Why we can never travel faster than the speed of light

[Mark Quayle]

The issue involves objects travelling in space-time rather than being carried by space-time expansion such as the recession velocities of distant galaxies which can exceed the speed of light and their velocities depend on the distance from the observer.

Suppose you are on a train moving at a constant velocity, there is no such thing as an absolute frame of reference, the train could be moving in a forward direction at a velocity v or the train is stationary and the countryside is moving backwards at a velocity of -v.
There is no preferred frame of reference and there is no physics experiment that can differentiate between the two for the observer on the train.
An observer on the ground however is stationary and the train is moving forward at a velocity v in his frame of reference.

This gets back to the original example of observers A, B, and C.
In observer A’s frame of reference, he may for all intents and purposes be stationary and the approach velocity is observer B approaching and similarly in observer B’s frame of reference he may be stationary and it is observer A that is approaching.
There is no preferred frame of reference and A and B’s frames of references are equally valid.
In observer C’s frame of reference known as the ground or laboratory frame, the symmetry of the velocities is observed as A and B are travelling towards each other at a speed of 0.6c but the approach speed is 0.88c not 1.2c as nothing can exceed the speed of light c for objects moving through space-time.

Well, one thing for sure, you and I agree on: I just don't get this.

 

[Hans Blaster]

@sjastro @Hans Blaster

Ok, well how about this then? If you took an object (with mass) in space, and constantly applied the exact same amount of thrust throughout to it, and didn't ever change it, to make that object accelerate and gain speed, are you saying that it would no longer ever gain any more speed at some point? Or just what exactly? Because that doesn't make very much practical or physical sense to me, etc? Yes? No?

Oh, and if so, then maybe also why please, etc?

Much thanks.

Take Care/God Bless.

For a fixed mass object under constant thrust (that is a constant applied force), the momentum would keep going up at a steady rate, but as you got closer to the speed of light, the change in velocity from that applied force would go down. This is where that relativistic factor 1/√(1-v²/c²) comes in. One way to think about is with a relativistic mass, M, which is that factor multiplied by the rest mass, M = 1/√(1-v²/c²) * m. As v gets closer to c, 1/√(1-v²/c²) becomes very large, so the F=Ma you were applying is constant, but M gets very large so acceleration, a, gets small.

F = 1/√(1-v²/c²) * m a

Accelerating further to increase velocity v, only makes the 1/√(1-v²/c²) factor even bigger. Every bit of applied force yields less and less change in velocity as v approaches c and it never gets there.

One way to think about it is with Zeno's paradox, which goes something like this..

If you start on one side of the room and at the top of every minute you walk half of the remaining distance and stop, will you ever get to the other wall? At 12:00:00, you walk half way and half is left, at 12:01:00, you walk half of the remaining 1/2 with half of the 1/2 remaining, and so on. Each time you stop before the wall, but get closer and closer. Acceleration near the speed of light is like that, any increase in speed makes getting the last bit to reach "c" harder and no matter how much force you apply you'll never get there.

 

[SelfSim]

Well, one thing for sure, you and I agree on: I just don't get this.

But my question is: 'Why not?'

 

[sjastro]

Yes, so, if I am hurtling through space at, say .6c away from the center of the galaxy, and a star is receding from the center of the galaxy in a trajectory opposite mine at .6c, we will not be able to see each other, because the light never arrives from the star to me, center of the galaxy irrelevant, except for the purposes of describing the situation.

This is why, as described by some authorities on the subject, there are, theoretically, distant galaxies we never see, not because they are too far and too faint, but because they are receding relative to us, too fast for the light from them to ever reach us.

No in your example galaxies are gravitationally bound systems which prevent spacetime expansion, you and the star will have a relative recession velocity of 0.88c, the same magnitude as the approach velocity.

Since the recession velocity is less than c, you and the star will remain causally connected despite the fact the star will eventually become invisible due to the inverse square law which is the relationship between the star’s brightness and its distance from you.
In this case it is not the light never reaching us but not reaching us within a particular time interval.

Likewise, then, (but for the general expansion of the universe) we could theoretically be on a collision course with a celestial body moving towards us so fast that we will never see it until the collision?

