What is the velocity of a photon?

[gbfmgbfm]

The 4-velocity of a photon is undefined, so does that mean the velocity of a photon is undefined?

What is the velocity of a photon?

 

[Meir Achuz]

The 3 velocity of a photon is c (or 1 in natural units).

 

[ghwellsjr]

The speed of a photon is defined to be c according to Einstein's second postulate since we cannot measure the one-way speed of light and photons only travel one way.

 

[PAllen]

The OP was a little vague on context. In flat spacetime (SR), it is c. In GR, it is c for a 'local measurement' - small distance. Otherwise, it will not necessarily be c, and will depend not only on the geometry, but also on how you intend to measure it (in particular, there isn't a unique definition of large distances; therefore there cannot be a unique definition of any speed as non-local measurement). Mathematically this manifests as 3-velocity of null geodesics in GR is coordinate dependent; there is no way to pick unique preferred coordinates; and no way to pick coordinates such that 3-velocity is c for all null geodesics.

 

[gbfmgbfm]

The OP was a little vague on context. In flat spacetime (SR), it is c. In GR, it is c for a 'local measurement' - small distance. Otherwise, it will not necessarily be c, and will depend not only on the geometry, but also on how you intend to measure it (in particular, there isn't a unique definition of large distances; therefore there cannot be a unique definition of any speed as non-local measurement). Mathematically this manifests as 3-velocity of null geodesics in GR is coordinate dependent; there is no way to pick unique preferred coordinates; and no way to pick coordinates such that 3-velocity is c for all null geodesics.

You write, "Otherwise, it will not necessarily be c, and will depend not only on the geometry, but also on how you intend to measure it."

So, you are saying that sometimes the velocity of light is not c in a vacuum? Who has experimentally proven this?

 

[PAllen]

You write, "Otherwise, it will not necessarily be c, and will depend not only on the geometry, but also on how you intend to measure it."

So, you are saying that sometimes the velocity of light is not c in a vacuum? Who has experimentally proven this?

The Shapiro time delay is normally interpreted as such a measurement: http://www.astro.ucla.edu/~wright/deflection-delay.html

 

[Naty1]

you are saying that sometimes the velocity of light is not c in a vacuum

no.

In flat spacetime (SR), it is c. In GR, it is c for a 'local measurement' - small distance.

locally, the speed of light is always c.

 

[gbfmgbfm]

no.



locally, the speed of light is always c.

So then you are saying that the velocity of a photon is c?

 

[gbfmgbfm]

The Shapiro time delay is normally interpreted as such a measurement: http://www.astro.ucla.edu/~wright/deflection-delay.html

So as you state that the velocity of light is not c in some cases (violating Einstein's second postulate), are you saying that Einstein's relativity is violated in these cases?

1. First postulate (principle of relativity)
The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of coordinates in uniform translatory motion.
2. Second postulate (invariance of c)
As measured in any inertial frame of reference, light is always propagated in empty space with a definite velocity c that is independent of the state of motion of the emitting body.

 

[PAllen]

So as you state that the velocity of light is not c in some cases (violating Einstein's second postulate), are you saying that Einstein's relativity is violated in these cases?

1. First postulate (principle of relativity)
The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of coordinates in uniform translatory motion.
2. Second postulate (invariance of c)
As measured in any inertial frame of reference, light is always propagated in empty space with a definite velocity c that is independent of the state of motion of the emitting body.

These are postulates of special relativity. For general relativity, they remain true locally, but are not even well defined globally.

 

[Passionflower]

These are postulates of special relativity. For general relativity, they remain true locally, but are not even well defined globally.

Indeed.

 

[gbfmgbfm]

These are postulates of special relativity. For general relativity, they remain true locally, but are not even well defined globally.

So locally, the velocity of a photon is always c. Then why is the four-velocity of a photon undefined?

 

[PAllen]

So locally, the velocity of a photon is always c. Then why is the four-velocity of a photon undefined?

Because 4 velocity is derivative with respect to proper time along a world line. Light follows a null world line, which means proper time is 0 between any two points on it, so said derivative cannot be defined.

As has already been said, 3-velocity is readily defined (derivative of spacelike coordinates by timelike coordinate), and is c for any inertial frame in SR (using reasonable coordinates). For GR, re-read what I've already said.

 

[robphy]

The 4-velocity of a photon is undefined, so does that mean the velocity of a photon is undefined?

What is the velocity of a photon?

To add to Meir Achuz's reply...

the spatial velocity of any particle, according to inertial observer Bob,
can be defined by taking the 4-momentum vector of that particle and
breaking it into temporal- and spatial-components (according to Bob),
then forming the ratio of the [vector] spatial-compenent to the [scalar] temporal-component.
The result is the spatial-velocity [sometimes called 3-velocity] of the particle according to Bob.
This works whether the particle is timelike or lightlike.

The 4-velocity, however, can be defined as the 4-momentum divided by its magnitude (the square-root of the square-norm)... i.e. it's a unit-vector. This is fine for a timelike particle. However, it doesn't work for a lightlike particle since the square-norm of its 4-momentum is zero. Thus, the 4-velocity cannot be defined for a lightlike particle.

