn't even have a four-vector that can be normalized in this way. Any tangent vector to the world-line of a ray of light has a magnitude of zero, so you can't scale it up or down to make it have a magnitude of c. For consistency, Greene would presumably have to say that a ray of light "moves through spacetime" at a speed of zero, which is obviously pretty silly.

6. Oct 1, 2015

### greswd

It is just a consequence of the Lorentz transformations that the 4-velocity can always have a constant magnitude.

It doesn't mean that the particle travels through space-time at a constant c. In fact, there's no "true" interpretation behind SR.

For the time being, it just appears to be the fundamental means by which space and time transform.

7. Oct 1, 2015

### Starlover

Thank you, everybody, for your replies. It's very kind of all of you. I simply might not be sophisticated enough for this forum.

BTW, Andrewkirk, what I meant by "he moved through time at XX speed" is
"XX amount of time elapsed during his trip (relative to time on Earth)." Is that of any help?