Why the Universe Is Not Expanding

Feb 9, 2023
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"Is the space inside, say, a galaxy growing but overcome by the gravitational attraction between the stars? The answer is no. Space within any gravitationally bound system is unaffected by the surrounding expansion."
View: https://youtu.be/bUHZ2k9DYHY?t=356


Sabine Hossenfelder: "The solution of general relativity that describes the expanding universe is a solution on average; it is good only on very large distances. But the solutions that describe galaxies are different - and just don't expand. It's not that galaxies expand unnoticeably, they just don't. The full solution, then, is both stitched together: Expanding space between non-expanding galaxies...It is only somewhere beyond the scales of galaxy clusters that expansion takes over." https://www.forbes.com/sites/starts...ont-actually-expand-in-an-expanding-universe/

So cosmologists apply the expansion solutions only to voids completely deprived of gravity; to galaxies and galactic clusters they apply nonexpansion solutions. Why do cosmologists resort to this trick? Because, if they applied expansion solutions to galaxies and galactic clusters, observations would immediately disprove the expansion theory. Here is why:

If expansion is actual inside galaxies and galactic clusters, the competition between expansion and gravitational attraction would distort those cosmic structures - e.g. fringes only weakly bound by gravity would succumb to expansion and fly away. And the theory, if it takes into account the intragalactic expansion, will have to predict the distortions.

But no distortions are observed - there is really no expansion inside galaxies and galactic clusters. And cosmologists, without much publicity, have decided to apply the expansion theory only to gravity-free space.

Since there is no expansion inside galaxies and galactic clusters, there is no expansion anywhere else.
 
write4u, you are delving into one of the biggest arenas in astrophysics and cosmology:

Let me offer you a variety of ideas on expansion, vacuum energy and space.

First we must examine if our universe is truly isotropic, looking the same in every direction you examine.

In April of 2020, using NASA’s Chandra X-ray Observatory and ESA’s XMM-Newton, “One of the pillars of cosmology – the study of the history and fate of the entire universe – is that the universe is ‘isotropic,’ meaning the same in all directions,” said Konstantinos Migkas of the University of Bonn in Germany, who led the new study, titled 'Doubts about basic assumption for the universe' “Our work shows there may be cracks in that pillar.”

Astronomers generally agree that after the Big Bang, the cosmos has continuously expanded. A commonly analogy is that this expansion is like a baking loaf of raisin bread. As the bread bakes, the raisins (which represent cosmic objects like galaxies and galaxy clusters) all move away from one another as the entire loaf (representing space) expands. With an even mix the expansion should be uniform in all directions, as it should be with an isotropic universe. But these new results may not fit that picture.

“Based on our cluster observations we may have found differences in how fast the universe is expanding depending on which way we looked,” said co-author Gerrit Schellenberger of the Center for Astrophysics | Harvard & Smithsonian (CfA) in Cambridge, Massachusetts. “This would contradict one of the most basic underlying assumptions we use in cosmology today.”

Scientists have previously conducted many tests of whether the universe is the same in all directions. These included using optical observations of exploded stars and infrared studies of galaxies. Some of these previous efforts have produced possible evidence that the universe is not isotropic, and some have not.

This latest test uses a powerful, novel and independent technique. It capitalizes on the relationship between the temperature of the hot gas pervading a galaxy cluster and the amount of X-rays it produces, known as the cluster's X-ray luminosity. The higher the temperature of the gas in a cluster, the higher the X-ray luminosity is. Once the temperature of the cluster gas is measured, the X-ray luminosity can be estimated. This method is independent of cosmological quantities, including the expansion speed of the universe.

Once they estimated the X-ray luminosities of their clusters using this technique, scientists then calculated luminosities using a different method that does depend on cosmological quantities, including the universe’s expansion speed. The results gave the researchers apparent expansion speeds across the whole sky – revealing that the universe appears to be moving away from us faster in some directions than others.

The team also compared this work with studies from other groups that have found indications of a lack of isotropy using different techniques. They found good agreement on the direction of the lowest expansion rate.