In equations...

Let $\vec p$ be the particle's 4-momentum.
If it's not lightlike, then we can define its 4-velocity $$\hat v = \frac{\vec p}{\sqrt{\vec p \cdot \vec p} }.$$(You could call it $\hat p$, if you wish.)

Let $\hat t$ be Bob's 4-velocity
so that the particle's 4-momentum (whether timelike or lightlike) can be written as
$\vec p= \vec p_{\| \hat t} + \vec p_{\perp \hat t}= E \hat t + \vec P = E(\hat t + \vec V)$.
(If it were timelike, then $\hat v=\gamma \hat t + \vec S = \gamma (\hat t + \vec V)$.)

The spatial-velocity of the particle according to Bob is the spatial vector (i.e. $\vec V \cdot \hat t=0)$:
$$\vec V=\frac{ \vec p_{\perp \hat t} } { \sqrt{\vec p_{\| \hat t}\cdot \vec p_{\| \hat t} } }= \frac{\vec P}{ E}$$

 

[gbfmgbfm]

Because 4 velocity is derivative with respect to proper time along a world line. Light follows a null world line, which means proper time is 0 between any two points on it, so said derivative cannot be defined.

As has already been said, 3-velocity is readily defined (derivative of spacelike coordinates by timelike coordinate), and is c for any inertial frame in SR (using reasonable coordinates). For GR, re-read what I've already said.

So the 3-velocity is c for a photon. Does this mean that in the three spatial dimensions, the velocity of light is c?

 

[gbfmgbfm]

To add to Meir Achuz's reply...

the spatial velocity of any particle, according to inertial observer Bob,
can be defined by taking the 4-momentum vector of that particle and
breaking it into temporal- and spatial-components (according to Bob),
then forming the ratio of the [vector] spatial-compenent to the [scalar] temporal-component.
The result is the spatial-velocity [sometimes called 3-velocity] of the particle according to Bob.
This works whether the particle is timelike or lightlike.

The 4-velocity, however, can be defined as the 4-momentum divided by its magnitude (the square-root of the square-norm)... i.e. it's a unit-vector. This is fine for a timelike particle. However, it doesn't work for a lightlike particle since the square-norm of its 4-momentum is zero. Thus, the 4-velocity cannot be defined for a lightlike particle.

In equations...

Let $\vec p$ be the particle's 4-momentum.
If it's not lightlike, then we can define its 4-velocity $$\hat v = \frac{\vec p}{\sqrt{\vec p \cdot \vec p} }.$$(You could call it $\hat p$, if you wish.)

Let $\hat t$ be Bob's 4-velocity
so that the particle's 4-momentum (whether timelike or lightlike) can be written as
$\vec p= \vec p_{\| \hat t} + \vec p_{\perp \hat t}= E \hat t + \vec P = E(\hat t + \vec V)$.
(If it were timelike, then $\hat v=\gamma \hat t + \vec S = \gamma (\hat t + \vec V)$.)

The spatial-velocity of the particle according to Bob is the spatial vector (i.e. $\vec V \cdot \hat t=0)$:
$$\vec V=\frac{ \vec p_{\perp \hat t} } { \sqrt{\vec p_{\| \hat t}\cdot \vec p_{\| \hat t} } }= \frac{\vec P}{ E}$$

So if the 4-velocity cannot be defined for a photon, can we say that the photon has no 4-velocity, but only a 3-velocity?

 

[robphy]

So if the 4-velocity cannot be defined for a photon, can we say that the photon has no 4-velocity, but only a 3-velocity?

A photon has no quantity called its 4-velocity.
According to an inertial observer, a photon has a 3-velocity whose magnitude is c.

 

[gbfmgbfm]

A photon has no quantity called its 4-velocity.
According to an inertial observer, a photon has a 3-velocity whose magnitude is c.

How does a photon move relative to the fourth dimension?

 

[robphy]

How does a photon move relative to the fourth dimension?

It traces a line in spacetime [as it would on an ordinary (3D-)position-vs-time graph].

 

[gbfmgbfm]

It traces a line in spacetime [as it would on an ordinary (3D-)position-vs-time graph].

What is the photon's velocity relative to the fourth dimension?

 

[robphy]

What is the photon's velocity relative to the fourth dimension?

That doesn't make any physical sense. Relative-velocities are defined between particles... in the simplest case, between two inertial worldlines that cross at an event.

 

[gbfmgbfm]

That doesn't make any physical sense. Relative-velocities are defined between particles... in the simplest case, between two inertial worldlines that cross at an event.

But can we not define the velocity of a photon relative to a point x1=0, x2=0, x3=0, where it starts from?

Do we not say that the photon's velocity is c relative to this point?

 

[Dale]

But can we not define the velocity of a photon relative to a point x1=0, x2=0, x3=0, where it starts from?

Do we not say that the photon's velocity is c relative to this point?

