[Just thinking]

Firstly, what do you mean by moving away from us at 10% the speed of light?

If light speed is one billion miles per hour the ten % of that one hundred million miles per hour. Thanks for your second post it's a lot to get my head around. When referring to the curvature of space I fail to understand. If we look at a galaxy that is one billion light years away we see the light as it was one billion years ago the galaxy is most likely in reality not in the location that we are seeing it in now it may be many degrees in any direction away from what we are seeing now. Is this related to the curviture of space or is it not related at all.

[Eternal Student]

Hi.

Quote from: Kryptid on Today at 00:14:49

    Energy is.

How is energy lost as a result of redshift?

   Obviously I'm not Kryptid.  I'm just assuming you won't mind an answer from anyone.   

   Don't consider light as a wave for a moment, put all your ideas about compressing a wave of electromagnetic radiation on hold for a moment.   Instead let's consider light as something comprised of particles.
    It it is possible to imagine a single photon travelling through space.  Let's not allow it to interact with anything, get absorbed by anything or do anything interesting other than travel through space.
   A photon carries a certain amount of energy  which is given by the formula   Energy = hf    where   h is plancks constant and  f is the frequency of the photon.
   The general idea is that if the photon redshifts as it travels through expanding space then this means it's frequency drops and the amount of energy it is carrying decreases.   Does that make sense or seem reasonable?
    We haven't allowed the photon to interact with anything, it's had no opportunity to pass on or transfer that energy to anything else.  Energy has just been lost as it travelled.

   So, just by considering the travel of a single, isolated photon,  we are in a situation where the redshift of that photon would imply that energy has been lost.  Conservation of energy seems to have been broken.   Does that make sense?   We don't really need to worry about other situations like a wide beam of light as you described earlier.  Yes, a wide beam might scatter more, spread out the wave or "thin it down" as you described earlier.  It could be exactly as you described earlier.   We don't need to worry too much about that.   The important thing is that there is at least ONE situation where the redshift of light from a distant galaxy would imply that energy was lost.  That's all we need to worry Physicist's.  The conservation of energy doesn't usually break in ANY situation.   Does that make sense?

   The redshift of photons as they travel through expanding space implies a situation where energy is not conserved.  The earlier posts in this thread all relate to explaining how this happens; why it happens;  how it can or can't be explained just by the movement of the source relative to receiver  etc. etc.
- - - - - - - - - - -
   

If we look at a galaxy that is one billion light years away we see the light as it was one billion years ago the galaxy is most likely in reality not in the location that we are seeing it in now it may be many degrees in any direction away from what we are seeing now.

  This is mostly true.  I haven't fact checked it, so it might be entirely true.

Is this related to the curviture of space or is it not related at all.

   It's not really related.   I mean all movement and evolution of the universe is related somehow but I don't think this is a useful way of understanding curvature.
   Explaining the curvature of spacetime is a big topic, it might be better in a new thread, although to be honest it might be faster just to look for some existing articles, books and/or videos that explain this.

Best Wishes.

[Kryptid]

Energy is.

How is energy lost as a result of redshift?

The energy of a photon is directly correlated with its wavelength. Greater wavelengths have less energy than shorter ones. So if red-shift causes a photon's wavelength to double, that photon now has only half of the energy that it started with.

[hamdani yusuf]

Energy is.

How is energy lost as a result of redshift?

The energy of a photon is directly correlated with its wavelength. Greater wavelengths have less energy than shorter ones. So if red-shift causes a photon's wavelength to double, that photon now has only half of the energy that it started with.

Where does the half of the energy go?
Let's say a transmitter send a pulse of laser beam containing 1 Joule of energy. A faraway receiver gets the pulse, but the wavelength has been doubled. Does it mean that only half Joule of energy arrives at the receiver? Where does the other half Joule go? Is it absorbed by the space between transmitter and reciever?

[Halc]

The energy of a photon is directly correlated with its wavelength. Greater wavelengths have less energy than shorter ones. So if red-shift causes a photon's wavelength to double, that photon now has only half of the energy that it started with.

