<p>The fact that photons and gravity (gravitons) travel at finite speeds has not escaped the attention of physicists.  It also has been noticed that Newtonian gravity, with no retardation of the forces, describes the motions of the solar system, so if GR, with propagation at light speed, does not imitate Newtonian with instantaneous propagation, at least in the weak limit and small scales, then GR would not explain how the solar system is stable.</p>
<p>There are terms in the potential of the Post Newtonian formulation (simplified GR theory for the weak limit cases) that depend on velocity and higher order derivatives that cancel the delay time effects. As a result, to at least first order, taking the short cut of assuming instantaneous positions also works.</p>
<p>In GR, if the source is moving, then the accelerations generated do not point back to the location of the source when the gravitons left the source. The gravitational accelerations are automatically corrected to the extrapolated instantaneous position.
For details you can see the paper:
&quot;<a href="https://ui.adsabs.harvard.edu/abs/2000PhLA..267...81C/abstract" rel="nofollow noreferrer">Aberration and the speed of gravity</a>&quot; by S. Carlip  (UC Davis).
It makes the interesting point that the simplest Lorentz Invariant potential is <span class="math-container">$\Phi = m/\vec{r}$</span> where <span class="math-container">$\vec{r}$</span> is the 4-dimensional separation vector rather than the spatial 3-vector, and acceleration, derived from its 4-vector gradient, contains velocity information.</p>
<p>The math for the electrodynamics case, in its gory details, was worked out in 1900 for electromagnetism by Liénard and Wiechert. You can go through the math <a href="https://en.m.wikipedia.org/wiki/Li%C3%A9nard%E2%80%93Wiechert_potential" rel="nofollow noreferrer">here</a>. It turns out that GR can be cast into a similar form as electromagnetism, see <a href="https://en.wikipedia.org/wiki/Gravitoelectromagnetism" rel="nofollow noreferrer">Gravitoelectromagnetism</a></p>
<p><a href="https://www.desy.de/user/projects/Physics/Relativity/GR/grav_speed.html" rel="nofollow noreferrer">Here</a> is a hand waving argument about some of this.</p>