The authors of this new study came up with two possible explanations for their results that involve cosmology. One of these explanations is that large groups of galaxy clusters might be moving together, but not because of cosmic expansion. For example, it is possible some nearby clusters are being pulled in the same direction by the gravity of groups of other galaxy clusters. If the motion is rapid enough it could lead to errors in estimating the luminosities of the clusters.

These sorts of correlated motions would give the appearance of different expansion rates in different directions. Astronomers have seen similar effects with relatively nearby galaxies, at distances typically less than 850 million light years, where mutual gravitational attraction is known to control the motion of objects. However, scientists expected the expansion of the universe to dominate the motion of clusters across larger distances, up to the 5 billion light years probed in this new study.

A second possible explanation is that the universe is not actually the same in all directions. One intriguing reason could be that dark energy – the mysterious force that seems to be driving acceleration of the expansion of the universe – is itself not uniform. In other words, the X-rays may reveal that dark energy is stronger in some parts of the universe than others, causing different expansion rates.

“This would be like if the yeast in the bread isn’t evenly mixed, causing it to expand faster in some places than in others,” said co-author Thomas Reiprich, also of the University of Bonn. "It would be remarkable if dark energy were found to have different strengths in different parts of the universe. However, much more evidence would be needed to rule out other explanations and make a convincing case."

Space is expanding between the galaxies. This means that one day, far in the future, inhabitants of our galaxy will not see any other galaxies in the night sky.

The above doesn’t mean that galaxies that are near to one another don’t interact, however.

Just as Earth’s gravity might pull on a nearby asteroid, sending it on a collision course with our planet, the Andromeda and Milky Way galaxies interact with each other gravitationally. This has resulted in the two galaxies falling toward each other at a rate of about 37 miles per second (60 km per second). Because Andromeda is 2.5 million light-years away — with 1 light-year equivalent to 5.9 trillion miles (9.5 trillion km) — this galactic crash won’t occur for 4.5 billion years. And thankfully, the space between stars is so great that it’s unlikely anything will truly collide. But the merger will change the paths of stars within each galaxy due to gravimetric interactions.

As it turns out, a majority of the Milky Way halo was “formed by the merging of numerous progenitor galaxies,” according to a paper published in The Astrophysical Journal earlier this year. Though the exact number of times our galaxy has merged with another is unclear, what is clear is that the Andromeda collision won’t be the first time the Milky Way has merged with another galaxy. Nor will it be the last.

Our galaxy is part of a galaxy cluster known as the Local Group. One day, this collection of nearly 100 galaxies may have all merged. And there’s even evidence that the Local Group might itself merge with the closest large galaxy cluster, the Virgo Cluster, where the local group is also moving!

In 2019, astronomers using NASA's Hubble Space Telescope say they have crossed an important threshold in revealing a discrepancy between the two key techniques for measuring the universe's expansion rate. The recent study strengthens the case that new theories may be needed to explain the forces that have shaped the cosmos.

A brief recap: The universe is getting bigger every second. The space between galaxies is stretching, like dough rising in the oven. But how fast is the universe expanding? As Hubble and other telescopes seek to answer this question, they have run into an intriguing difference between what scientists predict and what they observe.

Like most things in physics, it’s all a matter of scale. Just like the strong nuclear force that holds the nucleus of an atom together is stronger on very tiny scales, gravity is a more dominant force on scales out to about the size of a large galaxy cluster, plus that pesky mysterious dark matter. Cosmic expansion only comes into play when we’re looking out over a large cosmological scale.

Hubble measurements suggest a faster expansion rate in the modern universe than expected, based on how the universe appeared more than 13 billion years ago. These measurements of the early universe come from the European Space Agency's Planck satellite. This discrepancy has been identified in scientific papers over the last several years, but it has been unclear whether differences in measurement techniques are to blame, or whether the difference could result from unlucky measurements.

The latest Hubble data lower the possibility that the discrepancy is only a fluke to 1 in 100,000. This is a significant gain from an earlier estimate, less than a year ago, of a chance of 1 in 3,000.

