# Is motion through space or spacetime?

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.

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Mister T
Gold Member
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.

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).

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 $s$.

PeterDonis
Mentor
The reason we normalize velocity four-vectors for massive particles is that the length of a tangent vector has no compelling physical interpretation.

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.

• nnunn
robphy
Homework Helper
Gold Member
"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 cin relativistic units. But this normalization is optional in all cases, and impossible for massless particles."

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).

PAllen
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.
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.

which even after reading the first line immediately set off my "crackpot alarm" , and continuing to read the post further confirmed this.

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).

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)?!

• nnunn
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, xμ(s)xμ(s)x^\mu(s), where sss is proper time. So you can view that as a point particle "moving" through 4-D spacetime as a function of proper time sss.

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?

atyy
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?

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

A.T.
But doesn't a particle "move" from event to event along its worldline and so moves through spacetime?
To avoid confusion, I prefer to use another term than "move" here, like "advance in space-time".

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

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?!

A.T.
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?!
It's semantics.

It's semantics.

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?!

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.

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.

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?

martinbn
Yes.

Yes.

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?

PAllen

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)?!

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.

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.

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?

stevendaryl
Staff Emeritus
Another comment on this topic. Ultimately, it's a matter of taste whether you think of particles as moving through space as a function of time, or moving through spacetime as a function of proper time (or some other affine parameter). However, I think that the latter interpretation has certain advantages in helping to understand intuitively some of the features of Special and General Relativity.

Time dilation: Since velocity through space is a frame-dependent quantity, velocity-dependent time dilation is similarly frame-dependent. That makes it seem like it's not real. But it is really the case (in the twin paradox) that the traveling twin will end up younger than the stay-at-home twin. So it's a little complicated to intuitively understand how something that is frame-dependent (time dilation) can end up making a frame-independent difference (the traveling twin is older). Of course, it all works out, but it's a little complicated to see that it's all consistent, and respects the equivalence of all inertial frames.

In contrast, if you view the traveling twin as traveling through spacetime as a function of proper time, then it's clear that the amount of proper time to reach a particular destination in spacetime depends on the traveling twin's 4-velocity, in exactly the same way that the amount of coordinate time required to reach a particular destination in space depends on the 3-velocity.

Gravitation: Popularizers explain that gravity in GR is just curvature. It's easy to understand how curved space can end up deflecting the path of a particle in motion. However, if you think of velocity as moving through space as a function of time, it's not at all clear how curvature leads to an object falling when it is initially at rest. Curvature deflects paths, but if an object is at rest, it doesn't have a path to deflect. This puzzle is resolved if you think in terms of 4-velocities. What you think of as an object at "rest" is actually an object with a sizable 4-velocity in the t-direction. Curvature deflects the object's path, so that the 4-velocity is no longer purely in the t-direction, but has a component in the z-direction (towards the Earth).

• nnunn
Another comment on this topic. Ultimately, it's a matter of taste whether you think of particles as moving through space as a function of time, or moving through spacetime as a function of proper time (or some other affine parameter). However, I think that the latter interpretation has certain advantages in helping to understand intuitively some of the features of Special and General Relativity.

Time dilation: Since velocity through space is a frame-dependent quantity, velocity-dependent time dilation is similarly frame-dependent. That makes it seem like it's not real. But it is really the case (in the twin paradox) that the traveling twin will end up younger than the stay-at-home twin. So it's a little complicated to intuitively understand how something that is frame-dependent (time dilation) can end up making a frame-independent difference (the traveling twin is older). Of course, it all works out, but it's a little complicated to see that it's all consistent, and respects the equivalence of all inertial frames.

In contrast, if you view the traveling twin as traveling through spacetime as a function of proper time, then it's clear that the amount of proper time to reach a particular destination in spacetime depends on the traveling twin's 4-velocity, in exactly the same way that the amount of coordinate time required to reach a particular destination in space depends on the 3-velocity.

Gravitation: Popularizers explain that gravity in GR is just curvature. It's easy to understand how curved space can end up deflecting the path of a particle in motion. However, if you think of velocity as moving through space as a function of time, it's not at all clear how curvature leads to an object falling when it is initially at rest. Curvature deflects paths, but if an object is at rest, it doesn't have a path to deflect. This puzzle is resolved if you think in terms of 4-velocities. What you think of as an object at "rest" is actually an object with a sizable 4-velocity in the t-direction. Curvature deflects the object's path, so that the 4-velocity is no longer purely in the t-direction, but has a component in the z-direction (towards the Earth).

This makes a lot of sense and is how I've thought of it up until now, but at the moment I feel confused more than anything. At first I thought that the blog post that I linked into my first post was complete nonsense, but now I'm not so sure. Is what the author wrote correct at all?

I thought the whole point of introducing a 4-dimensional spacetime was exactly because there is no observer independent way to separate space and time coordinates and hence they must be considered as coordinates of a single four dimensional geometry? Given this, it seems to me that one should be able to have motion through time and motion through space (since an object at rest, i.e. only "moving" in time with respect to one frame will be observed to be moving through both space and time with respect to another). However, according to the person I spoke to, to move through spacetime would require introducing a "meta-time"?! In the introduction to special relativity that I've had, the four-velocity of a particle was introduced by noting that if we wanted an observer independent notion of velocity, we most compute the rate of change of an objects four-position with respect to a frame-independent quantity, this being the proper time. This made sense to me at the time, since the proper-time of a particle is a parameter, since it is absolute (i.e. frame-independent), and hence parametrises both spatial position and coordinate time.