CC BY-SA 4.0
2024-01-20T21:01:33.890Z
If your immediate thought from the heading was similar to Laplace's problem then I'm glad you thought of this right away, as this is an entirely fair point if it was the point I think I am making. According to my math (admittedly I didn't show this anyway) it would take many thousands of lifetimes of the universe for the Earth to decay rather than the 400 or so years according to Laplace.  The effect I am talking about doesn't deal with constant velocities as it assumes that there will be a decreasing acceleration with distance. I'd recommend reading the rest of my question to see what I mean
Gref
CC BY-SA 4.0
2024-01-20T16:53:58.883Z
2024-01-29T19:02:00.653Z
2024-01-29T19:02:00.653Z
eshaya
<p>I’ve got a question that has been bothering me for the past few years that of the few  sources that are only somewhat related to this, none of them say anything about this should be wrong so I can’t disprove it and every time I try to throw math at it, it just seems to get more and more reasonable. I’m going to phrase my questions as statements, so I’m sorry if it comes across as a little bit assertive, I just think it’s clearer what I’m asking (and how I’m wrong) as a statement, and if nothing else, at least it will feel better when you get to tell me I’m wrong. I’m also going to repeat the same things a few times in different ways just because otherwise I don't think it's clear what I'm saying. The first part seems negligible but it isn’t until I bring up the second part that the “majority” of gravity which acts on everything/us in the observable universe and binds it together is from the matter in the parts of the observable universe furthest from us which are, according to Hubble's law, also the parts moving the fastest away from us and should feel this effect more, that it makes sense. I've had a lot of fun with the math and physics of trying to disprove this (as you can tell by the length of the question) but have come up short so I figured I might as well let all of you try your hand at seeing if you can disprove it.</p>
<p><strong>Part one:</strong></p>
<p>Imagine a universe with only two milky way massed galaxies which are moving apart in it, one is our home galaxy and the other is the away galaxy which is 10 million lightyears from us and traveling away from our galaxy at half the speed of light. When using Newton’s law of gravitation as per normal we would predict that the amount of force being applied to each galaxy would be in constant equilibrium and therefore there wouldn’t be any acceleration and the two galaxies would continue traveling without redshifting at all, even though the force they exert on each other would be decreasing with distance/time. (I realize using general relativity is more accurate than Newton’s laws, especially at high relative velocities, but it works out to be roughly the same (as in it contributes somewhat to what I’m finding) either way and Newton’s laws make it much easier. You’ll see what I mean.)</p>
<p>What this prediction didn’t take into account was that by the time gravity travels from our galaxy to theirs, they will have moved an extra 5,000,000 lightyears and so will experience gravity as though they were 15,000,000 lightyears away (yeah, this is wrong, they would be further away because among the other things that I’m missing and will mention later, this is only how much further they would have traveled in the time for gravity to reach their original position and assumes they stop after that. However, this makes life/math easier despite being one of the major things working against what I’ve found) experiencing only 7.61 * 10^-15 N/kg [1] rather than the 17.1 * 10^-15 N/kg [2] that would normally be expected at 10,000,000 lightyears. This will create an equal and opposite amount of gravity back towards us and by the time that gravity travels back to us we will have traveled to be 22,500,000 lightyears away and only experience 3.38 * 10^-15 N/kg [3]. And since we know that the only situation where there is no acceleration is in the original prediction where the forces add up and cancel out, we know that in the second, slightly more accurate equation that there must be a tiny, but yes, still existing acceleration. This means that if we had instruments powerful enough (I guess in this universe there is no “totally necessary” military industrial complex to feed instead) and watched for long enough we would notice that overtime the spectral lines of the galaxy would not only begin to redshift by a small amount, but that the rate at which those spectral lines redshift would also begin to increase and then decrease (it would increase if the galaxy was currently close enough and decrease as it gets further away). (Also, this effect is separate to Laplace's delayed gravity problem, in case it reminded you of that.)</p>
<p>To the observer using Newton’s laws as per normal this redshift would be completely impossible to explain and would have to say that there is a certain invisible force acting upon the galaxies. But to the observer who takes into account the time for gravity to travel between the galaxies this is just the normal consequences of gravity and just depends on the distance, mass and velocity of the other galaxy and our own. Moreover, they would say that even while not accounting for acceleration that this would cause, any object whose speed exceeds the escape velocity of the gravitational system should not only escape the system but also begin to accelerate away from it. Just as an aside, the inverse of this, ie; when two objects are moving towards each other while being unable to escape each other's orbit, is the same thing that causes an acceleration/deceleration on binary blackholes/neutron stars to have their orbits decay and collide with each other through gravitational radiation, so I know this is at least somewhat a real effect. Another note is that this acceleration isn’t very dependent on our mass and should appear regardless of whether we are ourselves in a galaxy or if we were just floating through intergalactic space in a spacesuit by ourselves away from a galaxy and vice/versa, as long as one object has a significant amount of mass then it should work, if they had the same amount of mass then it would double. (In our case it would double since at large scales the universe is homogeneous, so the same thing that is happening to us from our furthest parts of the observable universe would also be happening to that other side of the observable universe from us, though it may take extra time for the gravitational effect to equalize between the two. I'm just going to keep it simple and assume that it doesn't double for my math though.)</p>
<p><strong>Part two:</strong></p>
<p>This is almost always a very useless and negligible piece of information as you need large  amounts of mass, distance and velocity for it to ever be relevant, luckily for us according to Hubble's law, the further out in the universe you look the faster it moves away from us and the more mass there is. Why is this important? Normally when we think of the amount of gravity that acts on us in the universe we think that it is the objects closest to us that exert the most amount of gravity on us. However, since gravity only decreases with the square of the distance and the volume of space which contains mass to create gravity increases cubically (like the surface area of a sphere against its volume), the sum of the objects furthest from us in the observable universe begin to produce a more notable amount of gravitational force on us than those closest (Yeah, sure, Earth and the local group produces more gravity on us, but that obviously doesn’t matter in this case since it is also applied to them.). For example, an oversimplified calculation of the total gravitational force applied to us by the region between 10-20% of the way to the edge of the observable universe only produces 2.1 * 10^-10 N/kg [4] despite being much closer than the furthest 10% of the observable universe which produces 3.3 * 10^-10 N/kg on us [5] (From 1,400 mpc to 2,800 mpc, vs From 12,800 mpc to 14,200 mpc.)</p>
<p>Though small on our scale, 3.3 * 10^-10 N/kg adds up to being 10 million m/s every billion years [6], an appreciable amount. The interesting thing is that this force being this large, is similar to the Friedmann equation, in that it hinges itself upon the idea that the galaxies furthest from us won’t have moved much in the time it takes for gravity to reach them when in reality even an observer in a non accelerating, coasting observable universe with no dark energy will have the parts furthest from its center move by an extra 70,000 m/s for every megaparsec away from them. This means that a person on the other side of the observable universe, 4.4 * 10^26 meters away, will have traveled an extra 14.6 * 10^26 meters by the time gravity reaches it according to Hubble’s law [7], quadrupling its original distance, meaning it will experience a 16th the amount of gravity that was originally predicted [8]. It will then send out an equal and opposite amount of gravitational force back towards us on the far side of their observable universe and by the time it reaches us we will be 2 * 10^28 m away, 100 times further than when we started, meaning we will experience even less gravitational force than we originally sent out, this means that there is a net imbalance in the gravitational forces acting on us and therefore there must be an apparent acceleration, all without needing dark energy. (Again, all of this is all kind of wrong but wrong in such a way that it mostly works against me while still proving a point.)</p>
<p>As for how much acceleration we can expect from this in total, the total classical gravitational force from all the galaxies acting on us across the universe is 36.8 * 10^-10 N/kg [9] (This is done piecewise with python code to make it more accurate/easier. I go over it in more detail in [9] of the math section and in the code itself which I put below) but when we take into account the added distance from the speed of gravity, this goes down to 8.43 * 10^-10 N/kg [10], a 4x decrease. This is equivalent to a net acceleration of 28.3 * 10^-10 N/kg [11] which means that the sum of the parts of the universe farthest from us should accelerate away from us by an extra 89,000,000 m/s every billion years. Though you would have to take off roughly ¾ from it since that is the force from every direction when it should really only be the sum of the positive forces in one direction (sort of like how you would get attracted towards a direction by a semi sphere of mass but it would still be less than if that mass was all concentrated into a small pipe that was directly in line with the desired direction) (It also requires some thought as to why you can ignore the other side of the universe/the negative forces in this example (Spacetime tomfoolery)). Even then we still find that 20 million m/s extra every billion years / 260 million m/s during the lifetime of the universe (I’m trying to be extremely conservative and include everything that should decrease the effect, the real figure is probably higher) and puts us well within the range of what would be expected from dark energy and that’s before you start adding in things that will increase this effect such as the extra acceleration itself, changing density etc. With the parts closer to us seeming to accelerate much less, for example the region 7000 mpc away has only got a net acceleration of 11.9 * 10^-10 N/kg [12].</p>
<p>It’s honestly kind of a bit uncanny how well this works if we repeat the calculation for something closer to us where there is both better real world data and my equations and assumptions are more accurate such as type 1a supernovae which are currently 3000 mpc away we find again that our prediction is extremely reasonable. If we assume the current acceleration has been happening for the last 13 billion years then we will find that these galaxies will have traveled an extra 214 mpc [13]. This is, while slightly lower, pretty much spot on with what we observe (around a little over 500 mpc [A]). Though as I've said before I would expect that the assumptions in my math should underestimate it by about that much. (Yeah, I also realize this is an egregious assumption when you put any level of actual thought into it, I’m just trying to show in a minimum case that the effect is within the ballpark and when you experiment with more accurate equations for distance and use the actual Hubble constant, distance and density at each time period in the past it gives a larger acceleration that is far more accurate and closer to what we observe anyways). Another interesting note is how it can not only predict an increasing rate of expansion both right where it is increasing with the effect falling off exactly as far away from us as where we observe.</p>
<p>This wouldn’t be a problem to me if it wasn’t for the fact that neither the Friedmann equation or any paper/source I could find make much of a note of the effect of us moving away from the edge of the universe. They all use something more suited to how we would calculate escape velocity for something small, such as a satellite leaving our Earth / Solar system where the distance traveled during the time for gravity to catch up would be negligible. This assumption is pretty understandable as even for something massive, somewhat far away and rapidly traveling towards us such as the Andromeda galaxy only has a 0.2% increase in the amount of gravity it feels due to this effect, so the scale of the observable universe is pretty much the one time there is anything fast, massive and far enough to be helpful, though it might be more helpful on the scale of electrodynamics etc (This is pretty much how electromagnetic radiation is made). It’s also understandable since the equation is really messy once you start factoring in acceleration, jerk, yank etc… and pretty much relies on brute force calculation until your answer is good enough for your needs which is why I left all of the math as simple as possible. I did manage to get a bit further with calculating the actual amount of acceleration than what I've shown here but this post is already way too long and all it does is increase the amount of acceleration to pretty much exactly what you would expect from dark energy.</p>