These most precise Hubble measurements to date bolster the idea that new physics may be needed to explain the mismatch.

"The Hubble tension between the early and late universe may be the most exciting development in cosmology in decades," said lead researcher and Nobel laureate Adam Riess of the Space Telescope Science Institute (STScI) and Johns Hopkins University, in Baltimore, Maryland. "This mismatch has been growing and has now reached a point that is really impossible to dismiss as a fluke. This disparity could not plausibly occur just by chance."

The Universe is expanding, as found by Saul Permutter's team using Type Ia supernovae (https://supernova.lbl.gov/PhysicsTodayArticle.pdf ) and it’s only apparent over galactic distances, where gravity no longer dominates over dark matter. In other words, astronomers have observed that distant galaxies that are not bound to each other by gravity are receding from each other.

There are galaxies that are bound to each other by gravity, and these are not receding from each other. For example, the Milky Way and the Andromeda belong to a group of galaxies bound to each other by gravity. This is the Local Group consisting of some 50 galaxies.

Cosmological redshift - which can be determined to measure distance in the Universe - is only reliable on larger distances where the expansion of the Universe is the dominating impact on the spectrum of the object. On short distances the speed of the galaxy with respect to us will ruin such measurements.

And, to further confuse issues with expansion, gravity and the Hubble constant, the universe doesn't have to be expanding into anything in order to expand. I know that sounds ridiculous, so here's a different example that is easier to understand.

Imagine that you have a line that goes on forever. On that line, you have a mark every inch. There are an infinite number of inches. Now move each marker so they are separated by two inches. The whole pattern has expanded. It still goes to infinity, but the markers are further apart. The pattern has expanded, but the length is still infinite.

Now a new example. Suppose you have a long piece of rubber, going all the way to infinity. (That piece of rubber represents the universe.) The rubber has marks on it every inch. Now stretch the rubber, until the markers are two inches apart. It still goes to infinity -- but it has expanded.

Physicists think of "space" not as emptiness, but similar to a piece of rubber. (But they don't call it rubber; they call it the "vacuum" which contains energy. "Particles", in physics, are just vibrations of the vacuum.) The vacuum can expand, just like the piece of rubber. But because it goes all the way to infinity, it doesn't need more space. A clever way to say it is that "there's lots of room at infinity".

Spacetime has a natural tendency to expand. Recall that Einstein's failure to recognise this idea resulted in his inserting a value into his equations to prevent this expansion because it was widely thought at the time that the Universe was in a steady state.

When a region of spacetime is homogeneous (the same everywhere, i.e., no lumps of matter such as a galaxy or a Galactic cluster) and isotropic (i.e., the same in every direction) the distance between two points tends to increase over time (the rate at which this happens depends on the distance between the two points and the vacuum energy).

There is no proper acceleration associated to this. So nothing is actually accelerating. And the concept of velocity is slightly different here from what we usually rely upon. (It's an apparent velocity, but it cannot reliably be transformed in a different frame of reference. And sometimes (if there is enough distance between the two points to start with) that apparent velocity is greater than the speed of light. But this is seen in a frame of reference that is not actually inertial.
Space does not thin out. On the contrary in a way. What happens is that where there were two light years between two points a year ago, there is now 2.5 ly (not the actual rate, only used for illustration purposes), and there will be 3.5 ly a year later. But there is no acceleration felt by the objects, and in that sense, no actual motion.

Now, looking through the history of the universe, we see that rate increasing over time in a manner that would be consistent with a constant energy density in space. Dark energy is what we called this apparent constant energy density.

This has nothing to do with dark matter, beyond the name. Dark matter is simply matter that does not interact with light (i.e., with electromagnetism), but only gravitationally. And we know dark matter exists because some galaxies (note the difference of scale here, we went from in-between Galactic clusters to within a single galaxy) do not behave as they would if their gravity was only impacted by the visible (in the sense interacting with light) matter within it. But not all of them do. So there is definitely something that some galaxies have and some other don't. It's not just gravity behaving differently from what we thought.

