# I Is motion through space or spacetime?

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1. May 9, 2016

### Frank Castle

I was chatting to someone recently about the motion of objects and whether they propagate through spacetime or not. They were/are convinced that motion through spacetime is simply not possible arguing something along the lines of the following:

"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.

Physicist's portrayal of this seems to come from their 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.

The reason we normalize velocity four-vectors for massive particles is that the length of a tangent vector has no compelling physical interpretation. Any two tangent vectors that are parallel represent a particle moving through space with the same velocity. Since the length doesn't matter, we might as well arbitrarily set it to some value. We might was well set it to 1, which is of course the value of $c$ in relativistic units. But this normalization is optional in all cases, and impossible for massless particles."

They also referred me to this blog post: http://scienceblogs.com/goodmath/2008/01/17/the-nasty-little-truth-about-i/ which even after reading the first line immediately set off my "crackpot alarm" , and continuing to read the post further confirmed this.

My opinion is that objects do propagate through spacetime. My reason being that, although an objects worldline is a fixed trajectory through space, it is not remaining at a single point on that trajectory, it is moving along it. To move along its worldline the object must have some velocity associated with it - even if it remain stationary in space it will still be moving in time.
I think the problem in understanding is related to self-referential definitions of time that crop up, indeed time is a very difficult quantity to define without any self reference, but it doesn't mean that it isn't a physical quantity.

I have to admit, I'm starting to doubt myself a bit. Is my understanding correct here, or is this other person correct?

If someone could clarify this and also provide a convincing argument I'd much appreciate it.

Last edited: May 9, 2016
2. May 9, 2016

### Mister T

We describe motion using the notion of velocity, which is the time rate of change of position. In other words, you need both space and time to describe it. I see the rest of the discussion as a matter of semantics. A worldline is a path through spacetime. A trajectory is a path through space.

3. May 9, 2016

### PAllen

I mostly agree with your initial quoted text. A world line represents motion - the motion different in different coordinate systems. The analogy I would make is to a curve on on piece of graph paper. The curve doesn't move, but there is change of one coordinate with respect to the other all along it. In the case of spacetime, we call change of position coordinates with respect to time (coordinate) velocity, and we can say the world line represents motion with respect a given space-time foliation. But the curve itself doesn't move, and no point (event) on the world line moves (with respect to what would it move? we would need to add another dimension for that).

4. May 9, 2016

### stevendaryl

Staff Emeritus
I think it's just a matter of how you prefer to look at it. The worldline of a point-particle can be described by a parametrized path, $x^\mu(s)$, where $s$ is proper time. So you can view that as a point particle "moving" through 4-D spacetime as a function of proper time $s$.

5. May 9, 2016

### Staff: Mentor

I would not agree with this particular part of the quoted text. The length of the tangent vector for massive particles does have an obvious physical interpretation: the rest mass of the particle. Knowing that a particle has a particular worldline in spacetime does not tell us all there is to know about the particle; it has other properties, of which rest mass is one, and the length of the tangent vector models that property.

6. May 9, 2016

### robphy

It seems to me that the reason we normalize velocity 4-vectors for massive particles is that we can think of it as $\hat t$, which can be interpreted as "the displacement one-tick later along an inertial worldline" and which can be dotted with a 4-vector to extract the time-component of that 4-vector according to that observer. In addition, the set of all of unit 4-velocities traces out the future unit-hyperbola (which Is a visualization of the metric tensor and whose asymptotes are along the light-cone).

7. May 10, 2016

### PAllen

Hmm. That's the length of the 4-momentum which is not the tangent vector. The tangent vector (in general) is just derivative of curve coordinates with respect to some parameter. Its norm is arbitrary, and can be varied by chaning parameter. Thus, we make a 4-velocity by making proper time the parameter.

8. May 10, 2016

### Frank Castle

I agree that the curve itself doesn't move, and that the events themselves don't move. This is fixed. But doesn't a particle "move" from event to event along its worldline and so moves through spacetime? I get that (neglecting GR and the expansion of the universe, that spacetime is essentially a static, absolute background, but what I don't see is why objects can't move through this background? The worldline of a particle still has a tangent vector to it at each spacetime point, so wouldn't this be describing the rate of change in the objects motion through spacetime (with respect to proper time)?!

9. May 10, 2016

### Frank Castle

This is how I was thinking of it, but would you say that I'm technically incorrect to say that objects move through spacetime? It seems to me that it would be possible, since even when an object is at rest in space, time is still increasing, but maybe I'm misunderstanding something here?

10. May 10, 2016

### atyy

I think stevendaryl gave the mathematics in post #4 that corresponds to this.

11. May 10, 2016

### A.T.

To avoid confusion, I prefer to use another term than "move" here, like "advance in space-time".

12. May 10, 2016

### Frank Castle

So what is correct then? Do particles advance in spacetime but propagate through space? I am left feeling confused now as to what is the correct understanding/interpretation?!

13. May 10, 2016

### A.T.

It's semantics.

14. May 10, 2016

### Frank Castle

But is it correct at all to think of a particle propagating along its worldline in spacetime? Following from what Stevendaryl wrote, if one parametrises a particles worldline by its proper time then it has a well-defined four velocity, so isn't it in this sense propagating in spacetime? Otherwise the whole point of introducing four velocity etc. seems pointless if what is actually physically correct is that the particle is propagating through space with time labelling each point along its trajectory such that it maps out a path in spacetime?!

15. May 10, 2016

### martinbn

Velocity in this context means tangent vector. Even in (pure)geometry people say velocity of the curve, when they mean the tangent vector, and there are no particles and no motion along the curve. It brings intuition from one area to help with the abstraction in another.

16. May 10, 2016

### Frank Castle

I understand that, but the tangent vector to the curve at any particular point is still quantifying the rate of change in the curve at that point (with respect to its parametrisation), right?

17. May 10, 2016

### martinbn

Yes.

18. May 10, 2016

### Frank Castle

But this is my point. Given that the curve has a velocity vector associated with it at each point, doesn't this mean that an object travelling along this curve will travel through spacetime as it advances along the curve?

19. May 10, 2016

### PAllen

Ultimately, as you've seen a variety of answers, this is a question of philosophy. The physics is in the math and how you relate observables to mathematical quantities. The intent of my post was to assert that most of the initial quote was not wrong and is similar to how I think about it, not that it the only correct way to think about it.

20. May 10, 2016

### Frank Castle

Fair enough.

Out of interest, what are your thoughts on the blog post that I linked in my first post? Is any of it correct or is the author misunderstanding things?