Kinetic energy relative or absolute?

[qftqed]

Hello folks! I have just come from a night at the pub with fellow students of life where over many a pint several mysteries of the universe were discussed. One of my friends asked me a question concerning special relativity that initially took me by surprise and which has me wondering about the nature of energy in relativity theory. The question was this:

How is it that something can be said to have kinetic energy when, in its own reference frame, it has zero velocity? If it too is relative, what are the implications of that?

Initially I didn't have a good response, because I'd never given it any thought, but after a length of time I cobbled together what I think the answer is, but I would love to hear what the answer actually is if it's out there somewhere. My answer is something like this:

Kinetic energy is not relative! It is the work done in accelerating an object from 0 m/s up to its final velocity. Even though a moving object has zero velocity in its inertial reference frame, it needed to accelerate to get there, and this was done with respect to absolute space-time. Or something like that. My mind is a bit boggled at the moment, but I feel like this answer doesn't quite tell the whole story.

So what is the actual state of affairs?

 

[WannabeNewton]

How is it that something can be said to have kinetic energy when, in its own reference frame, it has zero velocity?

This isn't true. It has zero kinetic energy in its rest frame.

Kinetic energy is not relative!

Oh but it is.

It is the work done in accelerating an object from 0 m/s up to its final velocity. Even though a moving object has zero velocity in its inertial reference frame, it needed to accelerate to get there, and this was done with respect to absolute space-time.

This has nothing to do with space-time or even special relativity. Kinetic energy is relative in Galilean relativity as well. If I place a pencil on my desk I have done work against gravity to lift it up from my bag and onto the desk. The pencil is going from zero gravitational potential energy to some non-zero gravitational potential energy given by the height of my desk above the ground and so in its initial and final states it has zero kinetic energy. My doing work doesn't change that-I just need to do work in order to lift up the pencil against gravity.

As an aside, things don't acceleration "relative to space-time". They simply accelerate. This is true both in Newtonian mechanics and in relativity due to the nature of inertia in these theories. The same goes for rotation.

 

[Nugatory]

Kinetic energy is not relative! It is the work done in accelerating an object from 0 m/s up to its final velocity

This isn't a relativity question, it's part of classical mechanics.

Despite the definition of kinetic energy as the work done in accelerating an object from zero meters/second to its final velocity, kinetic energy is still relative - because the definition of zero meters/second is itself relative and therefore everything that follows from it must also be relative. You won't get anything absolute until you look at how the moving object interacts with stuff around it.

An easy example: I shoot a 1000 kg elephant with an elephant rifle that fires a .1 kg bullet at a speed of one km/sec. Total kinetic energy if we take the elephant to be at rest is, by $E_k=(mv^2)/2$, $5x10^4$ Joules. Take the bullet to be at rest and the elephant to be moving towards it at one km/sec, and the kinetic energy is $5x10^8$ Joules.

The absolute quantity here is the amount of energy delivered to the body of the elephant by the collision. That quantity is the same whether we analyze the problem in terms of the bullet hitting the elephant or the elephant hitting the bullet.

 

[qftqed]

Ah, well now it seems obvious haha. I suppose having the question asked in the context of special relativity introduced unnecessary mystery to an otherwise simple problem. Even then, when I think about it, it's simple. Thanks (= case closed!