Gravity can quite easily be repulsive due excessive negative pressure as mentioned by @Stan Liou. Let us work this out. We will need two equations -
  1. Einstein's Equation
    𝑅𝜇𝜈=8𝜋𝐺(𝑇𝜇𝜈−12𝑇𝑔𝜇𝜈)Rμν=8πG(Tμν−12Tgμν)
    where 𝑇=𝑔𝜇𝜈𝑇𝜇𝜈T=gμνTμν. This equation describes how matter affects the curvature of space-time. In particular, we will use two forms of 𝑇𝜇𝜈Tμν. One for a perfect fluid of density 𝜌ρ and isotropic pressure 𝑝p
    𝑇𝑓𝑙𝑢𝑖𝑑𝜇𝜈=(𝜌+𝑝)𝑈𝜇𝑈𝜈+𝑝𝑔𝜇𝜈Tμνfluid=(ρ+p)UμUν+pgμν
    where 𝑈𝜇Uμ is the four velocity of the fluid itself. The other EM tensor that will be used is the one that contributes to a non-zero vacuum energy, or a cosmological constant, defined by
    𝑇𝑐𝑚𝜇𝜈=−𝜌𝑣𝑎𝑐𝑔𝜇𝜈Tμνcm=−ρvacgμν
  2. Raychaudhari's equation
    𝑑𝜃𝑑𝜏=2𝜔2−2𝜎2−13𝜃3−𝑅𝜇𝜈𝑈𝜇𝑈𝜈dθdτ=2ω2−2σ2−13θ3−RμνUμUν
    This equation is a purely geometric one. It is simply a statement about the behaviour of geodesics on a curved manifold. Here 𝜃θ is the "expansion parameter" and its value describes the affect of gravity on a ball. 𝜃>0θ>0 implies that the ball grows and 𝜃<0θ<0 implies that it shrinks. Here 𝑈𝜇Uμdenotes the 4-velocity of particles moving in the geodesic.
We can now work out the effect of gravity for the simplest possible situation. We consider a set of particles that are initial at rest w.r.t each other in a small region (so that we can approximate 𝑔𝜇𝜈=𝜂𝜇𝜈gμν=ημν) of spacetime (take this time to be 𝑡=0t=0), i.e. 𝑈𝜇=(1,0,0,0)Uμ=(1,0,0,0). At 𝑡=0t=0, 𝜔=𝜎=𝜃=0ω=σ=θ=0 (these quantities are like the stress, shear and expansion. Since the particles in the ball are not moving at all, all of the above are zero). Thus, at this time, the Raychaudhuri's equation takes the form
𝑑𝜃𝑑𝜏=−𝑅00dθdτ=−R00
We can also treat (in an approximation) the set of particles as a perfect fluid, in which case
𝑇00=𝜌, 𝑇=𝑔𝜇𝜈𝑇𝜇𝜈=−𝜌+3𝑝T00=ρ, T=gμνTμν=−ρ+3p
We can then use Einstein's equation to derive 𝑅00R00 for the system. We then get
𝑑𝜃𝑑𝜏=−𝑅00=−4𝜋𝐺(𝜌+3𝑝)dθdτ=−R00=−4πG(ρ+3p)
What we are really interested is the sign of 𝑑𝜃𝑑𝜏dθdτ. Usually energy densities are positive. For any positive pressure, 𝑑𝜃𝑑𝜏<0dθdτ<0 and the ball shrinks (doing exactly what we expect of gravity). In fact, even for negative pressure with magnitude as big as |𝑝|𝑚𝑎𝑥=𝜌3|p|max=ρ3 gravity always attracts. However, for any negative pressure with larger magnitude, we find that gravity indeed becomes a repulsive force!