<p>Just to clarify what I’m asking in a series of hypotheses. Again, sorry if they come across as a bit assertive.</p>
<p>1: The amount of gravity felt is determined by the distance of the object when gravity reaches it, not the distance when the gravity was emitted.</p>
<p>2: If the amount of distance has increased since it was emitted and gravity is thus weaker, it should cause the emitting galaxy to appear to accelerate/redshift from us.</p>
<p>3: This effect is only really useful with massive gravitational forces, massive velocities and over massive distances.</p>
<p>4: There is an increasing amount of gravitational work done on us by the objects furthest from us as gravity decreases slower than the volume of matter which produces gravity increases.</p>
<p>5: The matter furthest from us is massive enough, traveling fast enough away from us far enough such that it creates a seemingly large acceleration on us/them, enough to be said to be dark energy.</p>
<p>6: This effect isn’t somehow canceled out by the observable universe on the opposite side.</p>
<p>7: This effect isn’t covered in the Friedmann equations.</p>
<p>8: The scale factor (the closest thing I could find to being similar to this in the Friedmann equation) is separate to this as it covers how the density of the universe changes over time, not how gravity diminishes with changing distance. I can kind of confirm this by saying that there isn’t really an easy way to calculate the amount of extra distance that should be traveled when accounting for acceleration, jerk, yank etc… as it would require you to add each term in again and again which would make the equation very messy and something of a pain to do as well as all of the other factors that need to be accounted for. Also, while relativity takes into account the instantaneous relative velocity of an object, it doesn’t take into account the amount of distance it will travel in the time for light to travel and should maybe be adjusted?</p>
<p>9: I’m fairly certain the amount of acceleration kind of tapers off eventually.</p>
<p>10: My equations aren’t entirely right because they don’t take into account the acceleration that would happen during the travel time as well as the rate of acceleration increasing etc. (I found that just using velocity was good enough to be able to prove a point, all without needing to add in assumptions such as that there would be an acceleration)</p>
<p>11: Regardless, the current math I’ve done shows that by the time you add in the other details such as acceleration during travel, relativistic effects, changing density, it should happen both ways, the extra distance traveled after gravity has reached the original position, how taking this effect into account changes previous assumptions about the universe, etc. It should add up to a reasonably large amount of force, that should at the very least be worth mentioning, provided my previous few hypotheses were correct.</p>
<p>12: Also, this is entirely different to Laplace's problem where he found that factoring in a finite, light speed of gravity to calculations of the Earth orbiting the Sun would predict the two to collide in 400 years. (It works on the same principle that allows for photons to exist as a result of moving charges.)</p>
<p>13: There isn't some form of weird MOND style effect at large distances that causes this gravity to be negligible at that distance anyways.</p>
<p><strong>A few general notes:</strong></p>
<p>You really need to think about this not in terms of space and/or time but in terms of spacetime otherwise it won’t work.</p>
<p>You also need to think of it as us accelerating away from everything in the universe which is why you can ignore the other side of the universe.</p>
<p>I probably should have given my answer in terms of redshift/luminosity but... yeah. (This post is already incredibly long.)</p>
<p>A reasonably typical hamburger contains 5x more calories by weight than a m67 hand grenade, so if you're looking for a healthy alternative to McDonald's then you can always try eating a m67 hand grenade instead. (Not sure if this has anything to do with what I was saying. But, yeah.)</p>
<p>Also, I’ve noticed that most papers don't seem as interested in the total gravitational effect of distant objects. In fact, not only do I not think I’ve ever seen anyone else mention that most of the gravity binding the observable universe together is from its furthest parts, but when I specifically tried googling for people talking about it I couldn’t see any discussion of this. This is a shame as I think it should be mentioned more in general as I’m certain where the majority of gravity in the observable universe comes from would necessarily have at least some role in understanding whatever actually causes dark energy. (Maybe I'm wrong about how much people talk about it.)</p>
<p><strong>Things that I have tried to disprove this.</strong></p>
<p>The math is wrong: Yeah, this is definitely true, there are a bunch of assumptions and estimations that I’ve made that are not correct. That being said, most of my assumptions I tried to underestimate the amount of acceleration while making my math as simple as possible so that it would be obvious what I’m doing wrong. Every time I try to make the equations more accurate than what I've shown here it always makes it asymptotically closer to what we observe without going over what would be expected.</p>
<p>The acceleration should entirely cancel out from the gravity from the other side of the universe: That’s a large part of the point, a large part of it should cancel out, this is what allows the universe to begin accelerating away from us in all directions. The only thing is that the same amount is canceled out in terms of our velocity through space so we end up changing how we see our way through time in the same way that a person stuck inside a black hole that absorbs some energy through light or loses energy through Hawking radiation would have their perception of time change. The accelerating expansion of the universe in such a case can be thought of more so as us accelerating through time.</p>
<p>I’ve also looked through papers that simulate the universe and noticed that of the very few that take into account the amount of time for gravity to travel all of them do something that would ruin/lessen the result such as only calculating it in 2d or locally since they were only really trying for simpler/smaller objects such as binary neutron stars. Again this is likely due to it being an incredibly annoying thing to calculate, especially at large distances over large amounts of time and especially since everyone seems to think that factoring in the speed of gravity at that distance should be negligible anyway.</p>
<p><strong>Things that could easily disprove me:</strong></p>
<p>The math/physics in the first part with the two galaxies is wrong: Seeing as this is the foundation of the whole argument, if any of my physics/math are wrong it will be obvious here. I'd love to know what actually happens as a result of two masses moving apart and whether or not it actually creates an acceleration or if something else would happen.</p>
<p>The physics is right but the math is wrong: Maybe if someone repeated the calculations with better assumptions such as a better measurement of the density of the Universe or do the calculations piecewise for each time period in the past better than I did they'll find the acceleration isn't enough to be at the very least worth mentioning.</p>
<p>Also, I'd love to be disproven that it's annoying to calculate/approximate this to an acceptable level of accuracy on our universe's scale, there's probably something quite interesting and beautiful that I'm missing.</p>
<p><strong>A summary.</strong></p>
<p>The Friedmann equation and other attempts including n-body simulations to model the rate of expansion of the universe are using a “static” gravity universe. That is not to say a non moving/accelerating universe but instead a universe that does not take into account the amount of extra distance traveled during the time for gravity to travel across it. This is important as the parts of the observable universe which are furthest from other parts not only have a longer gravitational travel time but, according to Hubble’s law, are also traveling away from each other at an increasingly faster rate. The importance of this stems from the fact that the mass which creates the gravity that acts on us increases with the volume of the universe for a given radius while the gravitational force that volume creates only decreases with the square of the same radius. This means that the majority of gravity which binds our observable universe together is from those galaxies which are furthest from us and are moving away from each other the fastest and so experience less gravity than they create in the opposite way that binary neutron stars collide with each other overtime through gravitational radiation. When this factor is taken into account the results very neatly line up with what would be expected from dark energy. It would also be obvious if papers did take this into account since it’s so finicky that you would need to dedicate half your paper to discussion about how you managed to take it into account. Even if it isn't the entire answer, if my minimum calculation is somehow correct, it should still account for 30% of the dark energy/acceleration in the universe which is not a bad start and would at least deserve a side note on the dark energy Wikipedia page.</p>
<p>Ultimately, the question I’m trying to ask is that I'm not sure why we attempt to calculate the force that should be holding the universe together and slowing its rate of acceleration down while ignoring that the majority of gravity comes from the parts furthest from each other which also happen to be the parts traveling away from each other so fast that (according to some dark energy theories) their light/gravity will never even reach each other, or at the very least (according to the standard cosmological model) will have their distances massively increase during that travel time. Regardless, if you've somehow managed to read all that, thank you very much for putting up with my excessively long post. I hope you have a nice evening and clear skies.</p>
<p><strong>Math</strong></p>
<p>Assumptions.
(I sourced all of these off Wikipedia's observable universe page just to act as a standard. (lmao))</p>
<p>Density of universe = 9.9 * 10^-27 kg/m^3 (Using mass of total energy (including dark energy (might be wrong to use this one (Although maybe it's possible that this is the right number to use since it factors in the energy/mass from this effect))) I’m also assuming it’s constant throughout space which probably isn’t very accurate.)</p>
<p>Radius of universe = 4.4 * 10^26 m or roughly 14500 mpc</p>
<p>Hubble constant = 70000 m/s/mpc.</p>
<p>Mass of Milky Way = 2.3 * 10^42 kg</p>
<p>1 lyr = 9.46 * 10^15m (convert it later to make the math look more clean)</p>
<p>[1]: d = r + rv/c where d = current distance between galaxies. r = distance when gravity reaches it. v = velocity of object away from observer.</p>
<p>Fg = GMm/d^2 or it can be written Fg = GMm/(r + rv/c)^2</p>
<p>d = 10,000,000 lyr + (10,000,000 lyr * 149896229 / 299792458) = 1.419 * 10^23 m</p>
<p>Fg = 6.67 * 10^-11 * 2.3*10^42 * 1 / (1.419 * 10^23)^2  =  7.61 * 10^-15 N/kg.</p>
<p>[2] Same formula as for [1], just set v = 0, r = 10000000 lyr.</p>
<p>[3] Same formula as for [1] just set r = 22500000 lyr.</p>
<p>[4] This is an oversimplified calculation but works to demonstrate a point. Essentially take two radiuses of spheres, calculate their volume and subtract the smaller one from the bigger one. Then multiply the new volume by the density of the universe. Then calculate the force of gravity using the larger sphere’s radius.
Fg = G(V1-V2)*D/r1^2
Where V1 = larger sphere volume, V2 is the smaller sphere’s volume and D = density of the volume. And volume is calculated with V = 4 pi r^3/4
It’s reasonably self-evident from the equation that the numerator will grow faster than the denominator the larger the sphere.</p>
<p>[5] Same as [4], just with V1 and V2 being larger since their radius is larger.</p>
<p>[6] 3.3 * 10^-10 * seconds in a billion years = 10 million m/s.</p>
<p>[7] Same oversimplification as in [1] but this time v = r * h (radius times Hubble constant).
r+r^2h/c</p>
<p>[8] Force is inversely proportional to the square of the radius (1/r^2) so it should be reduced by a 16th for a quadrupled distance.</p>
<p>[9] Essentially I created a python script that divides the universe into layers stacked on top of each other like an onion. I then calculate that layer’s distance and volume then calculate its gravitational force. You can find the python code that I used below.</p>
<p>[10] I repeat the same calculation as in [9] with the volume/mass being determined using the original distance but then calculate the distance for the gravitational force to act over using the same method for calculating distance as in [7] (r + hr^2/c).</p>
<p>[11] I’m just kind of guessing subtracting one from the other is how you would calculate the acceleration.</p>
<p>[12] Repeat the same calculation as in [9], [10], [11] except for r = 7000 mpc</p>
<p>[13] Repeat the same calculation as in [12] but this time throw the acceleration into this formula delta x = (at^2) / 2</p>
<p>[A] Source for Type 1a supernovae distances: <a href="https://news.fnal.gov/2024/01/final-supernova-results-from-dark-energy-survey-offer-unique-insights-into-the-expansion-of-the-universe/" rel="nofollow noreferrer">https://news.fnal.gov/2024/01/final-supernova-results-from-dark-energy-survey-offer-unique-insights-into-the-expansion-of-the-universe/</a></p>
<p><strong>Python code that I used to help calculate the rate of expansion piecewise:</strong></p>
<p>(I had too many characters in this post so I had to upload it to github instead.)</p>
<p><a href="https://github.com/Yeboii1237/Delta-Gravity/tree/main" rel="nofollow noreferrer">https://github.com/Yeboii1237/Delta-Gravity/tree/main</a></p>

