# I Muons magnetic field time dilation

1. Jun 1, 2017

### kodama

muons mean lifetime of 2.2 µs

muons in a strong gravitational field or traveling at relativistic speeds experience time dilation

would a muon in a strong magnetic field, say near a magnetar experience additional time dilation more than a muon in an non-magnetic field with the same gravitational strength, accounting for the stress-energy tensor of the magnetic field?

2. Jun 1, 2017

### Staff: Mentor

A strong magnetic field has a high energy density which contributes to the gravitational potential. Yes, but the effect is tiny.

3. Jun 1, 2017

### kodama

but is it greater than a muon in gravitational potential of equal strength, but no magnetic field

4. Jun 1, 2017

### Staff: Mentor

It does not depend on the field strength, it depends on the potential, which depends on the overall field geometry, and the energy density of electromagnetic fields is small. As an example, the atmosphere of Earth has a density of 1.3 kg/m3 or 1.2*1018 J/m3. To get the same energy density with a magnetic field, you need 1.2 MT (Megatesla). To match the density of water, you need 34 MT.

With 10 GT, you get a density of 4.4*108 kg/m3. That sounds large, but it is smaller than even the neutron star crust density of ~109 kg/m3, and tiny compared to the average density of a few 1017 kg/m3.

5. Jun 1, 2017

### kodama

I understand that.

the reason i am asking is that gravitons are bosons of spin-2, photons gluons w z are bosons of spin-1

dealing only with a QFT description of gravity as gravitons, not curved spacetime as in GR,

is there a deep reason that gravitons bosons by themselves can cause time dilation but other types of bosons like photons gluons w-z by themselves don't cause time dilaton, only via their effects via energy density?

6. Jun 1, 2017

### Staff: Mentor

There is no fundamental difference between photons, gluons, W, Z and all the fermions in that aspect. They all just contribute via their energy.
I don't think "gravitons cause time dilation" is a meaningful description, because it is not the local field strength that matters.

7. Jun 1, 2017

### kodama

here we are describing gravity solely in terms of gravitons on flat QFT, not curved spacetime as in GR.

the interactions of a photon with a charged fermion by itself doesn't cause any time dilation, but interactions between gravitons with fermions via energy does cause time dilation.

so what special properties does a massless spin-2 boson cause time dilation that a massless spin-1 boson does not?

8. Jun 2, 2017

### ChrisVer

Can you please provide a reference dealing with what you are actually asking (the model in which framework we are talking)?

9. Jun 2, 2017

### mitchell porter

10. Jun 2, 2017

### kodama

doesn't it violate the equivalence principle that 1 ev energy worth of massless spin-1 bosons photons and gluons doesn't dilate time as much as 1ev worth of massless spin-2 gravitons for a test particle?

11. Jun 2, 2017

### Staff: Mentor

Who claims that?

You are mixing real and virtual particles here I think.

12. Jun 3, 2017

### mitchell porter

The core question is
If I consult the Feynman Lectures on Gravitation, as suggested in the link I posted, I find Feynman contrasting particle motion in an electromagnetic field and in a gravitational field, as follows: "the gravitational equation has a qualitatively distinct new feature; not only the gradients, but also the potentials themselves appear in the equations of motion" (5.2.1).

From this he derives the gravitational time dilation. It seems to be a "timelike" manifestation of the fact that gravity is sensitive, not just to potential energy differences ("the gradients"), but to the absolute amount of energy ("the potentials themselves").

Meanwhile, I believe this universality of gravity can be deduced quantum mechanically, by considering the amplitude for emitting a massless spin-2 particle in the limit where the particle has zero momentum, we show that the coupling to all forms of energy-momentum is the same (I am referring to "Weinberg's low-energy theorem"). The analogous argument for a massless spin-1 boson only implies charge conservation.

So, my schema for explaining this is, low-energy theorem implies graviton coupling is universal, which implies a kind of absolute sensitivity to energy-momentum, and in particular that the speed of a physical process depends on the gravitational potential.

I am not 100% sure that I have it right, even schematically; and there is probably also a more geometric account of this, based on spin-2 mapping to a metric, but spin-1 only to a connection. But the combination of Weinberg and Feynman, correctly interpreted, surely has the kernel of an answer.

13. Jun 8, 2017

### ChrisVer

I think that as a statement is kind of simplified... for example a scalar charged particle that interacts electromagnetically, will also have the EM potential in its equation of motion (which is given by replacing the partial derivatives by the covariant derivatives including the electromagnetic field). The introduction of covariant derivatives looks pretty similar to how they are introduced in general relativity, but for different topologies (eg in GR you have them introduced for covariance along translations of vectors over the spacetime manifold, while in the Electromagnetism case it does almost the same for covariance along redefinitions of the EM field allowed by the U(1) local symmetry). The EM field in this case looks very similar to the Christoffel symbols (the Connections).
$D_\mu D^\mu \phi = ( \partial_\mu + ig A_\mu )^2 \phi = m^2 \phi \Rightarrow \partial^2 \phi - ig \phi \partial_\mu A^\mu+ A^2 \phi + (\partial_\mu K^\mu)= m^2 \phi$