What does this mean about the relation to the vacuum energy? Note that 𝑇𝑐𝑚𝜇𝜈=𝑇𝑓𝑙𝑢𝑖𝑑𝜇𝜈(𝑝→−𝜌𝑣𝑎𝑐)Tμνcm=Tμνfluid(p→−ρvac). We then find
𝑑𝜃𝑑𝜏=−𝑅00=8𝜋𝐺𝜌𝑣𝑎𝑐dθdτ=−R00=8πGρvac

We then find that any positive energy density will cause gravity to act repulsively! The usual example of a spacetime with positive energy density is de Sitter space.

de Sitter spacetime is the maximally symmetric spacetime of constant positive curvature. It is a solution of the vacuum Einstein equations with a positive cosmological constant. It is directly relevant for observation, in two (as fas as we know unrelated!) ways.

First, there is evidence that the very early universe had a period of rapid expansion, ‘inflation’, well approximated by de Sitter spacetime.

Second, our tiny present-day cosmological constant currently accounts for about 68% of the energy density of the universe, and this fraction is growing as the universe continues to expand. This means we are entering a second de Sitter phase. The early, inflationary de Sitter phase had a large cosmological constant and correspondingly tiny radius of curvature. The future, dark energy de Sitter has an energy set by today’s cosmological constant, and enormous radius of curvature close to today’s Hubble scale.

Now here is something new that might confuse you, or might help. In the standard physics theory, the galaxies are all getting farther apart; that is the expansion of the Universe. Yet in the way the theory describes it (I mean in General Relativity Theory) none of the galaxies are actually moving. All that is happening is that the amount of space (vacuum) in between them is increasing.

No, you will not learn this in school, or even in college (unless you have an extraordinary professor). It is usually taught in graduate school, when you are earning a Ph.D. degree. At that point the language you will encounter is this: "In the Big Bang Theory, all galaxies have fixed coordinates. (That means they are not moving.) This 'expansion' is described by the 'metric tensor', which describes the distances between those fixed coordinates. In the Big Bang Theory, it is the metric tensor which is changing; that represents the expansion of the Universe, even though the galaxies aren't moving. The recent discovery of accelerated expansion means that the rate of expansion is increasing."

Not only is the mass of the black hole influenced by the stars it eats and any other black holes it merges with—it’s influenced by the expansion of the universe itself.

That’s not quite as wild as it seems. If a black hole expands along with the universe, that expansion generates energy. And because energy and mass are linked through everyone’s favorite physics equation, e=mc², if the energy increases, the mass increases with it.

With the idea of cosmological coupling in place, the question became how to prove it. The team needed to demonstrate that the expansion of the universe was strongly tied to the growth of black holes. They called the strength of this bond k.

To test the connection, the team observed five ancient populations of spiral galaxies—large galaxies that are billions of years old and have finished merging—to see if they were gaining significant mass. In those galaxies, the central black holes aren’t accumulating much mass through collisions with other black holes or through swallowing huge amounts of stars.

The only way they could be gaining large amounts of mass would be through a connection to the expansion of the universe. And boy, were they gaining mass. The team calculated that these black holes that formed billions of years ago are anywhere from seven to 20 times more massive than similar, recently formed black holes.

That huge difference in mass lead researchers to calculate that the strength of the connection between expansion and mass increase in black holes was about k=3, a solution that feels a bit esoteric and inaccessible until it’s put into context.

So, here’s the context: Back in 2019, it was calculated that a connection strength of k=3 meant that instead of a singularity, black holes contained what’s called vacuum energy—a type of dark energy.

If that’s true, these new observations show that all black holes are producing a constant amount of dark energy equal to our existing measurements of dark energy. All of the dark energy in the universe would be accounted for by the vacuum energy created by black holes.

This is the first time that observational data has been able to explain the entire existence of dark energy with no new sources. Everything in this study was already folded into Einstein’s theories.

It still needs to be confirmed, and researchers are careful to say that this is just a first step in fully understanding the source of dark energy.

We've known for about 20 years that the expansion of our universe is accelerating; every day, our cosmos grows bigger and bigger, doing so faster and faster. It's a subtle effect, and it takes extensive and deep cosmological surveys and studies for scientists to notice it. But multiple independent lines of evidence all point to the same conclusion: accelerating expansion.