Where does the half of the energy go?
Let's say a transmitter send a pulse of laser beam containing 1 Joule of energy. A faraway receiver gets the pulse, but the wavelength has been doubled. Does it mean that only half Joule of energy arrives at the receiver? Where does the other half Joule go? Is it absorbed by the space between transmitter and reciever?

Energy is conserved only in a static metric (one where space doesn't change over time) such as an inertial coordinate system. It is not conserved in an expanding metric. So it doesn't go anywhere. The total energy reduces over time relative to that metric.

When the space through which particles move is changing, the total energy of those particles is not conserved.

https://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/

So if the transmitter and receiver are both stationary, but with space between them expanding by a factor of 2 while the light is in transit, the pulse loses energy and arrives at the receiver at a quarter the power, twice the duration, and half the energy.

Same situation using inertial coordinates:
In the inertial frame of the receiver, the light leaves the receding transmitter already at half power and loses no energy along the way.
In the inertial frame of the transmitter, the light has full energy all the way but still takes twice the duration to hit the receding receiver, but the full dose of energy is delivered to it.

This is one more way to illustrate that energy of light and other things (kinetic energy of a rock for instance) is very frame dependent, and not actually a property of the thing itself.

The example above is idealized and takes into account neither the mass of anything nor dark energy, both of which complicate the arithmetic (and mostly cancel each other) but not invalidate it other than the fact that since these things do exist, an inertial frame is only an approximation over such non-local distances.

[Just thinking]

Thank all that have made an effort to explain this light/photon problem to me I think I am starting to see what has been said is this it. If we have a radio tower and it is transmitting a signal let's say 27 MHz and now we move the tower at a very high speed away from us the 27MHz will be transformed into a lower frequency say 20MHz we will still receive the transmission but at a different wavelength. Am I with it now or still off the page?

[hamdani yusuf]

So if the transmitter and receiver are both stationary, but with space between them expanding by a factor of 2 while the light is in transit, the pulse loses energy and arrives at the receiver at a quarter the power, twice the duration, and half the energy.

Will it make a difference if the change of wavelength is caused by expansion of space, compared to physical movement/velocity difference between transmitter and receiver?

[Eternal Student]

Hi.

Will it make a difference if the change of wavelength is caused by expansion of space, compared to physical movement/velocity difference between transmitter and receiver?

    This thread does seem to be repeating itself often,  I can only apologise if I say something again.

1.    The receiver just receives radiation.  If it gets a frequency of 100 Mhz for that radiation, that is all it cares about.  The radiation will be 100 MhZ radiation regradless of how it came to be that frequency.  We can redshift radiation in several ways.   The receiver doesn't care if it was caused by space expanding or a relative velocity between the source and the receiver.
    That is where half the problems or misconceptions come from about an expanding universe.   We notice that what should be spectral emission/absorption lines from elements like Hydrogen seem to be redshifted when we look out to distant galaxies.    We are reasonaby sure that they were Hydrogen emission lines, so we know what frequency they had at the point of creation and in the frame of reference where the atom was stationary.   We know that we observe them at lower frequency when we receive them here on earth (or on a satellite out in space near earth).   We want to explain that redshift.   

2.   It's natural to suggest that space behaves just like Minkowski space and the source was moving away from us but it just doesn't work well.  Our theories remain more consistent if we assume the redshift was caused by the expansion of space rather than some velocity of the source and/or receiver through space.  One of the main points of evidence for this would be the Hubble law.   This suggests that the recession speed of distant galaxies can exceed the speed of light.  We would prefer to believe that there is something "wrong" with the way we measure distances over astronomical scales.  What is wrong with the way we measure distances is that the geometry of spacetime is not flat.   General relativity offers an explanation for why the change in distance between us and distant galaxy with time can exceed the speed of light.   This quantity     is called the recession speed but it is NOT a velocity that anything has, it is just the rate of change of distance with time.   In flat space, that quantity would have to be the magnitude of a velocity - it is essentially the definition of what a velocity is.
    I'm not going to repeat any more of this, this general discussion is available in many other articles, threads, videos, books etc. 