Sure. That is a 3-velocity, not a 4-velocity.

Note, this does not contradict what robphy said. The point x1=0, x2=0, x3=0 can represent a particle, specifically a particle which is at rest at the origin. It may be hypothetical, i.e. there need not be an actual particle at rest there. The point is more that relative velocity is defined between two worldlines. Are you familiar with the term "worldline"?

 

[gbfmgbfm]

Sure. That is a 3-velocity, not a 4-velocity.

Note, this does not contradict what robphy said. The point x1=0, x2=0, x3=0 can represent a particle, specifically a particle which is at rest at the origin. It may be hypothetical, i.e. there need not be an actual particle at rest there. The point is more that relative velocity is defined between two worldlines. Are you familiar with the term "worldline"?

Hello Dale,

Yes, I know it is a 3-veolicty, and not a 4-velocity. Yes I understand worldines very well.

Why do you create a particle and then take it away, writing, "The point x1=0, x2=0, x3=0 can represent a particle, specifically a particle which is at rest at the origin. It may be hypothetical, i.e. there need not be an actual particle at rest there."?

In your words, "there need not be an actual particle at rest there." So why did you introduce it in the first place?

What's wrong with imagining a point in space-time, without a particle?

robphy states, "Relative-velocities are defined between particles." As you have shown, this is not true. :)

It is important to be precise.

 

[gbfmgbfm]

Sure. That is a 3-velocity, not a 4-velocity.

Note, this does not contradict what robphy said. The point x1=0, x2=0, x3=0 can represent a particle, specifically a particle which is at rest at the origin. It may be hypothetical, i.e. there need not be an actual particle at rest there. The point is more that relative velocity is defined between two worldlines. Are you familiar with the term "worldline"?

Dear Dale,

Suppose we define a point (x1=0, x2=0, x3=0, x4=0). What is a photon's velocity relative to that point?

 

[PAllen]

Dear Dale,

Suppose we define a point (x1=0, x2=0, x3=0, x4=0). What is a photon's velocity relative to that point?

That's not meaningful as stated. It can be made meaningful, as follows: what is speed of photon whose world line passes through (t,x,y,z)=(0,0,0,0) relative to a timelike worldline passing through the same event? The answer is c.

(My comments about light traveling different speed than c only apply to a non-local measurement in GR. For the only natural interpretation of your question, as given above, the answer is c, period).

 

[gbfmgbfm]

That's not meaningful as stated. It can be made meaningful, as follows: what is speed of photon whose world line passes through (t,x,y,z)=(0,0,0,0) relative to a timelike worldline passing through the same event? The answer is c.

(My comments about light traveling different speed than c only apply to a non-local measurement in GR. For the only natural interpretation of your question, as given above, the answer is c, period).

How is the following statement not meaningful?

"Suppose we define a point (x1=0, x2=0, x3=0, x4=0). What is a photon's velocity relative to that point?"

Where does the simple question err? What law is the question violating?

 

[PAllen]

How is the following statement not meaningful?

"Suppose we define a point (x1=0, x2=0, x3=0, x4=0). What is a photon's velocity relative to that point?"

Where does the simple question err? What law is the question violating?

What is your definition of velocity? When I say it makes no sense as stated I mean that without more qualification it fails to be any of the following:

1) A relative velocity - need two world lines for that, one timelike.
2) A coordinate velocity - well, you could make this apply, but then you 'relative to an event' makes no sense. You could ask for coordinate velocity of a photon passing through an event, at that event . Big surprise - the result is coordinate dependent in GR, and c for any reasonable coordinates for an inertial frame in SR.
3) A 4-velocity: impossible to define for a photon.

 

[gbfmgbfm]

What is your definition of velocity? When I say it makes no sense as stated I mean that without more qualification it fails to be any of the following:

1) A relative velocity - need two world lines for that, one timelike.
2) A coordinate velocity - well, you could make this apply, but then you 'relative to an event' makes no sense. You could ask for coordinate velocity of a photon passing through an event, at that event . Big surprise - the result is coordinate dependent in GR, and c for any reasonable coordinates for an inertial frame in SR.
3) A 4-velocity: impossible to define for a photon.

I define velocity as velocity. How do you define velocity? Perhaps this is the problem?

"Suppose we define a point (x1=0, x2=0, x3=0, x4=0). What is a photon's velocity relative to that point?"

The answer is definitively c. I never mentioned any gravitational mass, nor any accelerated frames, nor flying toaster ovens for the photon to bounce off of, nor glass for the photon to travel through, nor anything else you might imagine, so there is no need for you to interject anything nor assume anything I do not mention.

If you saw this question on an exam, what would you answer? "Suppose we define a point (x1=0, x2=0, x3=0, x4=0). What is a photon's velocity relative to that point?"

Something other than c?

 

[PAllen]

I define velocity as velocity. How do you define velocity?

This is nonsense. I gave 3 definitions of velocity. You say velocity is velocity. You want to discuss something for which you reject all standard definitions yet refuse to provide your own.

Provide your definition of velocity or there is nothing more to say.