Astronomers quickly cooked up a cool name for that accelerated expansion: dark energy. But now we’re left with the much harder job of finding a culprit — what's causing it?

use general relativity, Albert Einstein's magnum opus, to understand gravity in all its manifestations, including the expansion of the universe. But the theory's equations have some wiggle room. Specifically, they allow for a so-called "cosmological constant," a fixed term that can be appended to the end. Adding this constant doesn't change the theory's descriptions of normal, everyday gravitational interactions, but it does make itself known when you're calculating the expansion of the universe.

Our natural inclination would be to set this constant to zero and forget about it, but Einstein himself introduced it because he found that without it, his relativity predicted a dynamic universe. At the time, both physicists and the general public thought of the cosmos as static and unchanging, so Einstein set a value for the constant to prevent those dynamic predictions. And then astronomer Edwin Hubble showed everyone that we do indeed live in an expanding universe, and Einstein realized that he'd missed a golden opportunity to predict that revolutionary observation. Oh, well.

But nowadays, we're faced with accelerated expansion, and the simplest explanation we have for it is that is that dark energy is simply Einstein's original cosmological constant. But the constant by itself is just a number — what's its physical significance?

In the 1960s, Soviet astrophysicist and all-around-genius Yakov Zel'dovich made a startling connection. The cosmological constant that appears in Einstein's equations is none other than the vacuum energy that quantum field theory predicts.

According to that theory, a suite of quantum fields permeates all of space-time. Sometimes, portions of these fields get excited and move around, and this is what we identify as particles. But left unperturbed, the fields are still associated with an energy. In other words, the empty vacuum of space-time has a raw energy, and that energy can be identified a the cosmological constant in general relativity, which means it might be the dark energy itself.

There are games you can play to make the predicted value for dark energy not infinity, but no matter what you do, you always end up with a very large number. But what about the actual, observed amount of dark energy, the one calculated from the accelerated expansion rate? It's very small: the energy equivalent of a little less than 1 hydrogen atom per cubic meter (35 cubic feet).

That "minor" discrepancy between dark energy's predicted value and the observed expansion rate is one of the biggest puzzles in modern physics. And its full resolution will probably come only with a true reckoning between quantum mechanics and general relativity.

Until then, we should probably understand how a vacuum energy can accelerate expansion.

See: https://astronomy.com/magazine/ask-...-the-milky-way-and-andromeda-galaxies-collide

See: View: https://www.reddit.com/r/AskPhysics/comments/xts9hg/expansion_of_spacetime_mass_and_gravity/


See: https://www.nasa.gov/feature/goddar...-s-expansion-rate-widens-with-new-hubble-data

See: https://www.quora.com/Is-space-expanding-within-clusters-of-galaxies?share=1

See: https://physics.stackexchange.com/q...um-energy-cause-the-expansion-of-the-universe

See: http://www.hartmanhep.net/GR2017/desitter-lectures-v2.pdf><

See: https://www.popularmechanics.com/space/a42941836/scientists-find-source-of-dark-energy/

See: https://www.space.com/42178-bringing-dark-energy-into-the-light.html

Two key properties of a vacuum energy affect expansion. One is the vacuum's persistence; as the universe expands, there's more space, so there's more vacuum, so there's more vacuum energy. So, in our evolving cosmos, we find more and more dark energy lying around. The second vacuum property that's key for expansion is that the vacuum has tension (usually, confusingly, referred to as "negative pressure," but pretty much the same). This tension resists the expansion of the universe; it's trying to rein in the expanding cosmos.

Put these two properties together and you get the complete opposite of what you may expect. This is because the equations of general relativity count all sources of energy to determine the behavior of the expansion of the universe, and different sources of energy can contribute positive or negative effects. So, the raw energy of the vacuum gets counted, which would be an attractive contribution, slowing down the expansion of the universe. But so does the vacuum's tension, which actually contributes repulsively., In other words, in the math of general relativity, the tension from dark energy carries a minus sign with it, and contributes to accelerating the expansion of the universe).
Hartmann352
 
First of all, the word diffusion is a synonym of expansion. And the difference is that is that diffusion is the intermingling of the molecules of a fluid due to random thermal agitation) while expansion is equated with volumetric expansion.

