# Mass at equator versus mass at pole

1. Nov 29, 2009

### edpell

Has anybody ever done an experiment to measure the mass of an object at the equator and then at the pole to see if they differ due to the higher velocity at the equator?

Or at least during the part of the day when the Earths rotation velocity and the Earth sun orbit velocity add up. I guess a lesser mass when the two are out of phase.

Last edited: Nov 29, 2009
2. Nov 29, 2009

### mgb_phys

There is a noticable effect - but it's nothing to do with relativity, it's just centrifugal force

3. Nov 29, 2009

### edpell

I am asking about mass not weight. Yes I agree with you there is a centrifugal effect on the weight.

4. Nov 29, 2009

### bcrowell

Staff Emeritus
In terms of time dilation, there is no difference between the pole and the equator. The gravitational and SR effects on time dilation cancel out. The reason for the exact cancellation is that the earth's surface is an equipotential.

The answer in terms of mass depends on what convention you're using. Most modern books use the convention that mass is invariant, but $p=m \gamma v$ rather than p=mv. With this convention, the mass is the same in both cases by definition.

5. Nov 29, 2009

### D H

Staff Emeritus
You appear to be confusing mass and weight, or mass and relativistic mass, I'm not sure which.

Rest mass is an invariant quantity.

6. Nov 29, 2009

### edpell

OK in these terms has anyone measured the rest mass of a given mass M at the pole (in one inertial frame) and then moved the mass M to a new inertial frame in motion with respect to the first (the equator) and again measured the rest mass to show that it is independent of frame. I know it is supposed to be but has it been experimentally tested?

7. Nov 29, 2009

### bcrowell

Staff Emeritus
Well, some books define mass as an invariant and some define it as non-invariant. It's a matter of convention. Einstein originally defined it as non-invariant, but changed his mind later. The invariant rest mass convention is more common in modern books, and in books that are aimed at a higher level.

To rephrase the OP's question in a way that doesn't depend on the choice of convention, say we transfer 1 kg worth of mass from the pole to the equator. Does this change the earth's gravitational or inertial mass? Since the earth's surface is an equipotential, the transfer of the mass can be accomplished without any input of energy. By equivalence of mass and energy, this means that its gravitational and inertial mass are unaffected.

To get an example where there is an effect, suppose I drop a rock off a cliff. The rock hits the ground at the bottom, and all the energy ends up as heat, which is eventually radiated out into space. The earth's gravitational and inertial mass have been reduced, because mass-energy has been removed in the form of radiation. The externally measured gravitational field of the earth is smaller than it used to be.

8. Nov 29, 2009

### bcrowell

Staff Emeritus
This seems to me to be a completely different question than your initial one. You seem to be asking essentially whether anyone has searched for violations of Lorentz invariance by measuring whether the mass of atoms is different when both the observer and the atom are in a different inertial frame than before, but still at rest relative to one another. I think the answer is basically yes, but in reality, high-precision tests of Lorentz invariance may not give tests of this kind of thing that are really as conceptually simple as what you have in mind. If fundamental constants like the mass of the electron are going to be frame-dependent, then you can't necessarily pin down which constant has changed unless you can measure something unitless like the fine structure constant. People have for example done tests where they look at the spectra of distant galaxies and tried to tell whether the fine structure constant was the same as it is here and now.

9. Nov 29, 2009

### edpell

Yes a test of Lorentz invariance done on an object with mass. It seem like the Michelson Morley experiment nicely shows invariance (OK an invariance of speed not of rest mass) for a mass less photon but I would like to see results for an object with mass (for invariance of rest mass or further that there is no preferred frame).

The fine structure constant looks like a good way to see if electric charge varies with redshift. It would be great to find something like this for mass of a particular kind of particle.

10. Nov 29, 2009

### bcrowell

Staff Emeritus
I think if you look at something like the ratios of wavelengths emitted by helium to wavelengths emitted by hydrogen, you'd probably be sensitive to the mass of the proton divided by the mass of the electron. (Ratios of hydrogen wavelengths to hydrogen wavelengths don't test this, because changing the reduced mass just rescales all the wavelengths.) This is something that could be done with astronomical data. I don't know how accurate the measurements are.

The NIST measures all the fundamental constants, such as the mass of the electron. If the various unitless factors that could be derived from these constants were frame-dependent, presumably the NIST would see anomalous effects with periods of one day and one year.