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jonmtkisco
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I'm taking the liberty of revising and restating this topic which started in a separate thread. Comments are welcome.
A lively debate is underway today by mainstream cosmologists as to whether the expansion of the universe implies that empty space between galaxies is also expanding. When faced directly with the question, most cosmologists will say that galaxies are moving apart because they were previously moving apart, but decline to state flatly that space itself expands. And yet it has been customary for textbooks and technical literature to explain both superluminal recession and cosmological redshift only as the result of space itself expanding. What seems clear is that the observational predictions of GR must be precisely identical regardless of whether space itself expands. Therefore, at a minimum we need a comprehensive theoretical description of both superluminal recession and cosmological redshift that does not resort to the concept of expanding space. Here are some thoughts on that subject.
Accurate application of Special Relativity depends on having a global inertial reference frame, which may be arbitrarily selected, but which cannot be accelerating, and by the same token cannot include significant gravitational objects. On the other hand, our universe appears to be homogeneously filled with gravitating matter. This means that instead of one global reference frame, we have an infinite series of tightly packed local reference frames.
Superluminal Recession
In our gravitation-filled universe, the rule of SR that no object can exceed the speed of light, c, relative to any other object, simply doesn’t apply. Objects at rest in any two local reference frames which are in motion relative to each other may have a relative velocity exceeding c. This is true even if the two frames are immediately adjacent to each other.
One might be tempted to call this is a "license to steal", in the sense that the SR speed limit of c doesn't seem to apply hardly anywhere in our universe. But the reality isn't that dire. The degree by which the velocity of an object in a local frame can exceed c relative to any other local frame is dictated entirely by General Relativity’s applicable metric of gravity. If the gravitational density is low, the degree of "violation of the speed limit" in nearby frames is infinitesimal. If the gravitational density is high, this speed limit can be "violated" to a larger degree. Even a low gravitational density enables large violations of the speed limit if the objects are extremely distant from each other, currently in the range of z=1.6.
Consider our very early observable universe, a fraction of a second after inflation is theorized to have ended, which could be visualized as being the total size of a beachball. The FLRW metric (to the extent its equation of state doesn't require modification on account of the then-dominant quark-gluon plasma) calculates that matter particles located just millimeters away from each other were receding from each other at velocities many times faster than c. This demonstrates that a tiny distance between distinct local frames is no inhibitor to observing a massive "violation" of the SR speed limit. All that’s needed is truly astounding gravitational density -- which is what theory calculates for our very early universe.
Note that any pair of particles which are observed to have a given relative recession velocity now haven’t gained relative velocity over time as their mutual distance increased. On the contrary, their relative recession velocity was enormously higher in the very early universe. In early times, the self-gravity of the universe hugely decelerated every galaxy pair's mutual recession rate; in late times, dark energy has reaccelerated them but to a much lesser degree. Absent the competing effects of those two accelerations, each pair of particles would retain the same relative recessionary momentum they had in the very early universe.
One must confront the question whether superluminal recession is a “physically real” phenomenon or just an observational artifact. If it is physically real, then it will continue regardless of how low the cosmic gravity density declines in the far future. Consider a model universe with dark energy [edit: Lambda=0 doesn't work well here]. As time passes, the gravitational density declines, and in the limit will approach zero (except for the gravity of the dark energy itself, which is more than offset by its antigravity negative pressure effect). By then superluminally receding particles (which are, say, at z=3 today) will be many times further apart. Yet this clearly begs the question, when gravitational density drops to zero in a given large but finite region, as it eventually must, how can superluminal relative recession velocities remain possible from one end of that region to the other, regardless of how widely separated the two particles are? Within its bounds, that region has in fact become converted to a true SR-compliant global frame. This logic suggests the theoretical possibility that superluminal recession is not physically real and might be an observational artifact.
If superluminal recession is merely an observational artifact, then the explanation must lie in the cosmological redshift that occurs as photons emitted by the receding particle (or galaxy) transition from each local gravitational frame to the next such frame along their path (keeping in mind that local frames are confined to an infinitesimal point and lack discrete boundaries.)
