# Time dilation and tunnelling probability

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1. Mar 12, 2015

### DDov

Assume you have a 2 particles sitting inside a double potential well. Each particle occupies one side of the well and given enough time the particles may both be found on the same side or swap sides (tunnelling). Assume that the well is inside a gravitational field. It is positioned in such a way that one side of the well experiences higher gravitational pull than the other. If the object that generates the field is the earth, then one side is close to the earth whereas the other side is far from the earth.

If we take into account time dilation then the particle close to earth will experience a 'slower' time than the particle away from earth. I realize that the amounts involved are infinitesimally small, but before dismissing the question lets just limit ourselves to the 'though experiment' area (as opposed to what can be done in a lab).

If a particle experiences slower time then the probability (which is a count of something over time) of it tunnelling across the barrier is lower than the particle that experiences 'faster' time. We should then observe that the particles are found more often in the 'slow' time part of the well than in the 'fast' time part of the well. It will appear that they are pulled towards the body that exerts the gravitational field.

Usually we have electrons in potential wells. This is probably not the right particle for this thought experiment since the electric charge would be several orders of magnitude larger than the gravitational pull. We need to use other, not charged particles.

Any thoughts?

Last edited: Mar 12, 2015
2. Mar 12, 2015

### David Horgan

An atomic clock with 10−18 instability.

Science, Volume 341, Issue 6151, pp. 1215-1218 (2013).

or

http://arxiv.org/abs/1305.5869

3. Mar 12, 2015

### wabbit

This is only loosely related to your question, but it does bring to mind Rovelli & Haggard's considerations about equilibrium in general relativity and the idea of thermal time, as discussed for instance in http://arxiv.org/abs/1302.0724]Haggard[/PLAIN] [Broken] & Rovelli :
Death and resurrection of the zeroth principle of thermodynamics
.

Also
- Presumably the calculation of this effect should agree with the classical gravitational pull experienced by the particle? (I find it easier to think of your experiment with a single particle which can tunnel back and forth, rather than two (distinguishable? subject to Pauli exclusion ? ) particles)

Last edited by a moderator: May 7, 2017
4. Mar 12, 2015

### DDov

I think that you have a fair point. I think that the experiment can be simplified to a single particle. There is no reason why 2 should be used. In fact, my original description was unnecessarily complicated. Thanks for clearing this out for me.

Last edited by a moderator: May 7, 2017
5. Mar 12, 2015

### DDov

Yes, it would be nice if the calculation did agree with the gravitational pull. I have not put any numbers into this, but if anybody is willing then it would be great!

Last edited by a moderator: May 7, 2017
6. Mar 12, 2015

### wabbit

Hey you started it ! Now you do the dirty work : )

(That's as good an excuse as I could find to gloss over the fact that I wouldn't know how to do it)

(Naive question : isn't it enough to just include the classical gravitational potential in the one defining the problem, ie one of the two wells is a little deeper - then it would only remain to show that the time dilation interpretation is consistent with the gravitational potential view ? )

Last edited: Mar 12, 2015