B Doppler Shift of Neutron: Energy & Momentum

[J O Linton]

TL;DR Summary

How are particles such as neutrons affected by cosmological red shift?

When a photon emitted by a distant galaxy travels across space and is detected at a later time when the universe has expanded by a factor of 2, its wavelength, frequency, energy and momentum are all changed by a factor of 2. If a neutron is emitted with energy E and momentum p and is also detected at a later time when the universe has expanded by a factor of 2, how will the energy and momentum of the neutron be affected?

 

[PeroK]

Summary:: How are particles such as neutrons affected by cosmological red shift?

When a photon emitted by a distant galaxy travels across space and is detected at a later time when the universe has expanded by a factor of 2, its wavelength, frequency, energy and momentum are all changed by a factor of 2. If a neutron is emitted with energy E and momentum p and is also detected at a later time when the universe has expanded by a factor of 2, how will the energy and momentum of the neutron be affected?

You would need to do the calculations for the specific trajectory of the neutron. In general, a low energy neutron would fail to reach the distant galaxy, as it could not overcome the recessional velocity, so there would be a threshold for the initial speed in order for the neutron to complete the journey.

My guess is that the redshift for a photon would specify an upper limit for the energy of the incoming neutron (i.e. for a neutron that starts out at near light speed), but I'd like to see what comes out of the calculations!

 

[Vanadium 50]

I don't think your question is well defined. You can talk about the energy at emission and energy at absorption, but not "the energy of the neutron:".

This was gone over and over in your previous thread,

 

[Reggid]

Summary:: How are particles such as neutrons affected by cosmological red shift?

When a photon emitted by a distant galaxy travels across space and is detected at a later time when the universe has expanded by a factor of 2, its wavelength, frequency, energy and momentum are all changed by a factor of 2. If a neutron is emitted with energy E and momentum p and is also detected at a later time when the universe has expanded by a factor of 2, how will the energy and momentum of the neutron be affected?

If the particle gets emitted by a comoving observer at cosmic time $t_0$ with momentum $p$ and energy $E=\sqrt{p^2+m^2}$ as measured by that observer, then it gets detected by another comoving observer at time $t_1$ with momentum $\hat{p}=p\times \frac{a(t_0)}{a(t_1)}$ and energy $\hat{E}=\sqrt{\hat{p}^2+m^2}$, where $a$ ist the scale factor (in your case $a(t_0)/a(t_1)=2$)

 

[Orodruin]

If the particle gets emitted by a comoving observer at cosmic time $t_0$ with momentum $p$ and energy $E=\sqrt{p^2+m^2}$ as measured by that observer, then it gets detected by another comoving observer at time $t_1$ with momentum $\hat{p}=p\times \frac{a(t_0)}{a(t_1)}$ and energy $\hat{E}=\sqrt{\hat{p}^2+m^2}$, where $a$ ist the scale factor (in your case $a(t_0)/a(t_1)=2$)

Or, in short, the momentum redshifts and the energy still follows the dispersion relation.

 

[J O Linton]

Yes, that is what I thought would happen. It is curious, though, that in this instance, momentum seems to be a more fundamental property than energy. Is there some deep reason for this?

 

[PeroK]

Yes, that is what I thought would happen. It is curious, though, that in this instance, momentum seems to be a more fundamental property than energy. Is there some deep reason for this?

Energy-momentum is a four-vector, so both energy and three-momentum are part of the same physical quantity. Which one you calculate first is up to you.

 

[PeroK]

If the particle gets emitted by a comoving observer at cosmic time $t_0$ with momentum $p$ and energy $E=\sqrt{p^2+m^2}$ as measured by that observer, then it gets detected by another comoving observer at time $t_1$ with momentum $\hat{p}=p\times \frac{a(t_0)}{a(t_1)}$ and energy $\hat{E}=\sqrt{\hat{p}^2+m^2}$, where $a$ ist the scale factor (in your case $a(t_0)/a(t_1)=2$)

It should, of course, be $a(t_0)/a(t_1)=\frac 1 2$.

 

[Orodruin]

Yes, that is what I thought would happen. It is curious, though, that in this instance, momentum seems to be a more fundamental property than energy. Is there some deep reason for this?

The only thing at work here is a spacetime symmetry. As you have already been reminded, associating a momentum with a neutron at all requires that you define an observer relative to which that momentum is measured. In #4 this was done explicitly by introducing comoving observers.

In the particular case of a Robertson-Walker spacetime, the momentum of a particle in free fall is a special quantity as it is related by the scale factor to the conserved quantity related to the Killing fields of the spatial symmetries.

 

[anuttarasammyak]

Yes, that is what I thought would happen. It is curious, though, that in this instance, momentum seems to be a more fundamental property than energy. Is there some deep reason for this?

Proper velocity or proper momentum observed in co-moving coordinate or FLRW metric coordinate would decrease according to inflation of Universe as $pa=const.$ which is common to light red shift $\omega a = const.$

[Moderator's note: Off topic speculation deleted.]

 

[PeterDonis]

Though no reference I have, it seems obvious to me.

The behavior of skaters when they extend their arms is reasonably obvious, yes. Making an analogy between that and the expansion of the universe is not. This speculation of yours is off topic. Please do not post further about it.

 

[PeterDonis]

Moderator's note: Some off topic posts (and part of a post) have been deleted. Please bear in mind the forum rules about personal speculation.