No since the universe is expanding a celestial body moving towards us must be travelling through spacetime in which case it is travelling less than the speed of light.
Information however can travel at the speed of light and we should be able to observe an impending collision as light from the object will reach us beforehand as being blue-shifted as evidenced from its spectrum.

 

[Tinker Grey]

A classic metaphor, I think. [Actual scientists, please pardon me.]

Imagine dots on a balloon. Now, add air. Watch the dots move away from each other: spacetime expansion.

Now imagine an ant walking on the balloon but with an speed limit. You can imagine the balloon expanding such that the dots are moving away from each other faster than the ant can walk.

The subtlety here is that the dots far away from the ant are moving away from the ant than the dots close to the ant. Thus, while the ant may get to the dot closest to it, it will never get to the dots furthest from it.

In spacetime, light travels like the ant toward some dot. It must travel along the surface of balloon. But the fabric of spacetime (balloon) is expanding faster than that while no particle in the system exceeds the speed limit.

 

[sjastro]

Well, I will agree with you that the speed of light doesn't actually change, but I don't see why or how if you are moving straight away from one light source, and straight to or towards another, why what you would be seeing from either one those light sources wouldn't change, or be either slower or faster as you went respectively, etc? Care to explain how or why that isn't, or wouldn't be true maybe?

There are many things scientists cannot explain, the mechanism for gravity, why the mass of the proton 1836X heavier than the electron, and a myriad of other things including why the speed of light is invariant to all observers.

The reason we come to the conclusion the speed of light is invariant is because it is exactly 350 years since its speed was first measured and no one over the centuries has found the speed to change irrespective if the observer or the light source move towards or away from each other.
Another reason is that if the speed can be exceeded in an observer's frame of reference causality is violated and events are no longer ordered, you could die before being born.

 

[sfs]

But understanding why it always takes more and more work, or force, to the point of approaching infinity, and maybe why, could really, really help me in understanding this right now currently?

The short answer to "why?" is "because that's the universe we live in". The longer answer is a more mathematically precise way of saying the same thing.

 

[Neogaia777]

@sjastro @Hans Blaster

I thank you guys for putting up with me, and my annoying, child-like questions. I will probably need to study a whole lot more, and become much more educated about these subjects before I can get back to you with anything else. But I thank you for your time tough.

Take Care/God Bless.

 

[Mark Quayle]

No in your example galaxies are gravitationally bound systems which prevent spacetime expansion, you and the star will have a relative recession velocity of 0.88c, the same magnitude as the approach velocity.

So, space-time, wait ...huh? You are saying that there is a normal space that is different from space near gravitational forces? Spacetime does not expand within the galaxy, but between galactic groups it does? So, where spacetime does not expand, nothing can travel faster than light relative to anything else in the galaxy --not even theoretically, nevermind actually. But one galactic system can, relative to other galactic systems, because spacetime is different where there are no celestial bodies nearby? Is that what you are saying?

Since the recession velocity is less than c, you and the star will remain causally connected despite the fact the star will eventually become invisible due to the inverse square law which is the relationship between the star’s brightness and its distance from you.
In this case it is not the light never reaching us but not reaching us within a particular time interval.

I understand that, IF the recession velocity is less than c. So far, however, I don't see how it is not theoretically possible for it to be receding at more than the speed of light.

No since the universe is expanding a celestial body moving towards us must be travelling through spacetime in which case it is travelling less than the speed of light.

Wait. Now spacetime expansion by definition implies that nothing can be moving contrary to that expansion at greater than the speed of light compared to anything else? Is that because spacetime is expanding so fast, or what? Even if we are talking between galactic systems, and not within galaxies where "spacetime does not expand"?

Information however can travel at the speed of light and we should be able to observe an impending collision as light from the object will reach us beforehand as being blue-shifted as evidenced from its spectrum.

Only because the celestial body cannot be approaching us faster than the speed of light.

