# Is motion through space or spacetime?

• I
• Frank Castle
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."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,f

#### 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:
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.

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 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 $s$.

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
"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).

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?

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.

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?!

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?!

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?

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 traveling along this curve will travel through spacetime as it advances along the curve?

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?

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.

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
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.

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
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.

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)?!

Last edited:
I have since spoken to this person trying to argue this point

Instead of you arguing with them, you could suggest that they sign up for PF and have us argue with them instead. That said, there are a number of errors in what you have quoted from this person:

proper time is merely the number of reflections in your parallel-mirror light clock

No, it isn't. It's the invariant arc length along your worldline. If you choose units for that arc length appropriately, you can make the number you use to describe the arc length match the number of reflections in a light clock that is set up appropriately; but that doesn't make the two the same thing.

When you move fast through space, this reduces in line with the Lorentz factor which is derived from Pythagoras's theorem.

No, it isn't, except in the very, very weak sense that the spacetime interval involves the squares of coordinate differentials. But the spacetime interval has a minus sign in it where the Pythagorean theorem has all plus signs. That makes a big difference. To give just the most important difference: using the ordinary Euclidean distance (the one we get from the Pythagorean theorem), the distance between any two distinct points must be positive. But the spacetime interval between two distinct points can be zero; in fact, the squared interval can be positive, zero, or negative. There's no way to get that from the Pythagorean theorem.

if you could move as fast as a photon, there is no proper time

You can't "move as fast as a photon"; there is no way to make a timelike object move on a null worldline. The two kinds of things are fundamentally distinct.

Also, if by "there is no proper time" he means "the concept of proper time does not make sense for a photon", then he is correct. But I strongly doubt that is what he means; I strongly suspect he means "the proper time for a photon is zero", which is wrong. The arc length along a photon's worldline is zero (zero spacetime interval between any two points on the worldline), but that does not mean the proper time is zero; it means the concept of "proper time" does not make sense.

Time is a dimension in the sense of measure, not a dimension that offers freedom of movement.

If this were correct, it would prove that the three space dimensions also did not offer freedom of movement. Since that's obviously false, this argument must be incorrect. The error in it is to think that "a dimension in the sense of measure" is something different from "a dimension that offers freedom of movement". The two are the same thing--in fact the statement quoted above, once you observe that it applies to space as well as time, can be taken as a reductio ad absurdum argument showing that the two are the same thing.

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.

Exactly. And this is why the concept of "proper time" does not make sense for a photon. You could try this on the person you're talking to; but I doubt if they understand the concept of "affine parameter" well enough to grasp the point.

Proper time is a frame independent quantity and so is fundamentally different to coordinate time.

Yes. More precisely, arc length along a timelike worldline between two chosen events 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?

The issue is not that it can't be viewed as "propagating along its worldline". The issue is that the two statements "the particle is propagating along its worldline" and "the particle is described by its worldline and doesn't propagate at all, the worldline just exists as a curve in spacetime" both describe exactly the same math and exactly the same physical predictions. So there is no way to tell them apart within the domain of physics; as far as physics is concerned, they're equivalent, and neither one is more "right" or "wrong" than the other. They're just two different sets of words that describe the same physics. That is what other posters in this thread have been trying to tell you.

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)?!

Again, these two alternative sets of words--"the particle propagates along its worldline" vs. "the particle exists at all points along its worldline"--both describe exactly the same math and exactly the same physical predictions. So they are both the same as far as physics is concerned.

I understand that intuitively, the two sets of words seem to be describing different ways that things could be. But that's why we don't use ordinary language to describe physics when we want to be precise--because ordinary language can mislead us by making us think that two different sets of words that seem different to us must be describing different physics. That might be the case, but it might not--and in this case, not. The way to tell is to look at the actual math and physical predictions.

