# Stanford: objects in spacetime all move at constant speed c?

• I
In this Stanford University lecture on Relativity, it is stated:

Likewise, objects in spacetime all move at constant speed c in spacetime but if you change its direction, say by moving at speed v in the x direction, then spatial speed will change and so will the speed along the ct direction. Again, its total speed will still be c through spacetime.

http://web.stanford.edu/~oas/SI/SRGR/notes/SRGRLect6_2007.pdf

Brian Green of Columbia University also says the same thing, as do many others.

Are they right?​

andrewkirk
Homework Helper
Gold Member
Yes they are right, but it doesn't mean what it sounds like it means. In particular it means nothing like the same as moving at speed c through space, which only massless particles like photons can do.

In cases like this, the only exact presentation of the science is a mathematical one, which involve be saying something such as that the four-velocity vector of a massive particle has magnitude c (or magnitude 1, if we're using relativistic units). The linked pdf starts to talk about that around the bottom of p2.

What do you mean by this, "Yes they are right, but it doesn't mean what it sounds like it means. In particular it means nothing like the same as moving at speed c through space, which only massless particles like photons can do."?

They're not talking about space like you are.

stevendaryl
Staff Emeritus
What do you mean by this, "Yes they are right, but it doesn't mean what it sounds like it means. In particular it means nothing like the same as moving at speed c through space, which only massless particles like photons can do."?

They're not talking about space like you are.

The statement that all objects move at speed c through spacetime is equivalent to saying: Every second, you move one second into the future. It's true, but it's not that profound of a statement.

robphy
Homework Helper
Gold Member
Likewise, objects in spacetime all move at constant speed c in spacetime but if you change its direction, say by moving at speed v in the x direction, then spatial speed will change and so will the speed along the ct direction. Again, its total speed will still be c through spacetime.
A more correct statement would be:
Massive objects in spacetime all move at constant speed c in spacetime (i.e., is a vector with magnitude c) [and spatial speed v (also called ##\tanh\theta## [a slope]) in space]
but if you change its direction, say by moving at nonzero spatial speed v in the x direction,
then spatial component (##\gamma\ v## or ##c\sinh\theta##) will change and so will the temporal component (##\gamma\ c## or ##c\cosh\theta##).
Again, its total speed (##\sqrt{(\gamma\ c)^2- (\gamma\ v)^2}##) will still be c through spacetime.
Massless objects in spacetime all move at constant speed zero in spacetime [and spatial speed c in space].

A more correct statement would be:
Massive objects in spacetime all move at constant speed c in spacetime (i.e., is a vector with magnitude c) [and spatial speed v (also called ##\tanh\theta## [a slope]) in space]
but if you change its direction, say by moving at nonzero spatial speed v in the x direction,
then spatial component (##\gamma\ v## or ##c\sinh\theta##) will change and so will the temporal component (##\gamma\ c## or ##c\cosh\theta##).
Again, its total speed (##\sqrt{(\gamma\ c)^2- (\gamma\ v)^2}##) will still be c through spacetime.
Massless objects in spacetime all move at constant speed zero in spacetime [and spatial speed c in space].

So the velocity of light through spacetime is 0? I would argue that the velocity of light is c.

Here is the math:

##\left(\dfrac{ds}{d\tau}\right)^2=\left(\dfrac{c \, dt}{d\tau}\right)^2 - \left(\dfrac{d \vec x}{d\tau}\right)^2##,

where ##ds## is the (invariant) infinitesimal spacetime interval, ##d\tau## is the (invariant) infinitesimal proper time interval, ##dt## is the (frame-dependent) infinitesimal coordinate time interval, and ##d \vec x## is the (frame-dependent) infinitesimal displacement.

In plain English, the left side is the (squared) "speed through spacetime" of something with mass. All observers agree on its value. The right side has two terms whose value depends on an observer's velocity relative to the massive object in question: the (squared) "speed through time" and the (squared) "proper velocity."

But this expression can be simplified. First, ##ds/d\tau = c##. Second, ##dt/d\tau = 1/\sqrt{1-v^2/c^2} = \gamma##. Third, ##d \vec x / d \tau = \gamma \vec v## (where ##\vec v## is normal velocity). So we have:

##c^2=(\gamma c)^2 - (\gamma \vec v)^2##.