CC BY-SA 4.0
2024-01-20T18:17:58.873Z
Hello Gref,  please see [ask]  What you have here is a presentation of your ideas with a request to review them.  The answer to the headline is "It already does" and "That's how General Relativity works"  The exceptionally long body is not answerable in the style of this forum, as it doesn't contain a single clearly expressed question.
James K
CC BY-SA 4.0
2024-01-20T18:54:40.417Z
@JamesK Forgive me but what you said is both true and slightly less true (The only part I (and perhaps others in the future) want is an answer to whether part 1 and the early parts of part 2 is true.)) when you say "It already does" and "That's how General Relativity works" It doesn't answer the question that is fundamental to whether or not the Friedmann equation is able to account for the extra distance added during travel, (think about the equations you have given, they don't account for velocity/acceleration according to distance/time). you may need to think harder about this.
Gref
CC BY-SA 4.0
2024-01-20T19:22:59.697Z
Sorry, this is my first question here, If part 1 is wrong then so is everything else, I will edit my question anyway. I only want confirmation on whether two objects moving apart causes one to see the other as accelerating away from it according to their gravitational force on each other and in the second part if all galaxies at that distance sum together according to the volume that creates it. I didn't mean to ask people to do my math for me, only that relativity certainly does not take this into account as you seem to think it does (relativity only takes into account instantaneous velocity)
Gref
CC BY-SA 4.0
2024-01-20T19:39:09.627Z
Yes, the Friedmann equations, which are consequences of General Relativity do account for the finite speed of "gravity".  This is not something that is added into GR, but fundamental to everything in relativity.  A finite maximum speed, and the axiom that the speed of "light" (and any other massless particle) is this speed in any frame of reference is the very basis of relativity.
James K
CC BY-SA 4.0
2024-01-20T20:24:42.950Z
@JamesK Even this answer (while very good) does not satisfy what I find to be beautiful about this question. To put this bluntly a core tenet of general relativity is that a particle can't "know" about what gravity it put out when it receives the "amount of gravity it put out back" and so later on it might be farther and experience a different force back. Your answer is correct if I'm asking about things such as Mercury's orbit deviating from Newtonian predictions, however I don't think your answer would explain a neutron star's orbit decaying or its inverse. Only the instantaneous velocity.
Gref
CC BY-SA 4.0
2024-01-21T16:35:33.613Z
"Imagine a universe with only two milky way massed galaxies ..." such a universe would not be modelled by the FLRW metric and Friedmann equations.  "I realize using general relativity is more accurate than Newton’s laws...", yes, so use GR. Newtonian gravity cannot be used to explain the dynamics of the universe as a whole. The rest is a consequence of your attempts to use Newtonian gravitational ideas in regimes where it doesn;t work. The FLRW metric isn't static it has an explicit time-dependence through the evolution of the scale factor.
ProfRob
CC BY-SA 4.0
2024-01-21T16:49:53.423Z
Also you haven't applied Newtonian physics correctly. You talk about/calculate the gravitational forces exerted on us by "shells" of the universe at different distances. The answer in each case is precisely zero. This is the well-known "Newton's shell theorem". https://en.wikipedia.org/wiki/Shell_theorem.
ProfRob
CC BY-SA 4.0
2024-01-21T18:26:56.083Z
Umm, what IS the question?
Nilay Ghosh
<gravity><cosmology><general-relativity><redshift><dark-energy>
Shouldn't the Friedmann equation take into account the massive extra distance traveled during the time for gravity to travel across the universe?
Gref
# Shouldn't the Friedmann equation take into account the massive extra distance traveled during the time for gravity to travel across the universe?