A fluid is having particles that easily move and change their relative position without a separation of the mass and that easily yield to pressure : or is capable of flowing or expansion due to temperature and reduced or increased pressure.

Secondly, the absolute absence of a gravitational field in a region of space is plainly impossible, since in the universe there's at least one particle with mass which generates an infinite-reaching field (and in fact, there's obviously a lot of stuff out there, from planets to stars to galactic filaments to giant molecular clouds). The gravitational force is the weakest of the fundamental interactions of Nature, and as the electromagnetic force, it has an infinite range; but unlike the latter, there's no "negative mass" equivalent which can "neutralize" the field far away from the interacting bodies.

That's why, among other things, that there's no macroscopic manifestation of the astronomically strong electric field that would be generated if all the elementary charges of electrons and protons, which compose every atom in any everyday object, were turned apart and separated.

On the other hand, the gravitational force shapes the entire universe at the cosmological scale, and the electromagnetic interaction has no significant effect on large (neutral) objects like planets and stars.

In 1918, Einstein described Mach's principle as a philosophical pillar of general relativity, along with the physical principle of equivalence and the mathematical pillar of general covariance. This characterization is now widely regarded as wishful thinking. Einstein was undoubtedly inspired by Mach's relational views, and he hoped that his new theory of gravitation would "secure the relativization of inertia" by binding spacetime so tightly to matter that one could not exist without the other.

However, the equations of general relativity are perfectly consistent with spacetimes that contain no matter at all. Flat (Minkowski) spacetime is a trivial example, but empty spacetime can also be curved, as demonstrated by Willem de Sitter in 1916. There are even spacetimes whose distant reaches rotate endlessly around the sky relative to an observer's local inertial frame (as discovered by Kurt Gödel in 1949). The bare existence of such solutions in Einstein's theory shows that it cannot be Machian in the strict sense; matter and spacetime remain logically independent.

The term "general relativity" is thus something of a misnomer, as pointed out by Hermann Minkowski and others. The theory does not make spacetime more relative than it was in special relativity. Just the opposite is true: the absolute space and time of Newton are retained. They are merely amalgamated and endowed with a more flexible mathematical skeleton (the metric tensor).

Calculations by Hans Thirring (1888-1979), Josef Lense (1890-1985) and others have shown that a large rotating mass will "drag" an observer's inertial reference frame around with it. The same calculations suggest that, if the entire contents of the universe were to rotate, our local inertial frame would undergo "perfect dragging" — that is, we would not notice it, because we would be rotating too! In that sense, general relativity is indeed nearly as relational as Mach might have wished.

Some physicists (such as Julian Barbour) have gone further and asserted that general relativity is in fact perfectly Machian. If one goes beyond classical physics and into modern quantum field theory, then questions of absolute versus relational spacetime are rendered anachronistic by the fact that even "empty space" is populated by matter in the form of virtual particles, zero-point fields and more. Within the context of Einstein's universe, however, the majority view is perhaps best summed up as follows: Spacetime behaves relationally but exists absolutely.

The magnitude of diffusion damping is calculated as a damping factor or suppression factor, represented by the symbol, which figures into the Boltzmann equation, an equation which describes the amplitude of perturbations in the Cosmic Microwave Background (CMB).

The strength of the diffusion damping is chiefly governed by the distance photons travel before being scattered (diffusion length). What affect the diffusion length has are primarily the properties of the plasma in question: different sorts of plasma may experience different sorts of diffusion damping. The evolution of a plasma may also affect the damping process.