Cosmological Redshift
In order to explain cosmological redshift without resorting to the expansion of space itself, the only tools left in our kitbag are SR relativistic Doppler Effect and gravitational redshift. Since neither of these effects can do the job alone, the solution seems to lie in combining them properly.
A.B. Whiting may have been on the right track when he derived the gravitational component of cosmological redshift in a universe with static gravitational density by calculating the difference between the matter density now and zero matter density. As he says, just multiplying the SR Doppler redshift and the gravitational redshift together calculates the correct instantaneous cosmological redshift for a flat FLRW universe with static density.
I think the remaining step needed to extend his analysis into a general equation for cosmological redshift is to perform an integration of the SR Doppler redshift at each point between the emitter and receiver, multiplied by an integration of the gravitational redshift at each point between the emitter and receiver (with matter density varying from that at emission to zero now.) Something like this:
[tex]\frac{\lambda_{r}}{\lambda_{e}} = \int\begin{array}{cc} v^{r}\\v_{e} \end{array} SR \ Doppler \ redshift \\\ \int\begin{array}{cc} 0 \\\rho_{e} \end{array}\\\ gravitational \ redshift [/tex]
I want to emphasize that, unlike my earlier attempt at a solution, I do not think a separate element should be included to account for clock rate differentials. The change in matter density as a function of time does not cause any clock rate differential in the homogeneous FLRW metric. In normalized units, the FLRW metric can be simply written as:
ds2 = -dt2 + a2(t)(dx2 + dy2 + dz2)
The cosmic clock (t) is invariant for purely comoving observers as a function of the declining matter density. The cosmic clock is just the timelike spacetime distance orthogonal to a hypersurface of constant comoving physical distance, so:
ds2 = -dt2.
So in the same way that the declining cosmic matter density does not create any gravitational redshift, it also does not create any clock differential between the emitter and receiver.
Conclusion
Hopefully these thoughts are broadly consistent with the eventual solution I think must inevitably be derived to provide an comprehensive alternative explanation of superluminal recession and cosmological redshift without resorting to the concept that space itself is expanding.
Jon
A lively debate is underway today by mainstream cosmologists as to whether the expansion of the universe implies that empty space between galaxies is also expanding. When faced directly with the question, most cosmologists will say that galaxies are moving apart because they were previously moving apart, but decline to state flatly that space itself expands. And yet it has been customary for textbooks and technical literature to explain both superluminal recession and cosmological redshift only as the result of space itself expanding. What seems clear is that the observational predictions of GR must be precisely identical regardless of whether space itself expands. Therefore, at a minimum we need a comprehensive theoretical description of both superluminal recession and cosmological redshift that does not resort to the concept of expanding space. Here are some thoughts on that subject.
Accurate application of Special Relativity depends on having a global inertial reference frame, which may be arbitrarily selected, but which cannot be accelerating, and by the same token cannot include significant gravitational objects. On the other hand, our universe appears to be homogeneously filled with gravitating matter. This means that instead of one global reference frame, we have an infinite series of tightly packed local reference frames.
Superluminal Recession
In our gravitation-filled universe, the rule of SR that no object can exceed the speed of light, c, relative to any other object, simply doesn’t apply. Objects at rest in any two local reference frames which are in motion relative to each other may have a relative velocity exceeding c. This is true even if the two frames are immediately adjacent to each other.
One might be tempted to call this is a "license to steal", in the sense that the SR speed limit of c doesn't seem to apply hardly anywhere in our universe. But the reality isn't that dire. The degree by which the velocity of an object in a local frame can exceed c relative to any other local frame is dictated entirely by General Relativity’s applicable metric of gravity. If the gravitational density is low, the degree of "violation of the speed limit" in nearby frames is infinitesimal. If the gravitational density is high, this speed limit can be "violated" to a larger degree. Even a low gravitational density enables large violations of the speed limit if the objects are extremely distant from each other, currently in the range of z=1.6.
