Can Massive Particles Ever Reach the Speed of Light?

[masudr]

I don't have any SR texts to hand, and was wondering if someone could prove the following conjecture:

all particles with positive rest-mass must move at speed less than c.

I understand that there are a multitude of physical reasons that support the above conjecture. But is there a good (mathematical) reason why the boundary condition at $\sigma = 0$ (not $\tau,$ as that becomes a useless parameter in this case) the particle cannot be traveling at c already?

Thanks in advance.

 

[HallsofIvy]

Surely that's a misprint- you meant "if someone could prove that all particles with positive rest-mass must move at speed less than c".

An object with positive rest-mass, moving at speed c would have infinite gravitational mass, disrupting the entire universe!

 

[masudr]

Thank you for the correction; I have subsequently corrected my original post.

I am aware of the various physical arguments for the above conjecture. Howeever, I would prefer an answer relying on the mathematical machinery of SR only...

 

[robphy]

There are probably many demonstrations of this... which may or may not be satisfactory to you.

One approach is this:
a particle with positive rest-mass has a future-timelike 4-velocity, which means that its tangent vector always points into the interior of its future light cone... it can't travel at the speed of light.

Another approach:
in terms of the rapidity parameter, that is, the Minkowski arc-length on the future unit hyperbola,
the asymptotic point corresponding to its light-cone is infinitely far away from any point on that hyperbola, corresponding to the tip of a future-timelike 4-vector.

 

[HallsofIvy]

The "mathematical machinery" of SR? Like the Lorenz transformation formulas? That's exactly what says that if a body with non-zero rest mass were to go at light speed, its relativistic mass would be infinite. No such infinite mass has ever been observed!

 

[JustinLevy]

I am aware of the various physical arguments for the above conjecture. Howeever, I would prefer an answer relying on the mathematical machinery of SR only...

Well, the "infinite gravitational mass" argument doesn't really answer your question because you said: is there a good (mathematical) reason why the ... the particle cannot be traveling at c already?

To answer your question, there is nothing in SR's postulates that prevents "faster than light" particles. In fact, they have been written about quite a bit (see http://en.wikipedia.org/wiki/Tachyons for an introduction). However if you allow "physical arguments", I believe they would destroy an "invariant" concept of causality.

Some consider the notion of time order so important in physics, that they feel this DOES count as violating SR ( http://scienceworld.wolfram.com/physics/Tachyon.html ). However, many believe the fundamental equations of physics make no distinction of time order (all current fundamental laws already have this feature). So I would argue that strictly SR doesn't require the non-existence of tachyons (its postulates don't preclude it), but to preserve an "invariant" causality they would require you to adopt the notion of a "preferred" frame. Some physicists even go so far as to argue that since the physics would still be the same in each frame, it is just a "reinterpretation" of events, not a break of causality (but that sounds to me more like redefining the word).

 

[jtbell]

Using the definitions of relativistic momentum and energy

$$p = \frac{m_0 v}{\sqrt{1 - v^2/c^2}}$$

$$E = \frac{m_0 c^2}{\sqrt{1 - v^2/c^2}}$$

one can derive the following expression for the speed:

$$\frac{v}{c} = \frac{pc}{E}$$

Going a step further, using $E^2 = (pc)^2 + (m_0 c^2)^2$, gives us

$$\frac{v}{c} = \frac{\sqrt{E^2 - (m_0 c^2)^2}}{E} = \sqrt{1 - \left(\frac{m_0 c^2}{E}\right)^2}$$

Therefore, if $m_0 > 0$, then $v/c < 1$, that is, $v < c$.

 

[masudr]

Thanks for all your responses. I have some further questions.

robphy:
I liked your first approach (and not entirely sure if I understood your second approach). However it led to the following question: why must all massive particles move on timelike paths?

jtbell:
Haven't your definitions for p and E assumed that the particle was massive in the first place (thus ensuring that v < c)? I ask this because those definitions of p and E don't work for relativistic particles.

---

I understand that particles on timelike/spacelike/null paths remain on timelike/spacelike/null paths. I suppose, then, my new question is: is there a mathematical reason (in the context of SR kinematics only) why all massive particles are traveling along timelike paths in the first place?

 

[jtbell]

Haven't your definitions for p and E assumed that the particle was massive in the first place (thus ensuring that v < c)? I ask this because those definitions of p and E don't work for relativistic particles.

I think in the second sentence you meant to say "massless" rather than "relativistic."

Nevertheless, your original question was specifically about massive particles!

was wondering if someone could prove the following conjecture:

all particles with positive rest-mass must move at speed less than c.