(Asked by: username_0 on 2024-01-20)

I’ve got a question that has been bothering me for the past few years that of the few sources that are only somewhat related to this, none of them say anything about this should be wrong so I can’t disprove it and every time I try to throw math at it, it just seems to get more and more reasonable. I’m going to phrase my questions as statements, so I’m sorry if it comes across as a little bit assertive, I just think it’s clearer what I’m asking (and how I’m wrong) as a statement, and if nothing else, at least it will feel better when you get to tell me I’m wrong. I’m also going to repeat the same things a few times in different ways just because otherwise I don't think it's clear what I'm saying. The first part seems negligible but it isn’t until I bring up the second part that the “majority” of gravity which acts on everything/us in the observable universe and binds it together is from the matter in the parts of the observable universe furthest from us which are, according to Hubble's law, also the parts moving the fastest away from us and should feel this effect more, that it makes sense. I've had a lot of fun with the math and physics of trying to disprove this (as you can tell by the length of the question) but have come up short so I figured I might as well let all of you try your hand at seeing if you can disprove it.

**Part one:**

Imagine a universe with only two milky way massed galaxies which are moving apart in it, one is our home galaxy and the other is the away galaxy which is 10 million lightyears from us and traveling away from our galaxy at half the speed of light. When using Newton’s law of gravitation as per normal we would predict that the amount of force being applied to each galaxy would be in constant equilibrium and therefore there wouldn’t be any acceleration and the two galaxies would continue traveling without redshifting at all, even though the force they exert on each other would be decreasing with distance/time. (I realize using general relativity is more accurate than Newton’s laws, especially at high relative velocities, but it works out to be roughly the same (as in it contributes somewhat to what I’m finding) either way and Newton’s laws make it much easier. You’ll see what I mean.)

What this prediction didn’t take into account was that by the time gravity travels from our galaxy to theirs, they will have moved an extra 5,000,000 lightyears and so will experience gravity as though they were 15,000,000 lightyears away (yeah, this is wrong, they would be further away because among the other things that I’m missing and will mention later, this is only how much further they would have traveled in the time for gravity to reach their original position and assumes they stop after that. However, this makes life/math easier despite being one of the major things working against what I’ve found) experiencing only 7.61 \* 10^-15 N/kg [1] rather than the 17.1 \* 10^-15 N/kg [2] that would normally be expected at 10,000,000 lightyears. This will create an equal and opposite amount of gravity back towards us and by the time that gravity travels back to us we will have traveled to be 22,500,000 lightyears away and only experience 3.38 \* 10^-15 N/kg [3]. And since we know that the only situation where there is no acceleration is in the original prediction where the forces add up and cancel out, we know that in the second, slightly more accurate equation that there must be a tiny, but yes, still existing acceleration. This means that if we had instruments powerful enough (I guess in this universe there is no “totally necessary” military industrial complex to feed instead) and watched for long enough we would notice that overtime the spectral lines of the galaxy would not only begin to redshift by a small amount, but that the rate at which those spectral lines redshift would also begin to increase and then decrease (it would increase if the galaxy was currently close enough and decrease as it gets further away). (Also, this effect is separate to Laplace's delayed gravity problem, in case it reminded you of that.)

To the observer using Newton’s laws as per normal this redshift would be completely impossible to explain and would have to say that there is a certain invisible force acting upon the galaxies. But to the observer who takes into account the time for gravity to travel between the galaxies this is just the normal consequences of gravity and just depends on the distance, mass and velocity of the other galaxy and our own. Moreover, they would say that even while not accounting for acceleration that this would cause, any object whose speed exceeds the escape velocity of the gravitational system should not only escape the system but also begin to accelerate away from it. Just as an aside, the inverse of this, ie; when two objects are moving towards each other while being unable to escape each other's orbit, is the same thing that causes an acceleration/deceleration on binary blackholes/neutron stars to have their orbits decay and collide with each other through gravitational radiation, so I know this is at least somewhat a real effect. Another note is that this acceleration isn’t very dependent on our mass and should appear regardless of whether we are ourselves in a galaxy or if we were just floating through intergalactic space in a spacesuit by ourselves away from a galaxy and vice/versa, as long as one object has a significant amount of mass then it should work, if they had the same amount of mass then it would double. (In our case it would double since at large scales the universe is homogeneous, so the same thing that is happening to us from our furthest parts of the observable universe would also be happening to that other side of the observable universe from us, though it may take extra time for the gravitational effect to equalize between the two. I'm just going to keep it simple and assume that it doesn't double for my math though.)

**Part two:**

This is almost always a very useless and negligible piece of information as you need large amounts of mass, distance and velocity for it to ever be relevant, luckily for us according to Hubble's law, the further out in the universe you look the faster it moves away from us and the more mass there is. Why is this important? Normally when we think of the amount of gravity that acts on us in the universe we think that it is the objects closest to us that exert the most amount of gravity on us. However, since gravity only decreases with the square of the distance and the volume of space which contains mass to create gravity increases cubically (like the surface area of a sphere against its volume), the sum of the objects furthest from us in the observable universe begin to produce a more notable amount of gravitational force on us than those closest (Yeah, sure, Earth and the local group produces more gravity on us, but that obviously doesn’t matter in this case since it is also applied to them.). For example, an oversimplified calculation of the total gravitational force applied to us by the region between 10-20% of the way to the edge of the observable universe only produces 2.1 \* 10^-10 N/kg [4] despite being much closer than the furthest 10% of the observable universe which produces 3.3 \* 10^-10 N/kg on us [5] (From 1,400 mpc to 2,800 mpc, vs From 12,800 mpc to 14,200 mpc.)

