But the Einstein-Hilbert action seems to imply that CMB cannot lose energy because there is no place where that energy could go. Then the history would lose energy, but that is not allowed in a stationary point of the matter lagrangian L in the Einstein-Hilbert action.
Massive particles in the FLRW universe: they do not lose kinetic energy
The lagrangian for a single massive particle outside of potential fields is the familiar kinetic energy formula.
In the ΛCDM model, it is familarly assumed that as the time passes, fast moving particles "lose" kinetic energy relative to the spatial coordinates which are fixed to "static" expanding matter content.
newtonian explosion
"static" particle "static" particle
<- ● ● • ----> v ● ->
fast particle
This would really be the case if the expansion of the universe would be equivalent to a newtonian explosion where the "static" matter is expanding its volume in Minkowski space. When the fast particle reaches the rightmost "static" particle, the relative speed of the fast particle compared to the "static" particle is less.
Let us have a history H of the FLRW universe, such that fast particles gradually lose speed as the universe expands.
Let m be a fast body of mass such that its density is extremely small. Then the self-potential of m relative to its own gravity field is negligible.
Let v₁ be the initial velocity of m relative to the standard FLRW coordinates, and v₂ its final velocity in the history H.
Let us make the following variation H' of the history H which runs for a coordinate time interval Δt:
1. we keep the metric H constant;
2. we let m move at an almost uniform speed v during Δt, except that the speed at the start is v₁ at the start and v₂ at the end.
How much does the Einstein-Hilbert action change? For the (rest) mass m itself, the volume element
can be taken as essentially constant during Δt, because the stretching of the spatial metric does not affect the rest mass. The same holds for the kinetic energy of m.
The changes in the Einstein-Hilbert action:
1. The kinetic energy action of m in the matter lagrangian L (which is of the form kinetic energy - potential energy) is reduced because the velocity v is almost constant during Δt.
2. The position of m is displaced relative to its own gravity field: the volume element sqrt(-g) obtains a slightly larger value because m does not reside in its own gravity potential well. However, we can make the effect on the action value as small as we want by reducing the density of the body m. Thus, we can ignore this item.
We conclude that the variation H' reduces the value of the Einstein-Hilbert action relative to H. The history H was not a stationary point of the action. We proved that the speed of m cannot change as the FLRW universe expands.
Massless scalar field waves in the FLRW model
The massless scalar field (m = 0) has a very simple lagrangian density formula. Let us have a wave packet. We must determine what history of a wave packet is a stationary point of the action in an expanding FLRW universe.
Let us stretch spatial distances by a factor 2. The naive transformation of keeping the function φ(r) values the same for each coordinate vector r, will make the wave packet volume 8-fold and the squares of the derivatives of φ 1/4-fold. The energy of the wave packet would double. Clearly, we must find a better transformation.
Let us have a history H of the universe for a coordinate time interval Δt. We make the following variation H':
1. keep the metric g the same;
2. do a Noether time variation of the wave packet.
By making the energy density of the wave packet very small, we can ignore its potential in its own gravity field, just as we did in the previous section.
The Noether time variation at the start time speeds up the time development of the wave packet, and the variation at the end slows down the time variation of the wave packet.
Is it a problem that the wave packet moves faster than c when we speed it up? Let us for the moment assume that it is not a problem.
For the Noether time variation to keep the value of the action the same, the "energy" measured by the time variation must stay constant. The energy of the wave packet cannot change.
But ΛCDM claims that the energy of the packet changes as the universe expands.
What about doing a Noether time variation of the entire radiation content of the universe?
"ACuriousMind" on the Physics Stack Exchange claims that since the universe as a whole is not time translation invariant, a Noether time variation will not show that energy is conserved. Does this mean that the history of the universe, H, is not a stationary point of the action? If not, then how do we decide which is a valid possible history?
Our own argument above only does a local Noether time variation. There is nothing which prevents us from doing that. And the laws of nature which govern a local wave packet are time translation invariant.
