# Questions about kinetic energy and gravitation

1. Mar 10, 2007

### Zeit

Hello everybody,

Let’s say a rocket which is at rest relatively to an asteroid. The rocket engine start and the rocket is launched toward the asteroid’s neighborhood. When the rocket engine is working, the rocket accelerates. Few minutes later, the rocket engine turns off and the rocket is now travelling at 0.5c, let’s say.

1) My first question is : the rocket engine is used to propulse the rocket forward, the motion of the rocket is due to the reaction of the ejection of the fluid from within the rocket engine. When the rocket reaches 0.5c, the rocket’s reference frame becomes galilean and an observer at rest in the rocket would considers that the rocket is motionless and that it’s the asteroid which is moving. However, the asteroid is more massive than the rocket, so its kinetic energy is higher than the rocket’s one, since they have relatively to each other the same speed. Where does the energy of the asteroid come from?

2) My second question is : GR tells us that energy has gravitational effects and that the “amount” of energy is proportional to the gravitational force (even if there is no gravitational force in GR). So, I would say that the asteroid has more effective gravitational effects when it’s moving rather than when it’s at rest. Is it correct? Does this explain the difference between the GR’s calculation and the newtonian calculation of light bending, the fact to consider not only mass but energy too?

It makes so much time that I wonder about these questions. There are the result of my misunderstanding of the subject and I would like to clear that up for once. Thanks for your help.

Zeit

2. Mar 10, 2007

### pervect

Staff Emeritus
From a moving frame, the asteroid appears to have more energy. This energy doesn't "come from" anywhere. Assuming the asteroid is an isolated system, its energy and momentum transform as a 4-vector. The norm of this vector (the mass of the asteroid) stays constant. But the energy and momentum do not stay constant.

It's simpler to look at it from the Newtonian POV. The energy in any given inertial frame stays constant, but this does not mean that if one switches frames that the energy is the same in the new frame as in the old.

Both energy and pressure contribute to the gravitational field. One can get the total mass of a stationary object via an integral related to the "gravitational force", as per http://en.wikipedia.org/wiki/Komar_mass. One actually integrates not the gravitational force, but the "force at infinity" to get this answer.

However, one can't write down such an integral for a single moving mass. One major problem is even defining the meaning of "gravitational force" under such circumstances. In a static metric, there is a natural defintion of "holding an object in place" and of the force required to do this. In the non-static metric of a moving mass, there isn't any such natural defintion.

You are basically getting into the issue of the defintion of energy and its conservation in GR. This is a rather advanced subject - you shouldn't expect to understand it fully unless you have a lot more background. It turns out that energy as a conserved quantity in GR can only be defined in special cases, and that a key element to a sucessful defintion of energy in GR is Noether's theorem, which relates conserved energies to time translation symmetries.

3. Mar 10, 2007

### Mentz114

Good question. It this apparent paradox that led Einstein to discover that E=mc^2. If you measure energy as 'rest energy' + kinetic energy then that
quantity does not change under special relativistic velocity addition.

Consult a textbook on special relativity for a full account. Or browse the forum Special relativity section.

I'd like to point out that you've misunderstood the term Gallilean. A non-accelerating frame is usual called 'inertial'.

[Pervects post has landed simultaneously with mine]

4. Mar 10, 2007

### Zeit

Well, I understand now why I was confused, even if I'm not quite sure how the asteroid "gains" its kinetic energy. Probably I should wait a bit more to have a "richer" background .

Yes, I haven't enough knowledge to expect to understand what SR and GR imply. That's why, even if it's not the first time I hear about 4-vector and Killing vector, I can't understand fully what you're talking about. But it's so interesting Thanks again.

What does it mean? The "total" energy (which is $\gamma m c^{2}$ if I'm not wrong) is an invariant? Would you like to tell me more about this please?

Do you have anyone to suggest me? I've read Einstein's book about relativity, but even if I haven't all the background needed to fully understand SR, this book was only speaking about Lorentz-tranformations, a bit about velocity addition and Minkovskian space, but not anything else. I would appreciate to have a more in-depth textbook, to learn a bit more about relativity.

Thanks a lot to point me this. As you have probably seen, I'm not pretty at ease to communicate in English, so I thought "gallilean" and "inertial" were equal, as they are in french.

Zeit