This means that everything with mass always moves through spacetime with a "speed" of ##c##, where "speed" means the proper-time-derivative of the spacetime interval. As ##v## increases, both terms on the right side increase, and the first of them (the time contribution) is always the bigger of the two. So it's not quite true that "speed through time" decreases as ##v## increases.

What is true is that an object at rest has all of its "motion" through spacetime directed forward in time, whereas moving observers would say that the object moves through both space and time. All observers agree, however, that the difference of the squares of the space and time contributions equals ##c^2##, and that the object moves through spacetime at the ##\tau##-rate of ##c##.

Nugatory
Mentor
Are they right?
It's one plausible way of visualizing the relationship between motion in space and paths in spacetime; and it captures the idea that you're always moving forward in time no matter what you do. It's not perfect (for example, either the speed through spacetime is not ##c## but instead ##\sqrt{-c^2}## - note that square root of a negative number - or you have to adopt a sign convention in which your speed through space is the square root of a negative number) and it's no substitute for understanding the underlying math.

So you can use it to form a mental picture to go with the math, but you can't build any deeper understanding on top of it.

Chestermiller
robphy
Homework Helper
Gold Member
So the velocity of light through spacetime is 0? I would argue that the velocity of light is c.
Let me be more explicit in my post (#5 above).

Massive objects in spacetime all move at constant speed c in spacetime
(i.e., its 4-momentum vector can be normalized to a "unit" 4-velocity vector with Minkowski-norm c)
[and has spatial speed (slope) v in space].

Massless
objects in spacetime all move at constant speed zero in spacetime
(i.e., its 4-momentum vector has Minkowski-norm zero [and thus can't be normalized])
[and has spatial speed (slope) c in space].

Battlemage!
PeterDonis
Mentor
Are they right?

It depends. Brian Greene, at least, when he makes this pitch in his pop science books, says something that isn't right. He says that all objects, including light, "move at c" in spacetime. But as robphy has pointed out, this isn't correct; it switches between two different interpretations of the term "speed through spacetime"--the norm of the 4-velocity in the case of massive objects, but the coordinate speed through space in an inertial frame in the case of light. This is a good example of a scientist saying something in a pop science book that he would never get away with in an actual peer-reviewed paper.

A.T.
Are they right?
It's not about right or wrong , just about geometrical interpretations of the same math. The advance through space-time at const c can be visualized better in a space-propertime diagram:

robphy
Homework Helper
Gold Member
It's not about right or wrong , just about geometrical interpretations of the same math. The advance through space-time at const c can be visualized better in a space-propertime diagram:

Can the clock effect/twin paradox be displayed on this diagram? Or a uniformly accelerated observer?
Or is it restricted to inertial observers?

A.T.
Can the clock effect/twin paradox be displayed on this diagram? Or a uniformly accelerated observer?
Or is it restricted to inertial observers?
I think for a uniformly accelerated observer it would look cone-like, as shown here for a local observer hovering in a gravitational field:
http://demoweb.physics.ucla.edu/content/10-curved-spacetime
Also use gravity slider here:

Here the non-local picture for a radial line through the Schwarzshild-geometry (which is non-inertial).

I guess you can transform any space-coordinate diagram into a space-propetime diagram. Here for the twins:

For the twins with constant acceleration it should look somewhat like this:

The key is that the length of all worllines (in space-propertime) represents the passed coordiante time, so they all have equal length. This is the geometrical interpretation closest to the "all moves at c through space-time" meme, that I know of.

Last edited:
PAllen
For a non-inertial observer in such a diagram, why would you not construct it so the 'defining observer' world line is vertical??!!

It cannot be good for coincident events to be separated on diagram (this occurs even for the inertial frame twin scenario) ...

A.T.
For a non-inertial observer in such a diagram, why would you not construct it so the 'defining observer' world line is vertical??!!
His world line is shown curved, because he is non-inertial.

It cannot be good for coincident events to be separated on diagram

Dale
Mentor
2021 Award
They're not talking about space like you are.

So the velocity of light through spacetime is 0? I would argue that the velocity of light is c.
You are being inconsistent here. If you are talking about "velocity through space" then massive objects have ##0\le v < c## and massless objects have ##v=c##. If you are talking about "velocity through spacetime" then massive objects have ##"v"=c## and massless objects have ##"v"=0##. In neither case do you get ##c## for both massive and massless objects.