Where:
  • is the conformal time.
  • is the "differential optical depth for Thomson scattering". Thomson scattering is the scattering of electromagnetic radiation (light) by charged particles such as electrons.
  • is the wave number of the wave being suppressed.
  • is the visibility function.
The damping factor, when factored into the Boltzmann equation for the cosmic microwave background radiation (CMB), reduces the amplitude of perturbations:

Where:
  • is the conformal time at decoupling.
  • is the "monopole of the photon distribution function"
  • is a "gravitational-potential in the Newtonian gauge". The Newtonian gauge is a quantity with importance in the General Theory of Relativity.
  • is the effective temperature.
Mathematical calculations of the damping factor depend on, or the effective diffusion scale, which in turn depends on a crucial value, the diffusion length, . The diffusion length relates how far photons travel during diffusion, and comprises a finite number of short steps in random directions. The average of these steps is the Compton mean free path*, and is denoted by . As the direction of these steps are randomly taken, is approximately equal to, where is the number of steps the photon takes before the conformal time at decoupling .

The diffusion length increases at recombination because the mean free path does, with less photon scattering occurring; this increases the amount of diffusion and damping. The mean free path increases because the electron ionisation fraction, decreases as ionised hydrogen and helium bind with the free, charged electrons. As this occurs, the mean free path increases proportionally: . That is, the mean free path of the photons is inversely proportional to the electron ionisation fraction and the baryon number density . That means that the more baryons there were, and the more they were ionised, the shorter the average photon could travel before encountering one and being scattered. Small changes to these values before or during recombination can augment the damping effect considerably. This dependence on the baryon density by photon diffusion allows scientists to use analysis of the latter to investigate the former, in addition to the history of ionisation.

The effect of diffusion damping is greatly augmented by the finite width of the surface of last scattering (SLS). The finite width of the SLS means the CMB photons we see were not all emitted at the same time, and the fluctuations we see are not all in phase. It also means that during recombination, the diffusion length changed dramatically, as the ionisation fraction shifted.

We know that the expansion of the Universe isn't slowing down. It's speeding up. We call the unknown force behind this acceleration dark energy.

See: https://wikidiff.com/expansion/diffusion

See: https://einstein.stanford.edu/SPACETIME/spacetime2.html

See: http://scihi.org/hermann-minkowski-space-time/

See: https://www.liquisearch.com/diffusion_damping/magnitude

See: https://physics.stackexchange.com/questions/33154/can-gravity-be-absent

Nevertheless, Einstein's theory of gravity represents a major swing back toward the relational view of space and time, in that it answers the objections of the ancient Stoics. Space and time act on matter, by guiding the way it moves. And the mass of matter acts on spacetime by producing the curvature, what we feel as gravity. Beyond that, matter can act on spacetime in a manner that is very much in the spirit of Mach's principle.

A freely moving or falling particle always moves along a geodesic. In general relativity, gravity can be regarded as not a force but a consequence of a curved spacetime geometry where the source of curvature is the stress-energy tensor (representing matter, for instance).

The stress-energy tensor is the source term for Einstein's equation, kind of like the electric charge and current is the source term for Maxwell's equations. It represents the amounts of energy, momentum, pressure, and stress in the space. Roughly (https://physics.stackexchange.com/questions/28875/what-is-the-stress-energy-tensor):

Screenshot 2023-03-03 at 23.32.05.png

Here 𝑢u is the energy density, the 𝑝p's are momentum densities, 𝑃P's are pressures, and 𝜎σ's are shear stresses.

In its most "natural" physical intepretation, Einstein's equation 𝐺𝜇𝜈=8𝜋𝑇𝜇𝜈Gμν=8πTμν (in appropriate units) represents the fact that the curvature of space is determined by the stuff in it. To put that into practice, you measure the amount of stuff in your space, which tells you the components of the stress-energy tensor. Then you try to find a solution for the metric 𝑔𝜇𝜈gμν that gives the proper 𝐺𝜇𝜈Gμνsuch that the equation is satisfied. (The Einstein tensor 𝐺G is a function of the metric.) In other words, you're measuring 𝑇T and trying to solve the resulting equation for 𝐺G.

But you can also in principle measure the curvature of space and use that to determine 𝑇T, which tells you how much stuff is in the space. This is what cosmologists do when they try to figure out how the density of the universe compares to the critical density, for example.
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