Mass, Energy, and Space-Time: How Do They Interact?

[sqljunkey]

If I had two objects with the same mass, but one had more energy than the other, would it curve space-time more than the other?

 

[Nugatory]

It depends on what you mean by “more energy”. Can you describe a specific example of what you’re thinking about?

 

[sqljunkey]

thermal energy or mechanical energy

 

[Dale]

If I had two objects with the same mass, but one had more energy than the other, would it curve space-time more than the other?

No.

If you are talking about kinetic energy then this is just the same stress energy tensor in a different frame. If you are talking about any other energy then it is a self contradiction.

 

[Nugatory]

And do you mean that we have a spacetime containing two objects, or are you asking for a comparison between a spacetime containing one system with non-zero thermal energy and another with non-zero mechanical energy?

In any case, you should spend some time considering something you were told in an earlier thread: any energy that can be made to vanish by a coordinate transformation is not going to contribute to curvature.

 

[PAllen]

If you add thermal energy to a body, then it will have more effective mass than it did before and curve spacetime more (its mass measured from afar by test bodies will be greater). Similarly, if you compress a spring, it will have more mass and curve spacetime more. In SR, it would be said that the invariant mass of the body has increased by the energy added as measured in the body's rest frame. In GR, it becomes incorporated in the stress energy tensor, but measurement from afar would show an increase, and all of the special case GR masses would apply to this case, and all would increase (Komar, Bondi, and ADM).

The following paper even reanalyzes known data to support this statement observationally:

https://arxiv.org/abs/gr-qc/9909014

 

[sqljunkey]

I just wanted to revisit this for a bit since I was reading something about negative heat capacity. If I take away kinetic energy ( or slow down the system) that would mean I added more energy right? That means if I "heat up" a system like this it will get cooler but the perceived mass would be greater.

Now does that mean any kind of energy added to a system, will get "stored" in the curvature of space-time? And does that mean that after a while of adding energy to a system that system would cease to have any kinetic energy at all?

 

[Nugatory]

If I take away kinetic energy ( or slow down the system) that would mean I added more energy right?

No. The kinetic energy represented by the speed of the system is frame-dependent (you can always choose to analyze the system using a frame in which the kinetic energy is zero, and if you change the speed of the system that just means that you'll use a different frame if you want to analyze the system that way) and is completely unrelated to the heat capacity of the system.

 

[PAllen]

I just wanted to revisit this for a bit since I was reading something about negative heat capacity. If I take away kinetic energy ( or slow down the system) that would mean I added more energy right? That means if I "heat up" a system like this it will get cooler but the perceived mass would be greater.

Now does that mean any kind of energy added to a system, will get "stored" in the curvature of space-time? And does that mean that after a while of adding energy to a system that system would cease to have any kinetic energy at all?

@Nugatory already corrected your "or slow down the system" comment. As to the rest, they key is the net energy flow. Note the example I gave of a compressed spring versus uncompressed. There is not necessarily any change in temperature (average microscopic kinetic energy), but energy has been added to compress the spring, nonetheless. The negative heat capacity examples involve another energy 'reservoir' such that the increase in average kinetic energy in the COM frame is associated with a decrease in total energy. In the case of a gravitating system, this extra reservoir is (at least when you can use a Newtonian approximation) the potential energy. Thus, because energy has left the system, its effective gravitational mass decreases even though average kinetic energy increased.

This is all part of why the correct approach for gravity is to use the stress energy tensor to describe the system. Then, to define the effective gravitational mass of a non-stationary system (e.g. one emitting radiation), you use the Bondi mass (not the ADM mass). This will show the decrease in effective mass that would be measured by test particles as the radiation goes past them.