If you don't like ##"v"=0## for light then you may want to move on to other topics besides this "velocity through spacetime".

Battlemage! and Jeronimus
PeterDonis
Mentor
So the velocity of light through spacetime is 0? I would argue that the velocity of light is c.

And what would you base your argument on? The tangent vector to the worldline of a light ray, which is the only feasible interpretation of the term "velocity through spacetime", is null; that means it has zero norm, so light's "velocity through spacetime" is indeed zero under this interpretation. The tangent vector to the worldline of a massive object, OTOH, has norm ##c##; that's what justifies the statement that massive objects have a "velocity through spacetime" of ##c##.

Another interpretation would be that the concept of "velocity through spacetime" doesn't make sense at all for light. But there is no interpretation I'm aware of under which light's "velocity through spacetime" is ##c##. Light's velocity through space is ##c##; but as Dale pointed out, you yourself were saying we should be careful not to confuse space with spacetime.

The statement that all objects move at speed c through spacetime is equivalent to saying: Every second, you move one second into the future. It's true, but it's not that profound of a statement.
You are being inconsistent here. If you are talking about "velocity through space" then massive objects have ##0\le v < c## and massless objects have ##v=c##. If you are talking about "velocity through spacetime" then massive objects have ##"v"=c## and massless objects have ##"v"=0##. In neither case do you get ##c## for both massive and massless objects.

If you don't like ##"v"=0## for light then you may want to move on to other topics besides this "velocity through spacetime".

Can you elaborate a bit on this?

If as you say, when considering spacetime, massive objects move at c (always i assume), whereas light moves at 0 then it seems acceleration cannot exist any more in the sense of increasing or decreasing the speed of a massive object.

PeterDonis
Mentor
then it seems acceleration cannot exist any more in the sense of increasing or decreasing the speed of a massive object.

In the spacetime model, acceleration does not change the speed of a massive object; it only changes its direction. The norm of the object's 4-velocity never changes; it is always ##c##. But the direction in which the 4-velocity points in spacetime can change; that is what 4-acceleration does.

Battlemage! and Jeronimus
In the spacetime model, acceleration does not change the speed of a massive object; it only changes its direction. The norm of the object's 4-velocity never changes; it is always ##c##. But the direction in which the 4-velocity points in spacetime can change; that is what 4-acceleration does.

Why is something which does not affect the velocity of an object called acceleration? 4-acceleration in this case.

Why is something which does not affect the velocity of an object called acceleration? 4-acceleration in this case.

Velocity is a vector with both magnitude and direction. A change in either is change in velocity.

Battlemage!
Velocity is a vector with both magnitude and direction. A change in either is change in velocity.

But is that really what is going on? Just a simple change of the direction of the velocity vector? What would prevent one then from pointing the vector to go backwards in time?

But is that really what is going on? Just a simple change of the direction of the velocity vector? What would prevent one then from pointing the vector to go backwards in time?

What prevents it? Nature. We're free to move in whatever spatial direction we choose, but we can only move forward in time. That's just how it is.

Nugatory
Mentor
Why is something which does not affect the velocity of an object called acceleration? 4-acceleration in this case.
It does affect the velocity - the four-velocity and the four-acceleration are related by ##\vec{a}=\frac{d\vec{v}}{d\tau}## just as you'd expect. What doesn't change is the magnitude of the four-velocity, and that's because the four-acceleration is always perpendicular to the four-velocity.

Battlemage!
Dale
Mentor
2021 Award
But is that really what is going on? Just a simple change of the direction of the velocity vector? What would prevent one then from pointing the vector to go backwards in time?
The forward pointing in time directions are not smoothly connected to the backwards pointing directions. This is a result of the fact that there is only one time dimension compared to three space dimensions.

Last edited:
Jeronimus
Nugatory
Mentor
But is that really what is going on? Just a simple change of the direction of the velocity vector? What would prevent one then from pointing the vector to go backwards in time?
There is no four-acceleration that can turn a four-velocity that lies in the future light-cone into a four-velocity that lies in the past light cone.

We're getting into stuff that is hard to visualize without doing the math because the geometry of spacetime is Minkowskian not Euclidian.

Battlemage!