3.   The key issue being discussed in this thread seems to be the conservation of energy.   We already know that total energy is a somewhat arbitrary quantity.   We are only interested in CHANGES in energy and that is all that the conservation of energy concerns itself with.    For example, if we have a system that consisted of some atoms then the kinetic energy of the atoms depends on the inertial reference frame we wish to use.  We can give the atoms more energy just by using a different frame and boosting the velocity of all those atoms.   This doesn't matter, we are only concerned with changes in the energy that might happen as the atoms interact, we don't claim to know or care about the actual energy that the system really had.   Hopefully, that makes sense.   If we don't consider everything in our system under one consistent inertial frame then the conservation of energy doesn't apply.
    Now, imagine a small number of photons travelling through space, let's keep them all close together and travelling parallel just to make it easy to visualise, they were all made at the same time from one source and they will all (after say 3 million years have passed on a satellite called "the receiver") be absorbed or received at the same place.  We'll draw a box around a regon of the universe that includes the source and receiver and consider this as our closed system.  Now we expect the conservation of energy to apply to our system.  However we're going to be careless and apply a different reference frame after each millienium (million years).  On the first millenium we use a reference frame such that the source was stationary in that frame.  On the second millenium we'll use a reference frame that has velocity +0.1c relative to the day 1 frame.  On the final millenium we'll use a ref. fame with velocity +0.2c rel. to day 1 frame.  Obviously, the energies of the group of photons change each millenium (a different redshift applies).   We might claim that energy was somewhow being lost each millenium but of course someone will point out that if we had used a single consistent reference frame then this wouldn't have happened.
    One of the key ideas in General Relativity is that we CANNOT apply a single consistent inertial reference frame to cover all of the universe.  There are always Local inertial reference frames that we can find and use but (for a general geometry of space) we cannot extend these inertial frames beyond the local region.  So we can reply to those people telling us to use one consistent frame by just saying  "For an expanding universe, we can't.  There isn't one.   The geometry of space is not Euclidean."

4.  It was mentioned much earlier in this thread that the issue of redshift in expanding space isn't simply regarded as a challenge to the conservation of energy.  It is just not possible to determine the energy of a photon at the time of release by a distant galaxy as it would be measured in our own local frame of refeence here on earth.  So the wider issue is that energy cannot meaningfully be defined universally,  only locally.

Best Wishes.

[Eternal Student]

If we have a radio tower and it is transmitting a signal let's say 27 MHz and now we move the tower at a very high speed away from us the 27MHz will be transformed into a lower frequency say 20MHz we will still receive the transmission but at a different wavelength. Am I with it now or still off the page?

   Yes, that's correct.

[Halc]

Will it make a difference if the change of wavelength is caused by expansion of space, compared to physical movement/velocity difference between transmitter and receiver?

1.    The receiver just receives radiation.  If it gets a frequency of 100 Mhz for that radiation, that is all it cares about.  The radiation will be 100 MhZ radiation regradless of how it came to be that frequency.  We can redshift radiation in several ways.   The receiver doesn't care if it was caused by space expanding or a relative velocity between the source and the receiver.

Excellently put. This was the gist of my post above.

2.   It's natural to suggest that space behaves just like Minkowski space and the source was moving away from us but it just doesn't work well.

We seem to disagree on this point. It seems to work well enough if you keep it sufficiently local (where the scalefactor stays reasonably linear). A separation of 2 BLY seems to fall within this 'local' range, with the deviations being minor secondary effects.  At larger distances, I agree it falls apart. You go out 7 BLY and suddenly the secondary effects start being significant. The universe expansion was still slowing 7 billion years ago.  Out twice that far you start dealing with event horizons that cannot be described with an inertial frame. The secondary effects begin to dominate the primary ones, and the inertial model completely falls apart. I'm not proposing the Minkowski spacetime as a model for the universe at large. That is indeed quite easily falsified.

