# Is motion through space or spacetime?

PeterDonis
Mentor
2019 Award
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
2019 Award
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
2019 Award
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
2019 Award
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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PeterDonis
Mentor
2019 Award
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
PAllen
2019 Award
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.

PAllen
2019 Award
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##?!

Dale
Mentor
Sounds reasonable to me except that I would say "one could" rather than "one would". The point is that either interpretation is a matter of choice and the physicist should feel at liberty to pick either, both, or neither as occasion suits. The universe doesn't care either way.

Sounds reasonable to me except that I would say "one could" rather than "one would". The point is that either interpretation is a matter of choice and the physicist should feel at liberty to pick either, both, or neither as occasion suits. The universe doesn't care either way.
Good point.

vanhees71
Gold Member
2019 Award
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 also like to use the "normalized" four-velocity vector
$$u^{\mu}=\frac{\mathrm{d} x^{\mu}}{\mathrm{d} s}=\frac{1}{c} \frac{\mathrm{d} x^{\mu}}{\mathrm{d} \tau}=\frac{1}{mc} p^{\mu},$$
just because it's convenient, particularly in continuum mechanics it's nice to have the normalized flow-velocity vector to project between the temporal and spatial components of four vectors wrt. the local restframe of the heat bath, using the covariant projectors
$$P_{\parallel}^{\mu \nu}=u^{\mu} u^{\nu}, \quad P_{\perp}^{\mu \nu}=\eta^{\mu \nu}-u^{\mu} u^{\nu}.$$

For the motion of a point particle one should be aware that the covariant equation of motion only superficially has four independent components
$$\frac{\mathrm{d} p^{\mu}}{\mathrm{d} \tau}=K^{\mu},$$
because you have (for classical particles!) the on-shell contraint
$$p_{\mu} p^{\mu}=m^2 c^2 \; \Rightarrow \; p_{\mu} \frac{\mathrm{d} p^{\mu}}{\mathrm{d} \tau}=0 \; \Rightarrow \; p_{\mu} K^{\mu}=0.$$
Thus only three of the four equations are independent. You can solve for the three spatial components in the respective frame of reference ("calculational frame"), and the time-component follows. It's in some sense a generalized energy-work theorem:
$$u_{\mu} K^{\mu}=0 \; \Rightarrow \; u^0 K^0=\vec{u} \cdot \vec{K} \; \Rightarrow \; K^0=\frac{\vec{u}}{u^0} \vec{K} = \frac{\vec{v}}{c} \cdot \vec{K}.$$
So you have
$$\frac{\mathrm{d} p^0}{\mathrm{d} \tau}=\frac{1}{c} \frac{\mathrm{d} \mathcal{E}}{\mathrm{d} \tau}=K^0=\frac{\vec{v}}{c} \cdot \vec{K}.$$
Now you can rewrite this in terms of the coordinate time, using ##\mathrm{d} t/\mathrm{d} \tau=u^0=\gamma##.
$$\frac{\mathrm{d} \mathcal{E}}{\mathrm{d} t}=\vec{v} \cdot \vec{F}, \quad \vec{F}=\frac{\vec{K}}{u^0}.$$

PAllen
2019 Award
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##?!
Yes, except there is not much meaning to speed c. For example, personally, I normalize 4-velocities to 1 rather than c, even when I am not setting c=1. There is no physics whatsoever in varying choice of affine parameter (and you can get any positive 'speed' you want by suitable choice of affine parameter; you can choose 42 if you are galactic hitch-hiker).

• Frank Castle and Dale
vanhees71
Gold Member
2019 Award
Sure ##c## is only a conversion factor from inconvenient SI units to physical units (SCNR).

If the universe is a static 4 dimensional construct and there is no motion to it, then you would have to explain why we experience change still.

We certainly do not experience the whole universe at the same time.
The universe might be static and not moving, but our experience of it is moving along the worldlines the instance of our bodies reside in, in what we call the present.

Dale
Mentor
If the universe is a static 4 dimensional construct and there is no motion to it, then you would have to explain why we experience change still.
That is easy to explain. At each event on your worldline your experience is a function of the past light cone of that event. Since the past light cone changes along the worldline the experience also changes along the worldline.

We certainly do not experience the whole universe at the same time.
Of course not, most of the universe is outside the past light cone of any given moment, and even within our past light cone the closer events are more important in determining our experience.

The universe might be static and not moving, but our experience of it is moving along the worldlines the instance of our bodies reside in, in what we call the present.
At every event you have the experience of being present. So what we call "the present" is not a unique physically identifiable moment, but rather a psychological impression that applies equally to any moment.

Of course not, most of the universe is outside the past light cone of any given moment, and even within our past light cone the closer events are more important in determining our experience.
This would be why one can locally interpret a timelike object as propagating along its worldine and hence moving in spacetime, right? (if we could observe the universe as a whole, neglecting gravity, then one could observe the entire worldline of the timelike object and hence the object would be at all points along its worldline).

Mister T
Gold Member
We certainly do not experience the whole universe at the same time.
We don't experience the whole universe at ANY time. Every experience I ever have occurs at one specific place AND at one specific time. Experience is a very limited and personal thing.

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There are some very enlightening responses in this thread. I'd like to share how this dumbazz imagines a world line in space time. I think of calculus, and the moving triad of tnagent, normal, and binormal vectors moving along a parametized curve. In this case of SR, the curve is a function of proper time, and the the triad is 3D Euclidian space. You can think of a particle at rest within that space, moving along at proper time, or the particle can have momentum within that space, as well as moving along the timeline (i.e. world line). Just my \$0.02

• nnunn
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.
Hi Vitro. You look at it the way Einstein did. Here are a few Einstein quotes you might find most interesting:

<< Since there exists in this four dimensional structure [space-time] no longer any sections which represent "now" objectively, the concepts of happening and becoming are indeed not completely suspended, but yet complicated. It appears therefore more natural to think of physical reality as a four dimensional existence, instead of, as hitherto, the evolution of a three dimensional existence. >> (Albert Einstein, "Relativity", 1952).

<< From a "happening" in three-dimensional space, physics becomes, as it were, an "existence" in the four-dimensional "world". >> (Albert Einstein. "Relativity: The Special and the General Theory." 1916. Appendix II Minkowski's Four-Dimensional Space ("World") (supplementary to section 17 - last section of part 1 - Minkowski's Four-Dimensional Space).

• nnunn