# Why can't we achieve c

1. Apr 16, 2006

### shaviprem

I have heard that you need an infinite amount of energy to achieve the speed of light. But if you have a particle accelerator and if it exerts a constant force (by applying an electric field) why won't the electrons we have reach c. i have also heard that electrons will travel upto something like 0.9995 c. but won't simply achieve c.
Thanx in advance for the replies

2. Apr 16, 2006

### Giulio B.

As you sayd....you would have to impress an infinite energy to make a particle reach c... so, an accelerato would be able to do that just in a infinite time

3. Apr 16, 2006

Staff Emeritus

A constant force doesn/t impose a constant change of speed in relativity. The physics is just different and involves c, so that the closer you get to c the greater force it requires to add a delta-v, and in the limit as v approaches c, the force required goes to infinity.

A more detailed answer requires showing you the math. I would do this, but it's been done many times before on this forum, and you should look up the links we provide.

4. Apr 16, 2006

### arildno

The simplified view is that an object's resistance to change in its state of motion (that's what you'd call inertial mass) increases with its velocity. Thus, a constant force acting upon it would be less and less effective in changing the object's state of motion as its speed picked up.

In a Newtonian perspective, a given (material) particle ALWAYS have the same inertial mass, in contrast to the relativistic perspective.

Last edited: Apr 16, 2006
5. Apr 16, 2006

### bernhard.rothenstein

relativistic motion

dp/dt=F
where p is the relativistic momentum.

6. Apr 16, 2006

### masudr

I'd say it's:

$$\frac{d}{d\tau}p^\mu = f^\mu$$

7. Apr 16, 2006

### pervect

Staff Emeritus
One does not need to argue about "mass" to show why an object cannot travel at the speed of light by accelerating. In fact, this genearlly tends to obscure the physics.

Instead, consider how velocities add in SR. Suppose A is going at .1c relative to B, who is going at .1c relative to C, and so and.

The relativistic velocity addition formula tells us that the velocity from A to C is not given by (.1+.1) but instead by

v = v1 + v2 / (1+v1*v2)

A close inspection of the formula revelas that no matter how many times we add .1c to a velocity, that that velocity will be less than 'c'.

This is why one can accelerate indefinitely at any desired acceleration, and never reach the speed of light.

8. Apr 16, 2006

### arildno

True enough; but I don't think citing an "unobvious" formula for velocity addition is any more explanatory to a novice than saying the object's resistance to change of its state of motion increases as its velocity increases.

9. Apr 16, 2006

### pervect

Staff Emeritus
There are several reasons I prefer to use the velocity addition formula. Basically, it addresses the problem in terms of kinetmatics - how we describe motion, independent of forces and masses. It illustrates that the problem of reaching c is kinematical in nature, and not dynamical. People who are confused about the issue often think they can get around the limit of the speed of light by reducing the mass of the object somehow. This won't work, and the velocity addition explanation explains why.

The additional concepts needed for dynamics (masses and forces) can be introduced at a later date. The dynamical explanation for the speed of light limit is superfically attractive at first glance, but leads to later confusion - see any of the threads about "relativistic mass" vs "invariant mass" for the sorts of confusion generated.

Basically, it's better to treat dynamics properly, than to give an incorrect half-baked introduction to relativistic dyanamics to students too early, one that basically has to be "unlearned" because it was not properly built in the first place. The problem of why c is the limiting velocity does not have to use any of the concepts of dynamics at all - the concepts of kinematics are sufficient.

Last edited: Apr 16, 2006
10. Apr 16, 2006

### masudr

When I was learning this stuff, the thing that helped me the most was this:

$$E = \gamma mc^2, \mbox{ not } E = 1/2 mv^2 \mbox{ and}$$
$$p = \gamma mv, \mbox{ not } p = mv.$$

So energy and momentum are still conserved, but just because you keep increasing the energy and/or momentum of a particle doesn't mean it's speed will keep on increasing as non-relativistic physics will imply.

11. Apr 16, 2006

### bobbytkc

without using equations, I think the key idea to be implanted is that of the asymptote. You can carry on accelerating a particle indefinitely supposing unlimited resources, but each incremental addition would be smaller and smaller, so its velocity would get bigger and bigger ,true, but it will never reach c.

for example the sum of the series 1+0.1+0.01+.... keeps getting bigger, but will never reach 1.12. 1.11 is bigger than 1.1, and 1.111 is bigger than 1.11, so something can increase indefinitely, and be indefinitely close but never quite reach a figure, because each increment is smaller than the one before.

The mathematics is slightly more complicated in SR but the idea is the same. The theory says that each incremental addition to its velocity due to a force is smaller than the previous incremental addition, no matter the magnitude of the force, so the velocity WOULD increase indefinitely, and would approach a value c, just never reaching it.

Last edited: Apr 16, 2006
12. Apr 17, 2006

### arildno

Fair enough.

13. Apr 17, 2006

### SizarieldoR

So, from Newtonian point of view, a photon moves with the highest speed c, because it has no mass (and thereso F=m.a predicts that even the smallest force would make it accelerate to c)? Or am I wrong?