Consider our very early observable universe, a fraction of a second after inflation is theorized to have ended, which could be visualized as being the total size of a beachball. The FLRW metric (to the extent its equation of state doesn't require modification on account of the then-dominant quark-gluon plasma) calculates that matter particles located just millimeters away from each other were receding from each other at velocities many times faster than c. This demonstrates that a tiny distance between distinct local frames is no inhibitor to observing a massive "violation" of the SR speed limit. All that’s needed is truly astounding gravitational density -- which is what theory calculates for our very early universe.
Note that any pair of particles which are observed to have a given relative recession velocity now haven’t gained relative velocity over time as their mutual distance increased. On the contrary, their relative recession velocity was enormously higher in the very early universe. In early times, the self-gravity of the universe hugely decelerated every galaxy pair's mutual recession rate; in late times, dark energy has reaccelerated them but to a much lesser degree. Absent the competing effects of those two accelerations, each pair of particles would retain the same relative recessionary momentum they had in the very early universe.
One must confront the question whether superluminal recession is a “physically real” phenomenon or just an observational artifact. If it is physically real, then it will continue regardless of how low the cosmic gravity density declines in the far future. Consider a model universe with dark energy [edit: Lambda=0 doesn't work well here]. As time passes, the gravitational density declines, and in the limit will approach zero (except for the gravity of the dark energy itself, which is more than offset by its antigravity negative pressure effect). By then superluminally receding particles (which are, say, at z=3 today) will be many times further apart. Yet this clearly begs the question, when gravitational density drops to zero in a given large but finite region, as it eventually must, how can superluminal relative recession velocities remain possible from one end of that region to the other, regardless of how widely separated the two particles are? Within its bounds, that region has in fact become converted to a true SR-compliant global frame. This logic suggests the theoretical possibility that superluminal recession is not physically real and might be an observational artifact.
If superluminal recession is merely an observational artifact, then the explanation must lie in the cosmological redshift that occurs as photons emitted by the receding particle (or galaxy) transition from each local gravitational frame to the next such frame along their path (keeping in mind that local frames are confined to an infinitesimal point and lack discrete boundaries.)
Cosmological Redshift
In order to explain cosmological redshift without resorting to the expansion of space itself, the only tools left in our kitbag are SR relativistic Doppler Effect and gravitational redshift. Since neither of these effects can do the job alone, the solution seems to lie in combining them properly.
A.B. Whiting may have been on the right track when he derived the gravitational component of cosmological redshift in a universe with static gravitational density by calculating the difference between the matter density now and zero matter density. As he says, just multiplying the SR Doppler redshift and the gravitational redshift together calculates the correct instantaneous cosmological redshift for a flat FLRW universe with static density.
I think the remaining step needed to extend his analysis into a general equation for cosmological redshift is to perform an integration of the SR Doppler redshift at each point between the emitter and receiver, multiplied by an integration of the gravitational redshift at each point between the emitter and receiver (with matter density varying from that at emission to zero now.) Something like this:
[tex]\frac{\lambda_{r}}{\lambda_{e}} = \int\begin{array}{cc} v^{r}\\v_{e} \end{array} SR \ Doppler \ redshift \\\ \int\begin{array}{cc} 0 \\\rho_{e} \end{array}\\\ gravitational \ redshift [/tex]
I want to emphasize that, unlike my earlier attempt at a solution, I do not think a separate element should be included to account for clock rate differentials. The change in matter density as a function of time does not cause any clock rate differential in the homogeneous FLRW metric. In normalized units, the FLRW metric can be simply written as:
ds2 = -dt2 + a2(t)(dx2 + dy2 + dz2)
The cosmic clock (t) is invariant for purely comoving observers as a function of the declining matter density. The cosmic clock is just the timelike spacetime distance orthogonal to a hypersurface of constant comoving physical distance, so:
ds2 = -dt2.
So in the same way that the declining cosmic matter density does not create any gravitational redshift, it also does not create any clock differential between the emitter and receiver.
Conclusion
Hopefully these thoughts are broadly consistent with the eventual solution I think must inevitably be derived to provide an comprehensive alternative explanation of superluminal recession and cosmological redshift without resorting to the concept that space itself is expanding.
Jon
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