# Massless Particles and the Universal Speed Limit

1. Apr 1, 2014

### Dunbar

I was hoping someone could help me with a simple physics question I've been mulling over, I've failed to find an answer using online queries, I suspect my line of inquiry is flawed in some way. I would be grateful if someone could confirm if I've understood this correctly:

Particles without mass should hypothetically have infinite speed, but experimentally they do not?

2. Apr 1, 2014

### phinds

Why should they have infinite speed?

3. Apr 1, 2014

### Dunbar

Because they have no mass?

I'm aware the particles would blink out of existence instantaneously if they did in fact have infinite speed.

4. Apr 1, 2014

### phinds

Why does a lack of mass imply infinite speed?

5. Apr 1, 2014

### Dunbar

With lower mass it takes less force to accelerate a particle to higher velocities? With zero mass there is nothing to prevent a particle from instantaneously reaching an infinite speed. Hypothetically?

Experimentally, we have observed that there is indeed a speed limit.

btw, thank you for responding, I greatly appreciate it.

6. Apr 1, 2014

### phinds

Yes, it does take less force to accelerate something to a given speed, but I don't see how it then follows that massless implies infinite speed. The thing applying the force has to be traveling at the speed which is imparted to the object. As an object gets lighter, I can propel it with my arm faster and faster but there is a limit to that. If I put it on top of a bomb, I can make it go faster still, but only up to the speed of expansion of the gasses that are expelled when the bomb goes off. Both the bomb and my arm require less and less force as the object being propelled gets lighter, but it still isn't going to travel any faster than my arm or the gasses.

I'm trying to explain something without math that is trivially easy to explain with math, but the problem you come back to is, where does the math come from?

Basically, the math comes from our(*) having made observations and developed the math (General Relativity) that describes what happens in the real world. It doesn't explain why the observations are what they are.

* Actually "our having made ... " is a bit of a stretch since Einstein didn't get any help from me.

7. Apr 1, 2014

### Staff: Mentor

But there is something that prevents a massless particle from reaching an infinite speed...

...and this is it. The thing that prevents infinite speed is that there is a speed limit. The rules the universe must follow are set up in such a way as to prevent anything from traveling faster than c.

8. Apr 1, 2014

### The_Duck

If you use special relativity to calculate the speed of a massless particle, you find that massless particles always travel at speed c, the speed of light.

In special relativity there is a sense in which there *are no* speeds greater than c.

If you try to do the calculation using Newtonian mechanics, this is what you would get. But in fact Newtonian mechanics is wrong at large velocities and needs to be replaced with special relativity. In special relativity, forces, accelerations, and speeds work differently from what is intuitive and familiar. It turns out that massless particles can't actually speed up or slow down, and always travel at c, no matter what forces are applied to them.

Last edited: Apr 1, 2014
9. Apr 1, 2014

### Staff: Mentor

That depends on the hypothesis that you're working with. If you start with $F=ma$, you'll be tempted to set $m$ equal to zero and conclude that any non-zero force leads to an infinite acceleration and hence infinite speed.

However, $F=ma$ is a simplification (used when teaching physics to students who have not yet encountered calculus) of the actual relation, $F=\frac{\mbox{d}p}{\mbox{d}t}$ where $p$ is the momentum. The momentum in turn is connected not to the velocity, but to the total energy, by the relationship $E^2=(m_0c^2)^2+(pc)^2$, which makes it clear that a massless particle ($m_0=0$) can still have non-zero momentum without infinite velocity.

It's worth noting that $F=\frac{\mbox{d}p}{\mbox{d}t}$ and $E^2=(m_0c^2)^2+(pc)^2$ reduce to the familiar $F=ma$ and $E_k=mv^2/2$ for particles of non-zero mass moving at non-relativistic velocities with constant acceleration. That is, the physics you've already learned isn't wrong, it just doesn't extend to massless particles and relativistic velocities.

10. Apr 1, 2014

### Matterwave

Isn't that a postulate of special relativity, rather than something which is calculated?

11. Apr 1, 2014

### Staff: Mentor

No, the postulate that special relativity makes is that "light is always propagated in empty
space with a definite velocity $c$ which is independent of the state of motion of the emitting body".

The impossibility of traveling faster than $c$ is derived from that postulate and some other even more reasonable assumptions.

12. Apr 2, 2014

### Dunbar

Thank you all for taking the time to reply, my knowledge of physics is very limited, I appreciate your patience.

Momentum's connection to total energy requires a formula that evokes c? In Newtonian mechanics light has mass, therefore its speed is not a universal limit. Its speed is simply a property of light. A massless particle's speed has no relevance to c in a Newtonian model? So, the simplified formula, $F=ma$, is the still the most appropriate for massless particles?

(I understand I've probably got this wrong, and ultimately it's all moot; Newton's models aren't supported by observations of relativistic speeds.)

In addition, would I be right in saying: in special relativity a massless particle is travelling at infinite speed in its own frame of reference?

13. Apr 2, 2014

### Staff: Mentor

Yes... And you might find it interesting to see what that formula says about an object that is at rest ($p=0$).

In classical mechanics, there are no massless particles, so classical mechanics says nothing about them. Also in classical mechanics light does not have mass; there's no contradiction here because in classical mechanics light is a wave not a particle, and no one expects a wave to have mass or be subject to forces.

No. Massless particles don't have a frame of reference at all.

One thing that may be confusing you here is our bad habit of talking about "the frame of reference of <something>". Frames don't belong to objects, objects don't own frames, so strictly speaking we shouldn't be using the word "of" here to suggest that the object has its own reference frame. When you see the words "the frame of reference of <something>", you should read that as convenient shorthand for the more precise "a frame in which <something> is at rest". There are no frames in which a massless particle is at rest.