One of the main points of evidence for this would be the Hubble law. This suggests that the recession speed of distant galaxies can exceed the speed of light.

Under the Milne model (which has been falsified), Hubble's law still stands, where Hubbles constant is exactly 1/time at all times. Galaxies don't have velocities greater than light because there are no galaxies further away than a fixed figure.  Recession rates in the expanding metric can exceed c, but those rates are expressed as a rapidity, not as a velocity. Rapidity has no upper limit, so in expanding space, the distance between a pair of galaxies can increase at arbitrarily high rates. In the inertial frame, none of these galaxies has a relative velocity greater than c.
I'm saying this not because I support the Milne model, but because Hubble's law is not in fact evidence for expanding space.  The acceleration is. You can't have that in flat space. Deceleration either for that matter since the symmetry must be broken.

So real visible galaxies like GN-z11 have recession rates over 2c, and since it is far enough away that an inertial model simply cannot be applied, one cannot really assign an inertial velocity to that, but if you could, it would be around 0.98c

General relativity offers an explanation for why the change in distance between us and distant galaxy with time can exceed the speed of light.   This quantity     is called the recession speed but it is NOT a velocity that anything has, it is just the rate of change of distance with time.

We seem to be in agreement about that much.

In flat space, that quantity would have to be the magnitude of a velocity - it is essentially the definition of what a velocity is.

Not that quantity, no. To convert a recession rate r to a flat velocity v of one object relative to the other:  v=tanh(r), but I agree, it is fairly inappropriate to apply such a transformation over non-local distances since the transformation assumes spacetime is flat.
The pop articles never really explain the difference between recession rate and the speed something is moving, or more precisely, between rapidity and velocity.


3.   The key issue being discussed in this thread seems to be the conservation of energy.   We already know that total energy is a somewhat arbitrary quantity.   We are only interested in CHANGES in energy and that is all that the conservation of energy concerns itself with.    For example, if we have a system that consisted of some atoms then the kinetic energy of the atoms depends on the inertial reference frame we wish to use.  We can give the atoms more energy just by using a different frame and boosting the velocity of all those atoms.   This doesn't matter, we are only concerned with changes in the energy that might happen as the atoms interact, we don't claim to know or care about the actual energy that the system really had.   Hopefully, that makes sense.   If we don't consider everything in our system under one consistent inertial frame then the conservation of energy doesn't apply.[/quote]Exactly, and since the universe is not a system that can be considered under any one consisten inertial frame, there is no necessary conservation of energy in the universe, something which is rarely admitted, which is why I like the Carroll article, who comes right out and says it rather than trying to sweep the embarrassment under the rug.

[Eternal Student]

Hi Halc and thanks for your time.
   Obviously, overall there is a lot of agreement.  I didn't really consider the Milne model but that's probably because I didn't start studying cosmology until we had computers and the internet.  Don't get me wrong, I'm old enough to remember the days before this but I just wasn't studying cosmology then.  In recent times, the Milne model barely gets a mention in the textbooks.   Anyway, I've enjoyed spending a couple of hours to read something about it.

It seems to work well enough if you keep it sufficiently local (where the scalefactor stays reasonably linear)

  OK.  It does work well enough on small scales.

Under the Milne model

   OK.  Seems reasonable. 
   We could support the validity of General relativity by various observations and bits of evidence, like the procession of the orbit of mercury   etc. etc. 
   As far as General relativity is concerned, the Milne model could only describe space that was empty (or at the very least it is obtained as a limit of a FLRW model when energy density and pressure → 0 ).   I would assume the Milne model was largely ignored many years ago just because the universe does not seem to be empty  (and no one was really disputing General relativity any longer).

    In flat space, that quantity would have to be the magnitude of a velocity - it is essentially the definition of what a velocity is.

Not that quantity, no. To convert a recession rate r to a flat velocity v of one object relative to the other:  v=tanh(r), but I agree, it is fairly inappropriate to apply such a transformation over non-local distances since the transformation assumes spacetime is flat.
The pop articles never really explain the difference between recession rate and the speed something is moving, or more precisely, between rapidity and velocity.