A precise variation which changes the action value in the massless scalar field wave example
H history
| |
| |
• • absorptions
\ /
\ / decay into two wave packets
/
/
• --> v
m massive particle
^ t
|
------> x
Let us have a massive particle m which decays into two wave packets. The particle m initially moves at a speed v and then decays into two light-speed wave packets. Both packets are later absorbed by other particles.
We look at the history H during a time period Δt.
We let the FLRW universe expand so much that the energy of the wave packets is negligible at the end of the period Δt. The absorbing particles can be very lightweight, much lighter than m.
Let us vary the history H in such a way that we let the particle m move to the right at a faster speed than v for a short time after the start of Δt. At the end of the Δt, we let the other particles move for a short time to the right, so that they catch the wave packets. Then we let the other particles move to the left, so that the variation H' has the same end state as H.
Since the other particles have negligible masses and the wave packets have negligible energies at the end of Δt, the variation at the end changes the value of the action negligibly.
But the variation at the start of Δt changes the action substantially since it changes the kinetic energy of m.
We showed that an infinitesimal variation H' of the history H changed the action substantially (linearly with the variation). Then the history H is not a stationary point of the Einstein-Hilbert action.
The proof illustrates that if a "packet" of energy is destroyed almost completely in a history H, then it is very easy to construct a variation H' which changes the value of the action.
We did not need to vary the propagation speed of the wave packets. They propagate at the light speed c. There is no problem with faster-than-light signals in the variation H'.
We used the fact that the value of the action is not changed if we translate the middle part of the history H to the positive x direction. That is, we used the spatial translation symmetry.
Why the same argument does not work in a newtonian balloon model of an expanding universe? It has to be because the interaction of the balloon rubber and particles moving on it is more complicated than in general relativity. The spatial translation symmetry does not hold if the particle made a significant "dent" into the rubber. If we keep the history of the rubber as is, but move the path of the particle, then the action of the middle part changes significantly.
Why does the literature claim that energy in radiation is lost when the universe expands?
Some papers refer to the Richard Tolman book from the year 1934. Let us find out how Tolman derives the result.
On page 385, Tolman writes:
Tolman claims that a particle moving in an expanding universe will slow down. But in the first section of this blog post we proved from the Einstein-Hilbert action that the particle must keep its velocity. Let us analyze how Tolman arrives at a different result.
• ----> v
Tolman derives the path of the particle from the geodesic equation. But if the geodesic equation claims that the particle loses kinetic energy, then it clashes with the Einstein-Hilbert action. How can this be?
A possible reason is that the geodesic equation "measures" the velocity v of the particle relative to the location where it started from. Then there is no loss of kinetic energy.
This leads us to a fundamental problem: if we have a particle which has gone a long way, the expanding universe may make it to recede from its original location faster than light. What is the kinetic energy in that case?
In the Einstein-Hilbert action, is the kinetic energy of the particle reduced or not? The matter lagrangian would require the kinetic energy to be the same, but the Einstein field equations probably require it to be reduced?
All these problems stem from the fact that there are no canonical coordinates in general relativity. The kinetic energy of a particle is a fuzzy concept.
Is the geodesic equation itself ambiguous?
The Einstein-Hilbert action is the fundamental equation. The geodesic equation is more like a guess of how a particle would move under a metric. The guess seems to clash with the action.
Is there any way to make a consistent model where the topology of the spatial metric is isomorphic to a 3-sphere and the geodesic equation holds?
In this blog we have been touting the Milne model where space is Minkowski, and we avoid the contradictions produced by the 3-sphere topology.
The balloon analogy
The balloon analogy of an expanding universe has no problem handling radiation pressure and so on, because it is a model in newtonian mechanics. For example, if there is a pressure from "gas" on the surface of the balloon, the pressure does work as the balloon expands. The gas cools, and the energy is given to the rubber in the balloon, stretching the rubber and giving it kinetic energy outward. The action of this system conserves energy.
