- #1
jcap
- 166
- 12
Title should be: Does rest mass increase in the FRW metric?
The flat FRW metric can be written in conformal co-ordinates:
$$ds^2=a^2(\eta)(d\eta^2-dx^2-dy^2-dz^2)$$
where ##\eta## is conformal time. Let us assume that ##a(\eta_0)=1## where ##\eta_0## is the present conformal time.
Now the energy of a massive particle ##E## is given by:
$$E=P^\mu V_\mu=mg_{\mu \nu}U^\mu V^\nu$$
where the 4-momentum of the particle is ##P^\mu=mU^\mu## and ##U^\mu##,##V^\nu## are the 4-velocities of the particle and observer respectively.
Let us assume that both the particle and the observer are co-moving at the same conformal time ##\eta##. Therefore the spatial components of their 4-velocities are zero. As the 4-velocities must also be normalised we have:
$$g_{00}U^0 U^0=g_{00}V^0V^0=1$$
Therefore the 4-velocities of the co-moving particle and observer are given by:
$$U^\mu=V^\mu=(\frac{1}{a(\eta)},0,0,0)$$
Thus the energy ##E## of a co-moving particle at time ##\eta##, as measured by a co-moving observer at time ##\eta##, is given by:
$$E = m\ g_{00}\ U^0\ V^0=m\ a^2(\eta) \frac{1}{a(\eta)} \frac{1}{a(\eta)}=m$$
Thus, using this definition of energy, the energy of individual co-moving massive particles is constant. Therefore, for example, we can say that the mass density of cosmological "dust", used in the Friedmann equations, simply goes like ##\rho_m \propto 1/a^3##. This is the conventional viewpoint.
But we can define an energy ##E_0## which is the energy of a comoving particle at time ##\eta## with respect to a comoving observer at the present time ##\eta_0## when ##a(\eta_0)=1##:
$$E_0 = m\ g_{00}\ U^0\ V^0=m\ a^2(\eta) \frac{1}{a(\eta)} \frac{1}{1}=m\ a(\eta)$$
My question is this: Perhaps ##E_0## is the correct energy for a comoving particle with respect to the co-ordinates in which the metric is expressed?
It seems to me that the co-ordinate system, used in the FRW metric above, is the system of co-ordinates that corresponds not to an arbitrary co-moving observer at time ##\eta## but to ourselves who are co-moving observers at the present time ##\eta_0##.
(In standard co-ordinates in which ##g_{00}=1## both ##E## and ##E_0## are the same. So it seems that one could then argue that the difference between ##E## and ##E_0## does not matter. But I think that in arguing this way one is implicitly assuming that the standard FRW co-ordinate system has a timelike Killing vector which it doesn't have. By contrast when one makes the argument in conformal co-ordinates one is implicitly assuming a conformal timelike Killing vector which is in fact a correct assumption.)
Thus we should say that the rest mass/energy of a comoving massive particle at time ##\eta## is given by ##E_0=m\ a(\eta)##.
This would have consequences for Einstein's field equations ##G_{\mu \nu}=8\pi G\ T_{\mu \nu}## when applied to cosmology.
Newton's gravitational constant ##G## in natural units, where ##\hbar=c=1##, is given by:
$$G = \frac{1}{M_{Pl}^2}$$
where ##M_{Pl}## is the Planck mass.
If rest masses are increasing with the scale factor ##a## with respect to the co-ordinate system then the Planck mass should also increase with the factor ##a##. This would imply that Newton's constant actually varies with the scale factor:
$$G \propto \frac{1}{a^2}$$
Thus the Friedmann equations, in either conformal or standard co-ordinates, would have to be modified.
The flat FRW metric can be written in conformal co-ordinates:
$$ds^2=a^2(\eta)(d\eta^2-dx^2-dy^2-dz^2)$$
where ##\eta## is conformal time. Let us assume that ##a(\eta_0)=1## where ##\eta_0## is the present conformal time.
Now the energy of a massive particle ##E## is given by:
$$E=P^\mu V_\mu=mg_{\mu \nu}U^\mu V^\nu$$
where the 4-momentum of the particle is ##P^\mu=mU^\mu## and ##U^\mu##,##V^\nu## are the 4-velocities of the particle and observer respectively.
Let us assume that both the particle and the observer are co-moving at the same conformal time ##\eta##. Therefore the spatial components of their 4-velocities are zero. As the 4-velocities must also be normalised we have:
$$g_{00}U^0 U^0=g_{00}V^0V^0=1$$
Therefore the 4-velocities of the co-moving particle and observer are given by:
$$U^\mu=V^\mu=(\frac{1}{a(\eta)},0,0,0)$$
Thus the energy ##E## of a co-moving particle at time ##\eta##, as measured by a co-moving observer at time ##\eta##, is given by:
$$E = m\ g_{00}\ U^0\ V^0=m\ a^2(\eta) \frac{1}{a(\eta)} \frac{1}{a(\eta)}=m$$
Thus, using this definition of energy, the energy of individual co-moving massive particles is constant. Therefore, for example, we can say that the mass density of cosmological "dust", used in the Friedmann equations, simply goes like ##\rho_m \propto 1/a^3##. This is the conventional viewpoint.
But we can define an energy ##E_0## which is the energy of a comoving particle at time ##\eta## with respect to a comoving observer at the present time ##\eta_0## when ##a(\eta_0)=1##:
$$E_0 = m\ g_{00}\ U^0\ V^0=m\ a^2(\eta) \frac{1}{a(\eta)} \frac{1}{1}=m\ a(\eta)$$
My question is this: Perhaps ##E_0## is the correct energy for a comoving particle with respect to the co-ordinates in which the metric is expressed?
It seems to me that the co-ordinate system, used in the FRW metric above, is the system of co-ordinates that corresponds not to an arbitrary co-moving observer at time ##\eta## but to ourselves who are co-moving observers at the present time ##\eta_0##.
(In standard co-ordinates in which ##g_{00}=1## both ##E## and ##E_0## are the same. So it seems that one could then argue that the difference between ##E## and ##E_0## does not matter. But I think that in arguing this way one is implicitly assuming that the standard FRW co-ordinate system has a timelike Killing vector which it doesn't have. By contrast when one makes the argument in conformal co-ordinates one is implicitly assuming a conformal timelike Killing vector which is in fact a correct assumption.)
Thus we should say that the rest mass/energy of a comoving massive particle at time ##\eta## is given by ##E_0=m\ a(\eta)##.
This would have consequences for Einstein's field equations ##G_{\mu \nu}=8\pi G\ T_{\mu \nu}## when applied to cosmology.
Newton's gravitational constant ##G## in natural units, where ##\hbar=c=1##, is given by:
$$G = \frac{1}{M_{Pl}^2}$$
where ##M_{Pl}## is the Planck mass.
If rest masses are increasing with the scale factor ##a## with respect to the co-ordinate system then the Planck mass should also increase with the factor ##a##. This would imply that Newton's constant actually varies with the scale factor:
$$G \propto \frac{1}{a^2}$$
Thus the Friedmann equations, in either conformal or standard co-ordinates, would have to be modified.
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