I'm sorry. I'm ignorant enough not to know even if you are or are not right. But it doesn't add up to me. My intuitive use of space is 3 dimensional distance, whether occupied or not, regardless of how immense. Galaxies and galactic clusters are moving 'through' that 3 dimensional space. Some things make sense to me, such as the fact that a galaxy can be receding from us at more than the speed of light. Yet, apparently by the fact that it is happening, and other phenomena, such as redshift, affirming that these things are by far mostly moving away from us and from each other, we name that "expansion", as though it was what causes the increasing distances. Yet, here, 'locally', you tell me there can be no theoretical motion of two bodies relative to each other at more than the speed of light, while locally, the normal geometry of 3 dimensions works, but in the larger 'empty' space, it does not? I'm lost.

 

[Neogaia777]

So, space-time, wait ...huh? You are saying that there is a normal space that is different from space near gravitational forces? Spacetime does not expand within the galaxy, but between galactic groups it does? So, where spacetime does not expand, nothing can travel faster than light relative to anything else in the galaxy --not even theoretically, nevermind actually. But one galactic system can, relative to other galactic systems, because spacetime is different where there are no celestial bodies nearby? Is that what you are saying?

I understand that, IF the recession velocity is less than c. So far, however, I don't see how it is not theoretically possible for it to be receding at more than the speed of light.

Wait. Now spacetime expansion by definition implies that nothing can be moving contrary to that expansion at greater than the speed of light compared to anything else? Is that because spacetime is expanding so fast, or what? Even if we are talking between galactic systems, and not within galaxies where "spacetime does not expand"?

Only because the celestial body cannot be approaching us faster than the speed of light.

I'm sorry. I'm ignorant enough not to know even if you are or are not right. But it doesn't add up to me. My intuitive use of space is 3 dimensional distance, whether occupied or not, regardless of how immense. Galaxies and galactic clusters are moving 'through' that 3 dimensional space. Some things make sense to me, such as the fact that a galaxy can be receding from us at more than the speed of light. Yet, apparently by the fact that it is happening, and other phenomena, such as redshift, affirming that these things are by far mostly moving away from us and from each other, we name that "expansion", as though it was what causes the increasing distances. Yet, here, 'locally', you tell me there can be no theoretical motion of two bodies relative to each other at more than the speed of light, while locally, the normal geometry of 3 dimensions works, but in the larger 'empty' space, it does not? I'm lost.

I'll try to add my limited understanding.

Space (space-time) (distant galaxies, and galactic clusters, and such) appears to be moving (expanding) out away from us as the center, and the further out away from us it is with us as the center, faster the further it is out away from us it is as the center, etc, to the point that at a certain point (distance) it begins to appear to be moving (receding/expanding) out away from us as the center at speeds (velocities) beyond the speed of light, etc, which we cannot see beyond that after that everywhere, etc. But, and here's the kicker, it always appears that way no matter where you are in the universe, with wherever you are at in it, and always with you as the center of that always everywhere, etc. So how much/fast it is really happening has to be somewhere in-between. Or at least, that's been a conclusion of mine lately. I say it's all moving/growing/expanding/receding at the same rate/velocity/speed equally everywhere, etc. And at those scales, even if it was happening at the exact speed of light equally everywhere (which I don't think it is, but is slower than that maybe) (but I also don't know how fast or slow it is all happening for sure, etc) But and/or anyway, at those scales, even if it was all happening at the speed of light, etc, it still would not be happening "fast" at those scales, but is actually happening actually quite slow, or slowly, etc. But it's happening everywhere the same, and at the same equal "rate" equally everywhere, etc, evidenced by the fact that it always is/always was/always will, look like that everywhere, etc, no matter where you are at in it equally everywhere, etc. And again, that is always with you as the center with everything moving out away from you as the center and faster the further it is out away from you as the center, and it always, always looks like that no matter where you are in it always, equally everywhere, etc.

God Bless.

 

[Neogaia777]

I'll try to add my limited understanding.

