# What is the velocity of a photon?

1. Dec 1, 2011

### 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?

2. Dec 1, 2011

### Meir Achuz

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

3. Dec 1, 2011

### 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.

4. Dec 1, 2011

### 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.

5. Dec 1, 2011

### gbfmgbfm

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?

6. Dec 1, 2011

### PAllen

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

7. Dec 1, 2011

### Naty1

no.

locally, the speed of light is always c.

8. Dec 1, 2011

### gbfmgbfm

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

9. Dec 1, 2011

### gbfmgbfm

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.

10. Dec 1, 2011

### PAllen

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

11. Dec 1, 2011

### Passionflower

Indeed.

12. Dec 1, 2011

### gbfmgbfm

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

13. Dec 1, 2011

### PAllen

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.

14. Dec 1, 2011

### robphy

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. $\vec V \cdot \hat t=0)$:
$$\vec V=\frac{ \vec p_{\perp \hat t} } { \sqrt{\vec p_{\| \hat t}\cdot \vec p_{\| \hat t} } }= \frac{\vec P}{ E}$$

15. Dec 1, 2011

### gbfmgbfm

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

16. Dec 1, 2011

### gbfmgbfm

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?

17. Dec 1, 2011

### robphy

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

18. Dec 1, 2011

### gbfmgbfm

How does a photon move relative to the fourth dimension?

19. Dec 1, 2011

### robphy

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

20. Dec 1, 2011

### gbfmgbfm

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