# What happens to the energy?

1. Sep 28, 2008

### sanman

Hi,

I'd like to ask what happens to energy/work that is expended to accelerate an object that is already traveling at lightspeed or at some appreciable fraction of lightspeed.

My recollection is that continued application of force results in an asymptotic acceleration that tapers off as the limit approaches lightspeed.

So as the acceleration under the applied force deviates from linearity, where is that difference in kinetic energy going?
Is it being converted into something else? I forget -- is it being converted into radiation, or something?

2. Sep 28, 2008

### Staff: Mentor

Not sure what you mean. If you add energy to a moving object to accelerate it, it gains kinetic energy.
From an outside observer, yes. To a person inside a spacecraft, no - the acceleration stays constant if the force and mass stay constant.
No, the energy in the acceleration is the kinetic energy of the object. It doesn't "go" anywhere else.

Note: kinetic energy is not linear with speed even under Newton's understanding of it.

3. Sep 28, 2008

### Staff: Mentor

The work that you do in accelerating an object that is already moving at near c (in your reference frame) simply goes into increasing the object's kinetic energy, the same as at non-relativistic speeds.

When the object's speed is close to c, increasing its speed by a very small amount increases its kinetic energy by a very large amount. To put it another way, changing the kinetic energy by a given amount (i.e. doing a certain amount of work on it) changes the speed by only a very small amount. Try working out a few examples using the relativistic kinetic energy equation:

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

4. Sep 28, 2008

### Jonathan Scott

Energy is still conserved. The kinetic energy of an object approaching light speed is unlimited. If the rest mass is m, the total energy at speed v is mc^2/sqrt(1-v^2/c^2) and the difference is the kinetic energy (which is of course approximately mv^2/2 for non-relativistic speeds).

One way of thinking about it is that the "relativistic mass" (that is, the mass corresponding to the total energy) increases so that a given force has less and less effect with increasing speed.

5. Sep 28, 2008

### sanman

Hi, sorry if I didn't express myself clearly.

I meant the change in velocity is what's linear, for below-relativistic speeds, but of course that deviates towards the asymptotic behavior as we approach lightspeed.

But you're saying that from the traveler's perspective, they won't notice any deviation from that linearity, and will still perceive the same reduction in travel time from planet A to planet B, as if they really were traveling FTL.

Of course, if they took some round-trip and disembarked back at their origin-point, Planet A, they will then notice that much more time has passed. But purely from their onboard perspective, they will feel that they have traveled that distance at FTL speed. Right?

ie. if the distance between planet A and planet B was 2 lightyears, then they could apply enough acceleration to themselves so that from their perspective, they could cross that distance in just 1 year, for example?

6. Sep 28, 2008

### Staff: Mentor

No, they will not "feel that they have traveled that distance at FTL speed" unless they choose to do the calculations wrong.
There is no "was" in a speed calculation. The distance is or isn't 2 light years. If the distance is 2 light years in one reference frame and 1 light year in another, you have to pick the correct one (not to mention the correct time measurement) to plug into your speed calculation. "That distance" (2 ly) isn't the right distance.

7. Sep 28, 2008

### Staff: Mentor

A related example: You are flying in an airplane from Chicago to Philadelphia. The plane's top speed is 600 mph and the jet stream gives you a tail wind of 200 mph. You calculate that the airplane is flying at 800 mph and then ask us why there is no sonic boom.

8. Sep 28, 2008

### sanman

My apologies, Russ.

I was speaking in terms of the passage of time observed by the traveler.
If the distance appears to be 2 lightyears before they started their travel (ie. from the "stationary" planetary reference frame), and then they start their travel and apply enough acceleration, their velocity would progress along the aforementioned asymptote-line, such that they could cross that distance and experience, say, only 1 year passing while onboard -- right?
That's what I was trying to say.