Space (space-time) (distant galaxies, and galactic clusters, and such) appears to moving (expanding) out away from us as the center, and the further out away from us it is with us as the center faster the further it is out away from us as the center, to the point that at a certain point (distance) it begins to appear to be moving (receding/expanding) out away from us as the center at speeds (velocities) beyond the speed of light, etc, which we cannot see beyond after that everywhere, etc. But, and here's the kicker, it always appears that way no matter where you are in the universe, with wherever/whatever you are at in it, and always with you as the center of that always everywhere, etc. So how much/fast it is really happening has to be somewhere in-between. Or at least, that's been a conclusion of mine lately. I say it's all moving/growing/expanding/receding at the same rate/velocity/speed equally everywhere, etc. And at those scales, even if it was happening at the exact speed of light equally everywhere (which I don't think it is, but is slower than that maybe) (but I also don't know how fast or slow it is all happening for sure, etc) But and/or anyway, at those scales, even if it was all happening at the speed of light, etc, it still would not be happening "fast" at those scales, but is actually happening actually quite slow, or slowly, etc. But it's happening everywhere the same, and at the same equal "rate" equally everywhere, etc, evidenced by the fact that it always is/always was/will be, etc, always look like that everywhere, etc, no matter where you are at in it equally everywhere, etc. And again, that is always with you as the center with everything moving out away from you as the center and faster the further it is out away from you as the center, and it always, always looks like that no matter where you are in it, equally everywhere, etc.

God Bless.

@Mark Quayle

Now, obviously, this is not happening within galaxies, or like, between us and our nearest star, or whatever, etc. Space (space-time) is not expanding/growing in the distance(s) between us and say our nearest star at the speed of light or whatever, etc. But this mainly only applies, or is only happening, only in the spaces between galaxies, or between the clusters and strings of galaxies in the universe, because that is where the expansion is really happening or is growing, etc. And at those scales, even if it was happening at the speed of light in those places, it would still be happening really quite slowly in those places, etc. But at these places between the clusters and strings of galaxies, those places are growing at the same (relatively slow) same equal rate equally everywhere, etc. And the effect is cumulative if there are several of them between us and other places, etc. Which is why we will always see what we will always see in it, no matter where we are in it, equally everywhere and at every place in it always, etc.

Just thought I might need to clear that up, etc.

God Bless.

 

[Neogaia777]

@Mark Quayle

Now, obviously, this is not happening within galaxies, or like, between us and our nearest star, or whatever, etc. Space (space-time) is not expanding/growing in the distance(s) between us and say our nearest star at the speed of light or whatever, etc. But this mainly only applies, or is only happening, only in the spaces between galaxies, or between the clusters and strings of galaxies in the universe, or whatever, because that is where the expansion is really happening or is growing, etc. And at those scales, even if it was happening at the speed of light in those places, it would still be happening really quite slowly in those places, etc. But at these places between the clusters and strings of galaxies, those places are growing at the same (relatively slow) same equal rate equally everywhere, etc. And the effect is cumulative if there are several of them between us and other places, etc. Which is why we will always see what we will always see in it, no matter where we are in it, equally everywhere and at every place in it, etc.

Just thought I might need to clear that up, etc.

God Bless.

And really, the fact that we have to have several of them between us and other places, before those things that far away from us can appear to be moving out away from us at speeds at or beyond the speed of light in those other places, really tells us that those places are not growing at the speed of light in those places, but is actually a lot slower than that in those places, etc. Which at those scales, is pretty darn slow actually, etc. But it's effect is cumulative with those places, etc.

God Bless.

 

[Neogaia777]

 

[sjastro]

So, space-time, wait ...huh? You are saying that there is a normal space that is different from space near gravitational forces? Spacetime does not expand within the galaxy, but between galactic groups it does? So, where spacetime does not expand, nothing can travel faster than light relative to anything else in the galaxy --not even theoretically, nevermind actually. But one galactic system can, relative to other galactic systems, because spacetime is different where there are no celestial bodies nearby? Is that what you are saying?

Not only does spacetime expansion not occur in our Milky Way galaxy because it is a tightly bound gravitational system, but the local cluster which contains the Milky Way is not expanding either because gravitational effects are still very strong.

A galaxy can be Doppler shifted in which case it moving through spacetime due to gravitational effects, cosmologically redshifted due to expansion of spacetime or a combination of the two.
Doppler shift can be red or blue, where galaxy motion does not have be along the line of sight of the observer and does not follow the redshift/distance relationship of Hubble’s law.
Cosmological redshift is always the result of the galaxies moving along the line of sight of the observer and conforms to Hubble’s law.