• nnunn and Dale
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 move with respect to itself (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)?!
Even though I like working with block universe philosophy for relativity (irrespective of the unknowable question of whether it is true), your 'friend' is being tendentious in claiming a single valid point of view. Let's take two philosophies of Newonian physics, then relativity, and see that there are few differences.

1) Block Universe in Newtonian physics: A particle is a curve in a fiber bundle of space X time. There is no motion at all. There is a constraint that a trajectory of a particle (adopting Newton's corpuscular hypothesis, this includes light) cannot move back in time or along spatial bundle (e.g. teleport). Such curves are mathematically possible but not physically possible. For a trajectory, one can define the velocity at a point of a trajectory as position derivative by time. This is a geometric feature of the trajectory, not a motion, in this philosophy.

2) Evolving philosophy in Newtonian physics: The universe is progressing along time, and particles move through space as a function of time. A 'no time travel' constraint still needs to be added by saying a particle can't be in two or more places at once (with some of them merging at a certain time).

3) Block universe in relativity: A particle is a curve in spacetime. The whole curve and spacetime just exist. There is no motion at all. There is a constraint that particle curve must be timelike, and the curve of light pulse or massless particle must be a null curve. Note, this constraint (or something that implies it) is needed to rule out spacelike = effective teleport paths. Since time is not an absolute bundle parameter, curve parameters are arbitrary, but it is very usefull to use an affine parameter, and specifically proper time for timelike trajectories. One thus defines velocity as tangent using one of these parameters. It is not unique for light. Velocity is a geometric property of a trajectory curve at a point, not a motion.

4) Evolving philosophy in relativity: There is no unique global evolution of space by time [this is why many, but hardly all physicists drop this approach for relativity], but local experience is characterized by evolution in proper time (thus restricting particles to timelike curves). There is motion through spacetime in the sense of (proper) time evolution along a particle history. The requirement of 'local evolution by proper time' prevents spacelike trajectories. Light (null) paths are also allowed as well (because they respect causality); they don't have a time evolution but a light pulse or massless particle can be said to move through spacetime with respect to a (preferably) affine parameter.

• Frank Castle and PeterDonis

Yes, I shall suggest this.

No, it isn't. It's the invariant arc length along your worldline. If you choose units for that arc length appropriately, you can make the number you use to describe the arc length match the number of reflections in a light clock that is set up appropriately; but that doesn't make the two the same thing.

That is what I thought.

"the concept of proper time does not make sense for a photon",

This is what tried to convey, but like you put, I think he believes that the proper time for a photon is zero.

[this is why many, but hardly all physicists drop this approach for relativity]

Is it better to adopt the block universe approach in relativity then? I guess I like approach 4) because it is (to me at least) more intuitive.

Is it better to adopt the block universe approach in relativity then? I guess I like approach 4) because it is (to me at least) more intuitive.
Not if it bothers you. You also don't have to choose one of the other. The real advantage to the block universe conception in general relativity is when thinking about global features and theorems. Thus, a practical approach for you might be to adopt the local evolving point of view for local problems, and the block universe point of view for global problems.

Not if it bothers you. You also don't have to choose one of the other. The real advantage to the block universe conception in general relativity is when thinking about global features and theorems. Thus, a practical approach for you might be to adopt the local evolving point of view for local problems, and the block universe point of view for global problems.

So globally one would interpret spacetime as being static, in the sense that time does not flow within it, time is itself a coordinate labelling events (along with 3 spatial coordinates) of spacetime, hence any flow of time would require re-introducing time when it has already been "used up" to construct the spacetime in the first place. In this sense, the worldlines of objects exist throughout spacetime and are not themselves moving through spacetime - everything is static. However, locally, objects with timelike worldlines can be ascribed a proper-time which parametrises their worldline. Since proper-time is fundamentally distinct from coordinate time - it is frame independent - we can utilise it as a "meta-time" and in this sense interpret a timelike object as locally propagating through spacetime from point to point along its worldline and constant speed ##c##?!