Though small on our scale, 3.3 \* 10^-10 N/kg adds up to being 10 million m/s every billion years [6], an appreciable amount. The interesting thing is that this force being this large, is similar to the Friedmann equation, in that it hinges itself upon the idea that the galaxies furthest from us won’t have moved much in the time it takes for gravity to reach them when in reality even an observer in a non accelerating, coasting observable universe with no dark energy will have the parts furthest from its center move by an extra 70,000 m/s for every megaparsec away from them. This means that a person on the other side of the observable universe, 4.4 \* 10^26 meters away, will have traveled an extra 14.6 \* 10^26 meters by the time gravity reaches it according to Hubble’s law [7], quadrupling its original distance, meaning it will experience a 16th the amount of gravity that was originally predicted [8]. It will then send out an equal and opposite amount of gravitational force back towards us on the far side of their observable universe and by the time it reaches us we will be 2 \* 10^28 m away, 100 times further than when we started, meaning we will experience even less gravitational force than we originally sent out, this means that there is a net imbalance in the gravitational forces acting on us and therefore there must be an apparent acceleration, all without needing dark energy. (Again, all of this is all kind of wrong but wrong in such a way that it mostly works against me while still proving a point.)

As for how much acceleration we can expect from this in total, the total classical gravitational force from all the galaxies acting on us across the universe is 36.8 \* 10^-10 N/kg [9] (This is done piecewise with python code to make it more accurate/easier. I go over it in more detail in [9] of the math section and in the code itself which I put below) but when we take into account the added distance from the speed of gravity, this goes down to 8.43 \* 10^-10 N/kg [10], a 4x decrease. This is equivalent to a net acceleration of 28.3 \* 10^-10 N/kg [11] which means that the sum of the parts of the universe farthest from us should accelerate away from us by an extra 89,000,000 m/s every billion years. Though you would have to take off roughly ¾ from it since that is the force from every direction when it should really only be the sum of the positive forces in one direction (sort of like how you would get attracted towards a direction by a semi sphere of mass but it would still be less than if that mass was all concentrated into a small pipe that was directly in line with the desired direction) (It also requires some thought as to why you can ignore the other side of the universe/the negative forces in this example (Spacetime tomfoolery)). Even then we still find that 20 million m/s extra every billion years / 260 million m/s during the lifetime of the universe (I’m trying to be extremely conservative and include everything that should decrease the effect, the real figure is probably higher) and puts us well within the range of what would be expected from dark energy and that’s before you start adding in things that will increase this effect such as the extra acceleration itself, changing density etc. With the parts closer to us seeming to accelerate much less, for example the region 7000 mpc away has only got a net acceleration of 11.9 \* 10^-10 N/kg [12].

It’s honestly kind of a bit uncanny how well this works if we repeat the calculation for something closer to us where there is both better real world data and my equations and assumptions are more accurate such as type 1a supernovae which are currently 3000 mpc away we find again that our prediction is extremely reasonable. If we assume the current acceleration has been happening for the last 13 billion years then we will find that these galaxies will have traveled an extra 214 mpc [13]. This is, while slightly lower, pretty much spot on with what we observe (around a little over 500 mpc [A]). Though as I've said before I would expect that the assumptions in my math should underestimate it by about that much. (Yeah, I also realize this is an egregious assumption when you put any level of actual thought into it, I’m just trying to show in a minimum case that the effect is within the ballpark and when you experiment with more accurate equations for distance and use the actual Hubble constant, distance and density at each time period in the past it gives a larger acceleration that is far more accurate and closer to what we observe anyways). Another interesting note is how it can not only predict an increasing rate of expansion both right where it is increasing with the effect falling off exactly as far away from us as where we observe.

This wouldn’t be a problem to me if it wasn’t for the fact that neither the Friedmann equation or any paper/source I could find make much of a note of the effect of us moving away from the edge of the universe. They all use something more suited to how we would calculate escape velocity for something small, such as a satellite leaving our Earth / Solar system where the distance traveled during the time for gravity to catch up would be negligible. This assumption is pretty understandable as even for something massive, somewhat far away and rapidly traveling towards us such as the Andromeda galaxy only has a 0.2% increase in the amount of gravity it feels due to this effect, so the scale of the observable universe is pretty much the one time there is anything fast, massive and far enough to be helpful, though it might be more helpful on the scale of electrodynamics etc (This is pretty much how electromagnetic radiation is made). It’s also understandable since the equation is really messy once you start factoring in acceleration, jerk, yank etc… and pretty much relies on brute force calculation until your answer is good enough for your needs which is why I left all of the math as simple as possible. I did manage to get a bit further with calculating the actual amount of acceleration than what I've shown here but this post is already way too long and all it does is increase the amount of acceleration to pretty much exactly what you would expect from dark energy.

Just to clarify what I’m asking in a series of hypotheses. Again, sorry if they come across as a bit assertive.

1: The amount of gravity felt is determined by the distance of the object when gravity reaches it, not the distance when the gravity was emitted.

2: If the amount of distance has increased since it was emitted and gravity is thus weaker, it should cause the emitting galaxy to appear to accelerate/redshift from us.

3: This effect is only really useful with massive gravitational forces, massive velocities and over massive distances.

4: There is an increasing amount of gravitational work done on us by the objects furthest from us as gravity decreases slower than the volume of matter which produces gravity increases.

5: The matter furthest from us is massive enough, traveling fast enough away from us far enough such that it creates a seemingly large acceleration on us/them, enough to be said to be dark energy.

6: This effect isn’t somehow canceled out by the observable universe on the opposite side.

7: This effect isn’t covered in the Friedmann equations.

8: The scale factor (the closest thing I could find to being similar to this in the Friedmann equation) is separate to this as it covers how the density of the universe changes over time, not how gravity diminishes with changing distance. I can kind of confirm this by saying that there isn’t really an easy way to calculate the amount of extra distance that should be traveled when accounting for acceleration, jerk, yank etc… as it would require you to add each term in again and again which would make the equation very messy and something of a pain to do as well as all of the other factors that need to be accounted for. Also, while relativity takes into account the instantaneous relative velocity of an object, it doesn’t take into account the amount of distance it will travel in the time for light to travel and should maybe be adjusted?

9: I’m fairly certain the amount of acceleration kind of tapers off eventually.

10: My equations aren’t entirely right because they don’t take into account the acceleration that would happen during the travel time as well as the rate of acceleration increasing etc. (I found that just using velocity was good enough to be able to prove a point, all without needing to add in assumptions such as that there would be an acceleration)

11: Regardless, the current math I’ve done shows that by the time you add in the other details such as acceleration during travel, relativistic effects, changing density, it should happen both ways, the extra distance traveled after gravity has reached the original position, how taking this effect into account changes previous assumptions about the universe, etc. It should add up to a reasonably large amount of force, that should at the very least be worth mentioning, provided my previous few hypotheses were correct.