The local cluster is part of the Virgo supercluster which in turn is part of Laniakea.
At increasing distances to include these superclusters, the influence of cosmological redshift increases as gravitational effects diminish and dominates at distances beyond Laniakea.
At large cosmological scales gravitational effects are insignificant, galaxies are stationary in spacetime but being carried along by the Hubble flow of spacetime expansion.
Under these conditions recession velocities can exceed the speed of light.

I understand that, IF the recession velocity is less than c. So far, however, I don't see how it is not theoretically possible for it to be receding at more than the speed of light.

As has been explained in other posts stars and observers are like protons in a particle accelerator, to move them in spacetime up to the speed of light requires an infinite amount of energy.

Wait. Now spacetime expansion by definition implies that nothing can be moving contrary to that expansion at greater than the speed of light compared to anything else? Is that because spacetime is expanding so fast, or what? Even if we are talking between galactic systems, and not within galaxies where "spacetime does not expand"?

This is not how spacetime expansion works, where there are no gravitational effects, galaxies are stationary and it is the spacetime scale between galaxies which is increasing. The dots on a balloon serves as a useful analogy, they do not move along the surface as the balloon is blown up but distance between the dots increases.

Recession velocities are scale dependant, the speed of light is exceeded when the cosmological redshift of a galaxy is around z=1.46 and using current cosmological models this corresponds to a distance of around 5.5 billion light years from the observer. At these distances the contribution of gravitational effects is negligible as galaxies and clusters are so spread out.

You used an example of a celestial body approaching us which is Doppler blueshifted and has nothing to do with cosmological redshift.

Only because the celestial body cannot be approaching us faster than the speed of light.

I'm sorry. I'm ignorant enough not to know even if you are or are not right. But it doesn't add up to me. My intuitive use of space is 3 dimensional distance, whether occupied or not, regardless of how immense. Galaxies and galactic clusters are moving 'through' that 3 dimensional space. Some things make sense to me, such as the fact that a galaxy can be receding from us at more than the speed of light. Yet, apparently by the fact that it is happening, and other phenomena, such as redshift, affirming that these things are by far mostly moving away from us and from each other, we name that "expansion", as though it was what causes the increasing distances. Yet, here, 'locally', you tell me there can be no theoretical motion of two bodies relative to each other at more than the speed of light, while locally, the normal geometry of 3 dimensions works, but in the larger 'empty' space, it does not? I'm lost.

Without getting into the mathematics our universe is modelled as a manifold where at local scales it appears flat, globally it may not be.........
The earth serves as an example, despite the fact there are some here who think the earth is flat, at local scales we can use Euclidean geometry instead of spherical geometry which describes the earth's surface globally.

The same principles apply to the universe but when it comes to an expanding universe we can use broader terms such a cosmological and non cosmological scales.
At non cosmological scales the universe behaves as if it is static, in the local galaxy cluster the Doppler shift which includes both red and blue shift of individual galaxies indicates movement through spacetime where the speed of light is the limit for the reasons given in this thread.
At cosmological scales the universe is expanding and the larger the scale the faster the recession velocities.
At distances where the recession velocity is greater than 2c, photons emitted back to the observer will never reach the observer and the galaxy becomes causally disconnected from the observer.

 

[Neogaia777]

@Mark Quayle

As far as I know (but I could be wrong) the known/observable universe is still organized in clusters/superclusters and strings for as far as we can know or see, etc, all gravitationally bound to one another, etc. But between all of these clusters/superclusters and strings are mostly empty areas or pockets of space that are; very slowly at those scales, are growing larger, and is happening at the same equal rate, equally everywhere, etc, (significantly less that 1 times the speed of light, so very, very slowly at those scales or distances, etc) and these are causing what expansion there is in the universe, and are even "pushing out on" all of the other clusters/superclusters and strings that are all gravitationally bound to one another throughout, between and around those mostly empty pockets throughout, and that might even be partially responsible for the clusters/superclusters and strings being arranged the way they are maybe, etc. And whether these have or do, or do have/keep/maintain some kind of a "balance" or not, I do not know, but as far as I know, what we can see of the known/observable universe, is that, whether it is near or far, or very far, or very, very far, etc, it is all still arranged this way everywhere no matter what, etc, (but I still could maybe be wrong about that maybe) (I'm still just only an enthusiastic interested amateur after all, etc) (No formal learning or schooling or training about this, etc, other than what I have learned using the internet, and a picture(s) of the universe, and by using my own personal brain, in my own personal time, etc).

