# Flat potentials and mass

1. Dec 14, 2004

Inside a shell of matter I would experience a flat gravitational potential and hence no gravitational force. Is there any experiment I can do, short of leaving the shell, that could allow me to determine the existence of the potential? For example would my extra potential energy show up in the form of mass?

2. Dec 14, 2004

### pervect

Staff Emeritus
Your clocks would tick slower because you were deep in a potential well. But you wouldn't AFAIK have any way of knowing this via local experiments, you'd need to compare clocks with someone outside the potential well.

3. Dec 14, 2004

### pmb_phy

Let's assume we're talking about GR. If the shell were rotating then it would not be true that you'd experience no gravitational force. There would be a gravitational force which would drag you around in circles. If you were in free-fall inside the sphere then you wouldn't experience these forces unless you tried to remain at rest relative to an external observer who is at rest relative to the sphere. You'd think you were in a rotating frame of reference.

If by "mass" you mean "proper mass" then that'd remain invariant. If you mean "relativistic mass" then it'd change as determined by an external observer. This was interpreted by Einstein to be a confirmation of Mach's Principle.

Pete

4. Dec 14, 2004

Staff Emeritus
The situation inside a large nonrotating shell has been used to approximate Mach's principle. The shell represents "the remote stars". Of course there is no first order effect, because of symmetry, but the statement was there is a second order effect which mimics inertia, at least approximately. I have never seen the math for this, and I am not up to calculating it from first principles.

5. Dec 14, 2004

### pervect

Staff Emeritus
My impression is that second order effects of a surrounding sphere exist, but require the sphere to be rotating or otherwise accelerating to be detectable. It's possible to interpret this in terms of Mach's principle, but it's also possible to interpret it just as frame dragging.

If you are just sitting inside a hollow sphere that isn't rotating or accelerating, you won't notice much. If you have communications with an outside observer outside the potential well, you can notice that your clocks are ticking more slowly than his.

If you really want to, you might be able to interpret the more slowly ticking clocks as an xxxxxxx decrease in mass or energy, though I don't think I've seen this done. One way of looking at this interpretation is that energy and time are conjugate variables, so rather than regarding t slowing, you could regard t as being constant and E increasing.

Another way of looking at this is to look at the definitions of the SI units. To define time and distance, all we need to do is to take some cesium atoms along with us. We can use them to define our local time and distance standards.

To define the unit of mass, we need to take a replica of the standard kilogram with us. Being deep in a gravity well, if the necessary symmetries exist to localize energy, we can say that the standard kilogram is "lighter" in terms of total mass/energy as compared to the same standard kilogram outside the potential well.

Thus if we compare units with a distant observer who is not in a well, we find - clocks run slower, distances are shorter, masses are lighter.

But this is all a constant difference of scale - everything is getting "smaller" by the same amount (time, distance, mass). This sort of unit change (a change in scale) doesn['t have any clasical physical consequences.

We don't have a theory of quantum gravity at this point, but I would expect that a measurement of the fine structure constant would be the same inside the sphere as out, so that quantum experiments would also not be able to detect such a change in scale.

Last edited: Dec 14, 2004
6. Dec 15, 2004

mach and the flat potential

If by "mass" you mean "proper mass" then that'd remain invariant. If you mean "relativistic mass" then it'd change as determined by an external observer. This was interpreted by Einstein to be a confirmation of Mach's Principle.

Pete[/QUOTE]

This statement interests me very much as I was under the belief that Einstein was disapointed not to have come up with an explaination for interial mass in terms of GR. Do you happen to have a reference for Einsteins 'confirmation of Mach's Principle'? The mass I was refering to was the intertial mass of an object inside the shell.

Getting back to the shell, if we think of it as an analogy for the universe then taking the radius of the universe to be GM/c^2 the value of the gravitational potential in the 'shell' is -mc^2. Is there anything to this?

7. Dec 15, 2004

### pmb_phy

The term "inertial mass" can be many things. In physics it can mean the quantity $m = m_odt/d\tau$ where $m_o$ is the particle's proper mass or it can refer to $m_o$ which is the intrinsic mass of a particle. Einstein was speaking of m rather than $m_o$ when it came to Mach's principle. See The Meaning of Relativity, Albert Einstein, Princeton University Press, page 100 to page 102.
I don't know. I'm not sure that makes sense since potential is a reference to something. E.g. its a difference in potential between two points. I don't understand where the universe comes into play regarding this and if the universe has something to do with your question then I ask what this difference in potential refers to. I don't know what you mean by -mc^2. This has the dimensions of energy. As such it must be a potential energy. But potential energy is a function of both the mass of the source and the mass of the particle. There is only one mass in that equation. Gravitational potential is defined as gravitational energy per unit mass. Thus potential has the dimensions of (velocity)2.

Pete

Last edited: Dec 15, 2004