# Only one speed

1. Jun 9, 2009

### mtitta

Does everything move incrementally at the speed of light?

2. Jun 9, 2009

### DaveC426913

Can you clarify your question?

Everything doesn't move at the speed of light. Only massless particles do.
"Incrementally" means "in discrete steps".

3. Jun 9, 2009

### maverick_starstrider

I think you really need to clarify your questions a bit more and the level of complexity you're kind of thinking in. Depending on how advanced you're talking "only massless particles move at the speed of light" isn't really true and one could argue that everything moves at the speed of light, or at least electrons do. This is because the velocity eigenvalues of the Dirac Equation are always c. Although, in this case the group velocity of the electron wave packet is not. But bringing it back to my initial point I'm not sure if this is the level of discussion you were interested in. If you have no idea what i'm talking about let me know

4. Jun 9, 2009

### DaveC426913

Electrons do not move at the speed of light. No particles with mass move at the speed of light.

5. Jun 9, 2009

### maverick_starstrider

"It may be easily verified that a measurement of a component of the velocity must lead to the results +/-c in a relativistic theory..." -Principles of Quantum Mechanics (Paul Dirac).

Take it up with Dirac. However, these results of the Dirac Equation do not result in a violation of special relativity since, as I already mentioned, the observable of importance is momentum of a particle (not velocity of its constituent basis vectors) and this can be shown to comply with SR.

6. Jun 9, 2009

### maverick_starstrider

You may find it illuminating to wiki phase velocity vs. group velocity vs. signal velocity. Signal velocity never exceeds c, the other two, however can

7. Jun 9, 2009

### DaveC426913

Is this helping the OP?

8. Jun 9, 2009

### maverick_starstrider

Well, as I already asked, I don't know at what level of complexity the OP is. If they're a first year physics major and just confused then probably not but if they're a first year grad student (and some posters are) then yes. yes it is.

9. Jun 10, 2009

### A.T.

One possible geometrical interpretation of the special theory of relativity is that everything advances at c trough space-(proper)time:

10. Jun 10, 2009

### sylas

No; it isn't... and your link suggests nothing of the kind.

The quote from Dirac is about the uncertainty principle; and it does not actually mean an electron moves at the speed of light.

11. Jun 10, 2009

### mtitta

I guess it was a loaded question and I overgeneralized but that is what I do.I like to keep things at the most basic level as possible. So let me be more specific if all matter ultimately consists of subatomic particles and all subatomic particles have wave particle personalities would it be reasonable to think that all matter moves thru space as a wave (at C) but exists at rest as a particle. I am an electronics engeneer and I always found physics facinating so I hope I am not way off base on this one. It just seems logical that nature would keep it simple and only one speed would exist in the universe. ----Mike

12. Jun 10, 2009

### Gan_HOPE326

But even if matter is a wave, it's not an electromagnetic wave, right? So it's not like it has to move at C... I think...

13. Jun 10, 2009

### maverick_starstrider

a) that quote has nothing to do with the uncertainty principle. It relates to the velocity eigenvalues of the Dirac Equation (which is like the schrodinger equation for an electron if one considers relativistic effects. Also, it was used by Dirac to posit the existance of antiparticles and is the first time that electron spin has come naturally as a degree of freedom of the solution (as opposed to the schrodinger equation where it can be artificially tacked on to yield the pauli equation)

b) a "particle" is actually a wave packet progressing in time in quantum mechanics (as opposed to a dirac delta function like in classical) and, according to the dirac equation, the propogation speed of each basis function of your electron wavepacket does indeed have a velocity that is ALWAYS +/- c. See the relativistic quantum section of, say, Sakurai for a more complete discussion of this. In general, the velocity operator is considered to be a problematic concept in relativistic QM

However, if your question is aimed at a more basic level then the signal velocity (which is basically what one usually means when talking about velocity in a classical sense) can never be c and the amount of total energy needed to reach a given speed is given by $E=\frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}}$, which you can see has a infinite discontinuity when v=c given that m doesn't equal 0

Last edited: Jun 10, 2009
14. Jun 10, 2009

### sylas

It is related both to the eigenvalues and to the uncertainty principle. If I tried to explain it myself I'd get it wrong; so here is what appears in Dirac's book, where both ideas show up. The quote appears on page 262. In the discussion:
The $\dot{x}_1$ given by (24) has as eigenvalues ±c, corresponding to the eigenvalues ±1 of $\alpha_1$. As $\dot{x}_2$ and $\dot{x}_3$ are similar, we can conclude that a measurement of a component of the velocity of a free electron is certain to lead to the result ±c. This conclusion is easily seen to hold also when there is a field present.

Since the electrons are observed in practice to have velocities considerably less than that of light, it would seem that we have here a contradiction with experiment. The contradiction is not real, though, since the theoretical velocity in the above conclusion is the velocity at one instant of time while observed velocities are always average velocities through appreciable time intervals. We shall find upon further examination of the equations of motion that the velocity is not at all constant, but oscillates rapidly about a mean value which agrees with the observed value.

In may easily be verified that a measurement of a component of the velocity must lead to the result ±c in a relativistic theory, simple from an elementary application of the principle of uncertainty of §24. To measure the velocity we must measure the position at two slightly different times and then divide the change of position by the time interval. (It will not do to measure the momentum and apply a formula, as the ordinary connexion between velocity and momentum is not value.) In order that our measured velocity may be approximate to the instantaneous velocity, the time interval must be very short and hence these measurements must be very accurate. The great accuracy with which the position of electron is known during the time-interval must give rise, according to the principle of uncertainity, to an almost complete indeterminacy in its momentum. This means that almost all values of the momentum are equally probable, so that the momentum is almost certain to be infinite. An infinite value for a component of momentum corresponds to the value ±c for the corresponding component of velocity.

Cheers -- sylas