God Bless.

 

[sjastro]

As a human interest story I decided to give DeepSeek a shot at my response to @Mark Quayle in post #74.

I made the request as neutral as possible to avoid any bias.

Evaluate the statements made by Mark Quayle and the responses made by sjastro.

Apart from DeepSeek upsetting the status quo in the AI market and Trump since DeepSeek is a product of China, it employs a thinking mode where it shows the steps leading to its final conclusions which were @Mark Quayle is confused and @sjastro is bloody awful at explaining things (perhaps an exaggeration).

First of all the thinking mode.

DeepSeek seems to have a problem between singular and plural forms of expression, next post is the conclusion.

 

[sjastro]

DeepSeeks conclusion.

​

I'll leave to @Mark Quayle if DeepSeek's evaluation of his comments are accurate, as for mine using GR terminology in the form of metric tensors might only serve to complicate more than simplify.

 

[Mark Quayle]

All three of them.

I'll try to add my limited understanding.

Space (space-time) (distant galaxies, and galactic clusters, and such) appears to be moving (expanding) out away from us as the center, and the further out away from us it is with us as the center, faster the further it is out away from us it is as the center, etc, to the point that at a certain point (distance) it begins to appear to be moving (receding/expanding) out away from us as the center at speeds (velocities) beyond the speed of light, etc, which we cannot see beyond that after that everywhere, etc. But, and here's the kicker, it always appears that way no matter where you are in the universe, with wherever you are at in it, and always with you as the center of that always everywhere, etc. So how much/fast it is really happening has to be somewhere in-between. Or at least, that's been a conclusion of mine lately. I say it's all moving/growing/expanding/receding at the same rate/velocity/speed equally everywhere, etc. And at those scales, even if it was happening at the exact speed of light equally everywhere (which I don't think it is, but is slower than that maybe) (but I also don't know how fast or slow it is all happening for sure, etc) But and/or anyway, at those scales, even if it was all happening at the speed of light, etc, it still would not be happening "fast" at those scales, but is actually happening actually quite slow, or slowly, etc. But it's happening everywhere the same, and at the same equal "rate" equally everywhere, etc, evidenced by the fact that it always is/always was/always will, look like that everywhere, etc, no matter where you are at in it equally everywhere, etc. And again, that is always with you as the center with everything moving out away from you as the center and faster the further it is out away from you as the center, and it always, always looks like that no matter where you are in it always, equally everywhere, etc.

God Bless.

@Mark Quayle

Now, obviously, this is not happening within galaxies, or like, between us and our nearest star, or whatever, etc. Space (space-time) is not expanding/growing in the distance(s) between us and say our nearest star at the speed of light or whatever, etc. But this mainly only applies, or is only happening, only in the spaces between galaxies, or between the clusters and strings of galaxies in the universe, because that is where the expansion is really happening or is growing, etc. And at those scales, even if it was happening at the speed of light in those places, it would still be happening really quite slowly in those places, etc. But at these places between the clusters and strings of galaxies, those places are growing at the same (relatively slow) same equal rate equally everywhere, etc. And the effect is cumulative if there are several of them between us and other places, etc. Which is why we will always see what we will always see in it, no matter where we are in it, equally everywhere and at every place in it always, etc.

Just thought I might need to clear that up, etc.

God Bless.

And really, the fact that we have to have several of them between us and other places, before those things that far away from us can appear to be moving out away from us at speeds at or beyond the speed of light in those other places, really tells us that those places are not growing at the speed of light in those places, but is actually a lot slower than that in those places, etc. Which at those scales, is pretty darn slow actually, etc. But it's effect is cumulative with those places, etc.