12: Also, this is entirely different to Laplace's problem where he found that factoring in a finite, light speed of gravity to calculations of the Earth orbiting the Sun would predict the two to collide in 400 years. (It works on the same principle that allows for photons to exist as a result of moving charges.)

13: There isn't some form of weird MOND style effect at large distances that causes this gravity to be negligible at that distance anyways.

**A few general notes:**

You really need to think about this not in terms of space and/or time but in terms of spacetime otherwise it won’t work.

You also need to think of it as us accelerating away from everything in the universe which is why you can ignore the other side of the universe.

I probably should have given my answer in terms of redshift/luminosity but... yeah. (This post is already incredibly long.)

A reasonably typical hamburger contains 5x more calories by weight than a m67 hand grenade, so if you're looking for a healthy alternative to McDonald's then you can always try eating a m67 hand grenade instead. (Not sure if this has anything to do with what I was saying. But, yeah.)

Also, I’ve noticed that most papers don't seem as interested in the total gravitational effect of distant objects. In fact, not only do I not think I’ve ever seen anyone else mention that most of the gravity binding the observable universe together is from its furthest parts, but when I specifically tried googling for people talking about it I couldn’t see any discussion of this. This is a shame as I think it should be mentioned more in general as I’m certain where the majority of gravity in the observable universe comes from would necessarily have at least some role in understanding whatever actually causes dark energy. (Maybe I'm wrong about how much people talk about it.)

**Things that I have tried to disprove this.**

The math is wrong: Yeah, this is definitely true, there are a bunch of assumptions and estimations that I’ve made that are not correct. That being said, most of my assumptions I tried to underestimate the amount of acceleration while making my math as simple as possible so that it would be obvious what I’m doing wrong. Every time I try to make the equations more accurate than what I've shown here it always makes it asymptotically closer to what we observe without going over what would be expected.

The acceleration should entirely cancel out from the gravity from the other side of the universe: That’s a large part of the point, a large part of it should cancel out, this is what allows the universe to begin accelerating away from us in all directions. The only thing is that the same amount is canceled out in terms of our velocity through space so we end up changing how we see our way through time in the same way that a person stuck inside a black hole that absorbs some energy through light or loses energy through Hawking radiation would have their perception of time change. The accelerating expansion of the universe in such a case can be thought of more so as us accelerating through time.

I’ve also looked through papers that simulate the universe and noticed that of the very few that take into account the amount of time for gravity to travel all of them do something that would ruin/lessen the result such as only calculating it in 2d or locally since they were only really trying for simpler/smaller objects such as binary neutron stars. Again this is likely due to it being an incredibly annoying thing to calculate, especially at large distances over large amounts of time and especially since everyone seems to think that factoring in the speed of gravity at that distance should be negligible anyway.

**Things that could easily disprove me:**

The math/physics in the first part with the two galaxies is wrong: Seeing as this is the foundation of the whole argument, if any of my physics/math are wrong it will be obvious here. I'd love to know what actually happens as a result of two masses moving apart and whether or not it actually creates an acceleration or if something else would happen.

The physics is right but the math is wrong: Maybe if someone repeated the calculations with better assumptions such as a better measurement of the density of the Universe or do the calculations piecewise for each time period in the past better than I did they'll find the acceleration isn't enough to be at the very least worth mentioning.

Also, I'd love to be disproven that it's annoying to calculate/approximate this to an acceptable level of accuracy on our universe's scale, there's probably something quite interesting and beautiful that I'm missing.

**A summary.**

The Friedmann equation and other attempts including n-body simulations to model the rate of expansion of the universe are using a “static” gravity universe. That is not to say a non moving/accelerating universe but instead a universe that does not take into account the amount of extra distance traveled during the time for gravity to travel across it. This is important as the parts of the observable universe which are furthest from other parts not only have a longer gravitational travel time but, according to Hubble’s law, are also traveling away from each other at an increasingly faster rate. The importance of this stems from the fact that the mass which creates the gravity that acts on us increases with the volume of the universe for a given radius while the gravitational force that volume creates only decreases with the square of the same radius. This means that the majority of gravity which binds our observable universe together is from those galaxies which are furthest from us and are moving away from each other the fastest and so experience less gravity than they create in the opposite way that binary neutron stars collide with each other overtime through gravitational radiation. When this factor is taken into account the results very neatly line up with what would be expected from dark energy. It would also be obvious if papers did take this into account since it’s so finicky that you would need to dedicate half your paper to discussion about how you managed to take it into account. Even if it isn't the entire answer, if my minimum calculation is somehow correct, it should still account for 30% of the dark energy/acceleration in the universe which is not a bad start and would at least deserve a side note on the dark energy Wikipedia page.

Ultimately, the question I’m trying to ask is that I'm not sure why we attempt to calculate the force that should be holding the universe together and slowing its rate of acceleration down while ignoring that the majority of gravity comes from the parts furthest from each other which also happen to be the parts traveling away from each other so fast that (according to some dark energy theories) their light/gravity will never even reach each other, or at the very least (according to the standard cosmological model) will have their distances massively increase during that travel time. Regardless, if you've somehow managed to read all that, thank you very much for putting up with my excessively long post. I hope you have a nice evening and clear skies.

**Math**

Assumptions.
(I sourced all of these off Wikipedia's observable universe page just to act as a standard. (lmao))

Density of universe = 9.9 \* 10^-27 kg/m^3 (Using mass of total energy (including dark energy (might be wrong to use this one (Although maybe it's possible that this is the right number to use since it factors in the energy/mass from this effect))) I’m also assuming it’s constant throughout space which probably isn’t very accurate.)

Radius of universe = 4.4 \* 10^26 m or roughly 14500 mpc

Hubble constant = 70000 m/s/mpc.

Mass of Milky Way = 2.3 \* 10^42 kg

1 lyr = 9.46 \* 10^15m (convert it later to make the math look more clean)

[1]: d = r + rv/c where d = current distance between galaxies. r = distance when gravity reaches it. v = velocity of object away from observer.

Fg = GMm/d^2 or it can be written Fg = GMm/(r + rv/c)^2

d = 10,000,000 lyr + (10,000,000 lyr \* 149896229 / 299792458) = 1.419 \* 10^23 m

Fg = 6.67 \* 10^-11 \* 2.3\*10^42 \* 1 / (1.419 \* 10^23)^2 = 7.61 \* 10^-15 N/kg.

[2] Same formula as for [1], just set v = 0, r = 10000000 lyr.

[3] Same formula as for [1] just set r = 22500000 lyr.

[4] This is an oversimplified calculation but works to demonstrate a point. Essentially take two radiuses of spheres, calculate their volume and subtract the smaller one from the bigger one. Then multiply the new volume by the density of the universe. Then calculate the force of gravity using the larger sphere’s radius.
Fg = G(V1-V2)\*D/r1^2
Where V1 = larger sphere volume, V2 is the smaller sphere’s volume and D = density of the volume. And volume is calculated with V = 4 pi r^3/4
It’s reasonably self-evident from the equation that the numerator will grow faster than the denominator the larger the sphere.

