# Why does going faster than the speed of light require infinite energy?

1. Jun 8, 2012

### Reptile202

I honestly can't make any sense of this. I've been told traveling even one time faster the speed of light, the mass becomes infinite; therefore, infinite energy is required.

However, traveling just one time above speed of light isn't an infinite speed. The object isn't constantly increasing in speed, rather it's maintaining a certain speed.

If one was constantly picking up speed, and had no intentions of stopping, I could see why infinite energy would be required.

To make short, does traveling faster than the speed of light actually require infinite energy? Or is the supply just some much, we call it infinite...Perhaps because not even our universe altogether has the needed supply?

2. Jun 8, 2012

Traveling faster than the speed of light does not require infinite energy!
What requires infinite energy is to accelerate a massive body from a subluminal speed to the speed of light. You can approach the speed of light but you will never quite make it, because for each unit of energy you put into body the speed change is less and less than before.
And yes the energy required is infinite, not just a really large number, but true infinity. Physically that means that no matter how much you put in you will never make it.

Now the topic of traveling faster than the speed of light is even worse since only bodies with imaginary mass can travel faster than light. For them, the situation would be opposite. They can travel as fast as they want but they cannot slow down to the speed of light. But of course that's largely irrelevant 'cause, you know, imaginary mass.

3. Jun 8, 2012

### Reptile202

Technically, wouldn't that mean the speed of light, even with infinite energy, is impossible? Because even if you have infinite energy, the more you put into the object, the speed change is going to be less and less regardless of a unlimited supply?

4. Jun 8, 2012

### nitsuj

Yes, oddly that's a better understanding than the first post.

Don't forget that infinite is an undefined quantity, imo better said as; it's a concept. In particular when used in this context.

I am sure you understand the concept of infinity.

There are other variables that you could correctly use, such as you can go the speed of light, but it would require an infinite amount of time / distance / energy. And again back to the point of infinity not being a specific quantity.

One last note, and touching on your last post. It's not specifically a lack of distance, energy or time that "makes it impossible" for a massive object to go c. It's the "mechanics" of the motion of electrodynamic bodies.....

I'm not one to recommend using math to better understand something. But play with an equation that can be used to calculate gamma or whatever, but in this case it would help reduce the idea of "limiting speed" to what it "really" is, and not what particular scenario you can imagine (infinite energy ect).

Last edited: Jun 8, 2012
5. Jun 8, 2012

### Nabeshin

The use of infinity in such statements as 'it requires an infinite amount of energy to accelerate a massive body to c' is simply sloppy language. Most people will sort of intuitively understand what is meant, but that doesn't make it technically correct. The correct statement would be something like, 'the energy required to accelerate a massive object arbitrarily close to c is unbounded.'

6. Jun 8, 2012

### Reptile202

So in actuality, not even infinite energy will make an object go beyond the speed of light? Basically, infinite is just a term used to describe the impossibility?

7. Jun 8, 2012

### nitsuj

But there is no such thing as "infinite" energy. I should still say yes to answer though. For the last question, no it's not.

Infinite in this case describes the "diminishing return" (i have a finance background) of energy input to a "motion" output. You can put in more and more energy into an object, as it gets towards the speed of c you (who is measuring the speed) will notice the object take on different characteristics other than acceleration (and infinitely so), despite efforts to accelerate the object.

Though this isn't a very good train of thought to better understand this concept. If you are interested and can put effort into this, look into calculating gamma via pythagorean theorem.

It's a small exercise for a big improvement in understanding in this fascinating regime.

8. Jun 8, 2012

### Naty1

You likely learned early on that everyone observes 'c' when they measure the local speed of light. A better way to think about the issue is that no matter how fast you go, light always passes you a speed 'c'. You can speed up as much as you like, expend as much energy as you wish accelerating, no matter how fast you peddle your bike nor press down on the accelerator, light will still blaze by you at the same old 'c'.

But exactly why that is the case, why electromagnetic waves travel at 'c' is not at all clear. We have no set of first principles from we can derive the speed of light, the charge of the electron and so forth. For now, they are what we observe.

9. Jun 8, 2012

### Reptile202

Thanks.

I believe I'm starting to make some sense of all of this.

"As the body's speed approaches c, light speed, the mass becomes huge and approaches infinity."

The above is something I found; however, it didn't go into the details I'd like to know. Basically, the mass would start increasing and never stop, which would require infinite energy? That is to say even if you maintained an exact speed around that level, the mass would still increase regardless?

10. Jun 8, 2012

### nitsuj

Yes that's generally correct (the part in quotes) Note there is a difference between the two "masses" here.

"rest mass" & "relativistic mass". With "rest mass" being typical (what's weighed by a scale), and the other what you had referred to above. Someone may mention a more popular term for "relativistic mass".