God Bless.

You seem to be talking about natural conclusions drawn from what we see. My question is more theoretical --not why we don't see such things, and not even to speculate as to why we see (or don't see) what we do (or don't), but rather, why it would not be possible for faster-than-light travel within a galaxy, vs what seems obvious, that there is such a thing as faster than light travel in the space between distant galaxy clusters relative to two moving bodies.

You didn't address, for example, why @sjastro keeps referring to 0.88c, instead of the intuitive 1.2c relative to each other, when two light sources depart each other in exactly opposite directions at a speed of 0.6c each.

 

[Mark Quayle]

Not only does spacetime expansion not occur in our Milky Way galaxy because it is a tightly bound gravitational system, but the local cluster which contains the Milky Way is not expanding either because gravitational effects are still very strong.

A galaxy can be Doppler shifted in which case it moving through spacetime due to gravitational effects, cosmologically redshifted due to expansion of spacetime or a combination of the two.
Doppler shift can be red or blue, where galaxy motion does not have be along the line of sight of the observer and does not follow the redshift/distance relationship of Hubble’s law.
Cosmological redshift is always the result of the galaxies moving along the line of sight of the observer and conforms to Hubble’s law.

The local cluster is part of the Virgo supercluster which in turn is part of Laniakea.
At increasing distances to include these superclusters, the influence of cosmological redshift increases as gravitational effects diminish and dominates at distances beyond Laniakea.
At large cosmological scales gravitational effects are insignificant, galaxies are stationary in spacetime but being carried along by the Hubble flow of spacetime expansion.
Under these conditions recession velocities can exceed the speed of light.

As has been explained in other posts stars and observers are like protons in a particle accelerator, to move them in spacetime up to the speed of light requires an infinite amount of energy.


This is not how spacetime expansion works, where there are no gravitational effects, galaxies are stationary and it is the spacetime scale between galaxies which is increasing. The dots on a balloon serves as a useful analogy, they do not move along the surface as the balloon is blown up but distance between the dots increases.

Recession velocities are scale dependant, the speed of light is exceeded when the cosmological redshift of a galaxy is around z=1.46 and using current cosmological models this corresponds to a distance of around 5.5 billion light years from the observer. At these distances the contribution of gravitational effects is negligible as galaxies and clusters are so spread out.

You used an example of a celestial body approaching us which is Doppler blueshifted and has nothing to do with cosmological redshift.


Without getting into the mathematics our universe is modelled as a manifold where at local scales it appears flat, globally it may not be.........
The earth serves as an example, despite the fact there are some here who think the earth is flat, at local scales we can use Euclidean geometry instead of spherical geometry which describes the earth's surface globally.

The same principles apply to the universe but when it comes to an expanding universe we can use broader terms such a cosmological and non cosmological scales.
At non cosmological scales the universe behaves as if it is static, in the local galaxy cluster the Doppler shift which includes both red and blue shift of individual galaxies indicates movement through spacetime where the speed of light is the limit for the reasons given in this thread.
At cosmological scales the universe is expanding and the larger the scale the faster the recession velocities.
At distances where the recession velocity is greater than 2c, photons emitted back to the observer will never reach the observer and the galaxy becomes causally disconnected from the observer.

Oh well. Thanks for trying, once again.

 

[Mark Quayle]

DeepSeeks conclusion.

View attachment 360483
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I'll leave to @Mark Quayle if DeepSeek's evaluation of his comments are accurate, as for mine using GR terminology in the form of metric tensors might only serve to complicate more than simplify.

Thanks. Close enough. It did not pick up on (not at all to say that it should have picked up on) what it demonstrated, that I have had no explanation that I can understand or even to identify that it does explain, HOW what it calls Global and I call between galactic [clusters] has a [real] difference of spacetime compared to local spacetime within a galaxy. Maybe the same question asked differently is why metric expansion is a property of spacetime itself, and not mere motion within a 3-d metric. Is spacetime a medium? a force? or just a mental grid expanding to represent the receding distances?

As before, I just don't get it. So. I don't mean to tax anyone further. Thanks for your patience.