Sorry if I'm rambling a bit here, but I just feel a little bit lost and trying to make sense of something I thought I understood, but know I'm not so sure.

stevendaryl
Staff Emeritus
However, according to the person I spoke to, to move through spacetime would require introducing a "meta-time"?!

I don't understand that complaint, since "proper time" already is a kind of "meta-time".

I don't understand that complaint, since "proper time" already is a kind of "meta-time".

That's what I thought, since proper-time is an additional independent parameter, so it can be used to parametrise time as well, right?

As a personal preference, while some may think of a world-line (-surface, -tube) as a mere path or trajectory of an object through space-time I like to think of it as being the object itself, a 4D object. What we perceive as the "object" is then just a slice of the actual 4D object intersected with an arbitrary 3D "now" surface. The 4D object itself is fixed and unchanging in this view.

• weirdoguy
PeterDonis
Mentor
That's the length of the 4-momentum which is not the tangent vector.

There is no such thing as "the" tangent vector, because of the normalization issue. But as long as the invariant mass of the object is constant, the 4-momentum is certainly a tangent vector; you just use proper time divided by invariant mass as the parameter (instead of proper time by itself).

It is true, though, that this no longer works for the case of changing invariant mass.

PeterDonis
Mentor
since proper-time is an additional independent parameter, so it can be used to parametrise time as well, right?

Yes. In fact, that is what the 4-velocity vector is doing. It is a 4-vector, meaning it has 4 components, and each component is the derivative of the corresponding coordinate with respect to proper time, i.e., with respect to the curve parameter. (This is assuming we have a timelike curve and are using proper time as the parameter, which is the most natural thing to do.) So the ##t## component of the 4-velocity is the derivative of ##t##, the "time" coordinate, with respect to the curve parameter. Integrating the ##t## component of the 4-velocity gives you the ##t## coordinate as a function of the curve parameter.

• nnunn
PAllen
There is no such thing as "the" tangent vector, because of the normalization issue. But as long as the invariant mass of the object is constant, the 4-momentum is certainly a tangent vector; you just use proper time divided by invariant mass as the parameter (instead of proper time by itself).

It is true, though, that this no longer works for the case of changing invariant mass.

But the post I was replying to said:

"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."

I argued basically as you do above, if you include my full post:

"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."

You were the one making a claim for "the tangent vector", and my post said the norm is arbitrary and depends on the parameter you use. Of course I agree that a particular normalization will produce 4-momentum for a particle whose mass isn't changing, thus 4-momentum can be 'a' tangent vector, and my wording above is slightly too strong. But you have to put the mass into the tangent vector to get it out, effectively. It's not a (geometrically) natural normalization of the curve (though it is an affine parameter), as 4-velocity is. Of course, there is another quibble - based on a Lagrangian derivation, many authors argue the 4-momentum must be a covector, while the 4-velocity is a vector. In this case, the 4-momentum is not even an instance of a tangent vector.

PeterDonis
Mentor
You were the one making a claim for "the tangent vector"

Yes, good catch. I should have clarified that I was assuming that by "the" tangent vector, the quoted text meant the 4-momentum, i.e., the vector whose invariant length is the object's rest mass. (And, of course, this also assumes that the rest mass is constant.)

It's not a (geometrically) natural normalization of the curve (though it is an affine parameter), as 4-velocity is.

If the invariant mass is constant, then "proper time divided by invariant mass" is just a rescaling of the unit of time, so it's just as "geometrically natural" as any other choice for a unit of time. I think the question is whether invariant mass itself can be considered a "geometric" property. In the context of SR, I don't think it can be, because, as you say, it has to be basically put in by hand. In the context of GR, there are cases where invariant mass can be considered a geometric property, but only for objects or systems that are considered sources of gravity, not for "test objects".

Yes. In fact, that is what the 4-velocity vector is doing. It is a 4-vector, meaning it has 4 components, and each component is the derivative of the corresponding coordinate with respect to proper time, i.e., with respect to the curve parameter. (This is assuming we have a timelike curve and are using proper time as the parameter, which is the most natural thing to do.) So the ##t## component of the 4-velocity is the derivative of ##t##, the "time" coordinate, with respect to the curve parameter. Integrating the ##t## component of the 4-velocity gives you the ##t## coordinate as a function of the curve parameter.

I have since spoken to this person trying to argue this point and they retorted with the following: " proper time is merely the number of reflections in your parallel-mirror light clock. When you move fast through space, this reduces in line with the Lorentz factor which is derived from Pythagoras's theorem. And if you could move as fast as a photon, there is no proper time.
Time is a dimension in the sense of measure, not a dimension that offers freedom of movement. You can't move through it, or through spacetime either."

Now, as I understand it, proper-time and therefore four-velocity is simply not defined for light-like worldline. To parametrise a photons worldline one has to choose an alternative affine parameter.
However, I disagree with the last line. Proper time is a frame independent quantity and so is fundamentally different to coordinate time. It can be used to parametrise the worldline of a particle and to construct a well-defined four-velocity, so in this sense, why can't a massive particle be propagating along its worldline? I get that the spacetime background is itself static (neglecting gravity), coordinate time and spatial coordinates have been "used up" to construct a 4D spacetime, so I see why one can't propagate through spacetime with respect to coordinate time since is part of the fabric of spacetime and hence cannot be reintroduced "from the outside" to describe motion within spacetime (this would be analogous to saying that in Newtonian 3D space one can propagate through ##(x,y,z)## with respect to ##x##). I also see that a particles worldline is fixed in spacetime (i.e. it doesn't move in spacetime), but if the worldline is timelike, then why can't the particle propagate along its worldline with respect to its proper time (otherwise does a particle exist at all points along its worldline)?!

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