Tuomo Suntola's Dynamic Universe has the 3-sphere expanding to a fourth spatial dimension, and the kinetic energy of a particle for the velocity in the fourth dimension is included. It is a step toward the newtonian balloon model. We have to check how Suntola's model handles energy conservation.
A locally Minkowski space?
What about using "canonical" coordinates which are locally Minkowski at each small "patch" of the the universe? We will worry later about gluing these coordinate systems together.
The kinetic energy of a particle would be defined relative to these canonical coordinates. Energy would be conserved.
The problems start when a particle moves from one patch to another. Suddenly, the kinetic energy of the particle changes.
A reverse Oppenheimer-Snyder collapse: make that the new ΛCDM model?
In the Oppenheimer-Snyder collapse, uniform dust collapses to form a black hole. If we run the collapse in reverse, it is much like an expanding FLRW universe. However there are natural canonical coordinates (the Schwarzschild coordinates), and we avoid the various problem with the topology of the 3-sphere in the FLRW model.
A radiation pressure really does work in that model, accelerating the explosion of the matter outward. Energy is conserved.
Maybe we should abandon the 3-sphere, and claim that the natural expanding universe model in general relativity is the reverse Oppenheimer-Snyder collapse? The spacetime is asymptotically Minkowski. There may be several Big Bangs at various locations of the asymptotic Minkowski space.
In our blog we have been claiming that the Oppenheimer-Snyder collapse "freezes" when a horizon forms. No singularity can form. But we see that our own universe has had a much larger matter density in the past, and we would have been inside the horizon. There was no freezing. The universe kept expanding. Currently, the observable universe would be outside the horizon, if there is not much matter which is farther than the observable volume.
We have supported the Milne model in this blog, to avoid the freezing. If there is an equal amount of negative gravity charge (in dark matter) as there is positive charge, then no horizon forms and the Milne model is the one.
Another option is to design a model where somehow the Oppenheimer-Snyder collapse can go past the forming horizon. We could claim that the Big Bang is the only case where such a behavior is possible, and it cannot happen in an ordinary black hole. As if an infinite time could pass in this special, Big Bang process, and take matter past the horizon. That is very ad hoc.
Conclusions
Let us close this blog post. In subsequent posts, we will further analyze the problem of missing canonical coordinates in the ΛCDM model.
Currently, our best bet is to replace the 3-sphere spatial topology of ΛCDM with a reverse Oppenheimer-Snyder collapse. The Schwarzschild coordinates would act as canonical coordinates against which we measure the kinetic energy of a particle.
If the ADM formalism is correct, then energy is conserved in a reverse Oppenheimer-Snyder collapse. We avoid the problems of energy non-conservation in the 3-sphere spatial topology.
In our blog we have been claiming that the gravity field merely simulates the "metric" of spacetime. The metric is not fundamental, and the true underlying spacetime metric and topology is the flat Minkowski space. We have said that the 3-sphere spatial topology of ΛCDM cannot exist. In this blog post we showed that the 3-sphere spatial topology causes serious, maybe insurmountable, problems for ΛCDM.
On the Internet there is a lot of discussion about energy non-conservation in expanding models of the universe. However, no one seems to have analyzed what the Einstein-Hilbert action says about energy non-conservation: it prohibits that. We are probably the first to recognize the serious problem in the ΛCDM model, and in the 3-sphere spatial topology.
We have noted in this blog that while there is ample empirical evidence that gravity can simulate "curved" spacetime, there is no empirical evidence that gravity could alter the topology of spacetime, like it does in the ΛCDM model. The hypothesis that gravity can alter the topology is a bold one. Maybe the hypothesis is incorrect.
It may be that an action in physics can truly work only if energy is conserved. Energy non-conservation in ΛCDM may be a fatal error in the model.
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