[5] Same as [4], just with V1 and V2 being larger since their radius is larger.

[6] 3.3 \* 10^-10 \* seconds in a billion years = 10 million m/s.

[7] Same oversimplification as in [1] but this time v = r \* h (radius times Hubble constant).
r+r^2h/c

[8] Force is inversely proportional to the square of the radius (1/r^2) so it should be reduced by a 16th for a quadrupled distance.

[9] Essentially I created a python script that divides the universe into layers stacked on top of each other like an onion. I then calculate that layer’s distance and volume then calculate its gravitational force. You can find the python code that I used below.

[10] I repeat the same calculation as in [9] with the volume/mass being determined using the original distance but then calculate the distance for the gravitational force to act over using the same method for calculating distance as in [7] (r + hr^2/c).

[11] I’m just kind of guessing subtracting one from the other is how you would calculate the acceleration.

[12] Repeat the same calculation as in [9], [10], [11] except for r = 7000 mpc

[13] Repeat the same calculation as in [12] but this time throw the acceleration into this formula delta x = (at^2) / 2

[A] Source for Type 1a supernovae distances: <https://news.fnal.gov/2024/01/final-supernova-results-from-dark-energy-survey-offer-unique-insights-into-the-expansion-of-the-universe/>

**Python code that I used to help calculate the rate of expansion piecewise:**

(I had too many characters in this post so I had to upload it to github instead.)

<https://github.com/Yeboii1237/Delta-Gravity/tree/main>

**Tags:** gravity, cosmology, general-relativity, redshift, dark-energy
> **Comments:**
> **username_1**: Hello Gref,  please see [ask]  What you have here is a presentation of your ideas with a request to review them.  The answer to the headline is "It already does" and "That's how General Relativity works"  The exceptionally long body is not answerable in the style of this forum, as it doesn't contain a single clearly expressed question.  
> **username_0**: @JamesK Forgive me but what you said is both true and slightly less true (The only part I (and perhaps others in the future) want is an answer to whether part 1 and the early parts of part 2 is true.)) when you say "It already does" and "That's how General Relativity works" It doesn't answer the question that is fundamental to whether or not the Friedmann equation is able to account for the extra distance added during travel, (think about the equations you have given, they don't account for velocity/acceleration according to distance/time). you may need to think harder about this.  
> **username_0**: Sorry, this is my first question here, If part 1 is wrong then so is everything else, I will edit my question anyway. I only want confirmation on whether two objects moving apart causes one to see the other as accelerating away from it according to their gravitational force on each other and in the second part if all galaxies at that distance sum together according to the volume that creates it. I didn't mean to ask people to do my math for me, only that relativity certainly does not take this into account as you seem to think it does (relativity only takes into account instantaneous velocity)  
> **username_1**: Yes, the Friedmann equations, which are consequences of General Relativity do account for the finite speed of "gravity".  This is not something that is added into GR, but fundamental to everything in relativity.  A finite maximum speed, and the axiom that the speed of "light" (and any other massless particle) is this speed in any frame of reference is the very basis of relativity.  
> **username_0**: @JamesK Even this answer (while very good) does not satisfy what I find to be beautiful about this question. To put this bluntly a core tenet of general relativity is that a particle can't "know" about what gravity it put out when it receives the "amount of gravity it put out back" and so later on it might be farther and experience a different force back. Your answer is correct if I'm asking about things such as Mercury's orbit deviating from Newtonian predictions, however I don't think your answer would explain a neutron star's orbit decaying or its inverse. Only the instantaneous velocity.  
> **username_2**: "Imagine a universe with only two milky way massed galaxies ..." such a universe would not be modelled by the FLRW metric and Friedmann equations.  "I realize using general relativity is more accurate than Newton’s laws...", yes, so use GR. Newtonian gravity cannot be used to explain the dynamics of the universe as a whole. The rest is a consequence of your attempts to use Newtonian gravitational ideas in regimes where it doesn;t work. The FLRW metric isn't static it has an explicit time-dependence through the evolution of the scale factor.  
> **username_2**: Also you haven't applied Newtonian physics correctly. You talk about/calculate the gravitational forces exerted on us by "shells" of the universe at different distances. The answer in each case is precisely zero. This is the well-known "Newton's shell theorem". https://en.wikipedia.org/wiki/Shell_theorem.  
> **username_3**: Umm, what IS the question?  

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## 1 answers

### Answer by username_4 on 2024-01-20. Score: 1

The fact that photons and gravity (gravitons) travel at finite speeds has not escaped the attention of physicists. It also has been noticed that Newtonian gravity, with no retardation of the forces, describes the motions of the solar system, so if GR, with propagation at light speed, does not imitate Newtonian with instantaneous propagation, at least in the weak limit and small scales, then GR would not explain how the solar system is stable.

There are terms in the potential of the Post Newtonian formulation (simplified GR theory for the weak limit cases) that depend on velocity and higher order derivatives that cancel the delay time effects. As a result, to at least first order, taking the short cut of assuming instantaneous positions also works.

In GR, if the source is moving, then the accelerations generated do not point back to the location of the source when the gravitons left the source. The gravitational accelerations are automatically corrected to the extrapolated instantaneous position.
For details you can see the paper:
"[Aberration and the speed of gravity](https://ui.adsabs.harvard.edu/abs/2000PhLA..267...81C/abstract)" by S. Carlip (UC Davis).
It makes the interesting point that the simplest Lorentz Invariant potential is $\Phi = m/\vec{r}$ where $\vec{r}$ is the 4-dimensional separation vector rather than the spatial 3-vector, and acceleration, derived from its 4-vector gradient, contains velocity information.

The math for the electrodynamics case, in its gory details, was worked out in 1900 for electromagnetism by Liénard and Wiechert. You can go through the math [here](https://en.m.wikipedia.org/wiki/Li%C3%A9nard%E2%80%93Wiechert_potential). It turns out that GR can be cast into a similar form as electromagnetism, see [Gravitoelectromagnetism](https://en.wikipedia.org/wiki/Gravitoelectromagnetism)

[Here](https://www.desy.de/user/projects/Physics/Relativity/GR/grav_speed.html) is a hand waving argument about some of this.

> **Comments:**
> **username_0**: If your immediate thought from the heading was similar to Laplace's problem then I'm glad you thought of this right away, as this is an entirely fair point if it was the point I think I am making. According to my math (admittedly I didn't show this anyway) it would take many thousands of lifetimes of the universe for the Earth to decay rather than the 400 or so years according to Laplace.  The effect I am talking about doesn't deal with constant velocities as it assumes that there will be a decreasing acceleration with distance. I'd recommend reading the rest of my question to see what I mean  

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