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Does rest mass increase in conformal co-ordinates?

  1. Nov 16, 2015 #1
    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.
     
    Last edited: Nov 16, 2015
  2. jcsd
  3. Nov 16, 2015 #2

    Chalnoth

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    Science Advisor

    Rest mass is a coordinate-invariant quantity, given by [itex]m^2 = P_\mu P^\mu[/itex]. No choice of coordinates has any impact on its value.
     
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