Your last question the answer is no. And Im not sure what is asked in the first. But maybe a good time to consider the idea that energy doesn't "disappear". If the body is near c and more and more energy is applied to the object it must "go somewhere". In this case an increase in "relativistic mass"/kinetic energy. ( i maybe wrong on the last part regarding increased kinetic energy, just seems right from concepts)

Last edited: Jun 8, 2012
11. Jun 8, 2012

### Reptile202

Sorry for the confusion.

Well, it said the mass would approach infinity; therefore, wouldn't the mass of the object have to start increasing with no end since it's infinite?

12. Jun 8, 2012

### nitsuj

yea for sure but as a result of increased energy input. Again appreciate that this is not Mass in the "at rest" sense, but in the kinetic sense. ( again I'm pretty sure that's right).

13. Jun 8, 2012

### Reptile202

So, even if an object maintained a speed, neither increasing or decreasing, the mass will continue to increase and increase with no end? And in order to keep the object moving, nothing less than unlimited energy will be needed?

14. Jun 8, 2012

### nitsuj

Shoot, that's what I was trying to steer you away from, you hinted this idea in post #9.

if the object moving near c relative to you is inertial (not accelerating) than it's "relativistic mass"/kinetic energy will also remain the same.

If you apply energy in hopes of accelerating the object, you would notice it mostly just gets harder to accelerate (because of increased "relativistic mass"/kinetic energy).

Keep in mind the "law" that something in motion stays in motion.

And for kicks to throw in a "loop", motion is relative. To someone else that object you have moving near the speed of light maybe going pretty slow. A cure for that, is to consider all of this we are discussing is about measurements that take place "across" spacetime.

Last edited: Jun 8, 2012
15. Jun 8, 2012

### Staff: Mentor

If the object maintains its speed, neither increasing nor decreasing, it will maintain the same mass. If you add more energy, it will speed up some and gain some more mass. But it will always get at least a little bit more speed with each increase in energy - you can't turn all the added energy into speed. This may be easier if you actually look at the equations (which are ONLY VALID for speeds below that of light!):

$m=m_{0}/\sqrt{1-(v/c)^{2}}$
$E=mc^{2}$

Where $m_{0}$ is the mass that will be measured when at rest relative to the observer doing the measuring, and v is the velocity of the object when moving. Plug some numbers in, and you'll see that you can't change m without also changing v, and no matter how large you make E and m, you can't get v to come out above c.

16. Jun 8, 2012

### Reptile202

:)

So it's impossible to go faster than light because too much of the energy would be used as mass?

17. Jun 8, 2012

### Staff: Mentor

To be very precise, it is impossible to take an object that has non-zero mass and is travelling at less than the speed of light, and accelerate to a speed equal to or greater than the speed of light.
The more energy you pour into it (push it, strap giant huge rockets to it, or whatever) the more of the added energy will go into increased mass and the less will go into increased velocity. Although it can get arbitrarily close to the speed of light (try plugging v=.999c, or v=.9999c, or v=.99999c into the equations above), it will never quite get there.

There's a different set of equations that govern the behavior of massless particles such as photons. They can ONLY travel at the speed of light, and there's no way of slowing them down.

18. Jun 8, 2012

### DrGreg

In Newtonian physics, when you accelerate an object it gains kinetic energy which is given by$$E = \tfrac{1}{2}m_0 v^2 \mbox{ ..................................(Newton)}$$But this is only an approximation to the true relativistic kinetic energy$$E = m_0 c^2 \left( \frac{1}{\sqrt{1-v^2/c^2}} - 1 \right)\mbox{ .........(Einstein)}$$This value "diverges to infinity", i.e. keeps on increasing without limit, as v gets closer to c. If it were possible for an object (with non-zero rest mass) to reach the speed of light it would have infinite kinetic energy. Where would that come from?

A simpler argument, without mentioning energy, is to note that all observers measure the same value for the speed of light. So if you have reached 90% of the speed of light relative to your starting point, according to you the speed of light is still 299,792,458 m/s, not 29,979,245.8 m/s, as you might expect. In other words you are no nearer your target and still going 299,792,458 m/s too slow. And you remain going 299,792,458 m/s too slow, from your own point of view, no matter how long you accelerate.

19. Jun 8, 2012

### nitsuj

I'd agree with that, seems well said given the context.
(as in the energy "put into" the object in order to try and accelerate it would be measured "more and more so" as an increase in mass as calculated above)

20. Jun 8, 2012

### Reptile202

Ah, thanks!

So the reason for not being able to travel at certain speeds is because the energy is being converted into mass? So really...infinite energy is irrelevant, because it too will be converted into mass as well?