   It's taken a while to see where you were getting this from.  The formula v=tanh(r)  was coming from the use of a Minkowski metric, rapidity and continued reference to the Milne model, I think.   So, yes actually you'd be right.

   I think the problem is from my use of the word "flat space".   I made the mistake here, I should have said "Euclidean space", or the thing everyone studied in school.

   My original sentence was meant to be more disposable and light weight.
In Euclidean space (which we all studied at school),   where  (Δs) = distance   = √( (Δx)² + (Δy)² + (Δz)²)       then     you would have been told that speed is defined to be  .  Hence, if the distance between two objects was changing at a rate given by  v = ,  then it would be inescapable that v is velocity that one of the those objects has relative to the other object.
- - - - - - - - -

    About the Hubble Law:
 

Hubble's law is not in fact evidence for expanding space.

    There are other models that could be consistent with the Hubble law.  You've mentioned the Milne model.   However, General relativity and a universe with a FLRW metric that has an increasing scale factor is the best candidate.
    I mean let's be fair the Hubble law is more properly called the   "Hubble-LeMaitre  law"   since  Georges LeMaitre was predicting Hubble's law from General Relativity years before Hubble made made those observations.
    The Milne model seems like a desperate attempt to show that special relativity and not general relativity might be enough to explain some things.

Best Wishes.

[Halc]

It seems to work well enough if you keep it sufficiently local (where the scalefactor stays reasonably linear)

OK.  It does work well enough on small scales.

Which is why I reference it. It works on scales small enough where scalefactor is reasonably linear, and since mass and dark energy are reasonably balanced in the recent past, that scalefactor is sufficiently linear that the mathematics makes a good approximation.

We could support the validity of General relativity by various observations and bits of evidence, like the procession of the orbit of mercury   etc. etc.

You can trash the Milne model just by the very existence of Mercury (or us).

It's taken a while to see where you were getting this from.  The formula v=tanh(r)  was coming from the use of a Minkowski metric, rapidity and continued reference to the Milne model, I think.   So, yes actually you'd be right.

   I think the problem is from my use of the word "flat space".   I made the mistake here, I should have said "Euclidean space", or the thing everyone studied in school.

I always wondered if 'Euclidian space' is appropriate when applied to spacetime. Yes, flat space is Euclidean if spatial triangles add up to 180°, but Euclidean spacetime would seem to be more the Newtonian concept with the intervals being computed as the sum of the squares of the 4 components instead of subtracting space distance from the time (t²-d²) that Minkowskian spacetime does. In other words, does that difference make Minkowskian spacetime non-Euclidean? I don't know the rules. You're more the formal math guy and I'd take your word on it over my guess.

In Euclidean space (which we all studied at school), where  (Δs) = distance = √( (Δx)² + (Δy)² + (Δz)²) then you would have been told that speed is defined to be  .  Hence, if the distance between two objects was changing at a rate given by  v = ,  then it would be inescapable that v is velocity that one of the those objects has relative to the other object.

But ds/dt is still frame dependent. Is an expanding metric applied over Euclidean spacetime (a transformation done by Milne) make it no longer Euclidean? It's just that distance (Δs) is not measured the same way in an inertial frame as it is in an expanding frame over the same space. There's different simultaneity between the two objects, and hence a very different proper separation between the two of them at any give time.
For instance, in comoving coordinates, stationary galaxy X might be 20 BLY away from stationary us (proper separation along the line of constant cosmological time) and that distance increasing at a rate of about 1.45c, and a clock on galaxy X reads the same time (since the BB) as here on Earth.
Relative to Earth's inertial frame, that same galaxy is currently a proper distance of only 12.35 BLY and moving at a rate of about .895c and a clock at galaxy X currently reads only 6.15 BY.
Both measure a proper distance, but to two completely different events separated by over 7 billion years in X time. Yay relativity of simultaneity. Yes, I know I've totally violated my definition of 'local enough' with these large figures, but we were talking Euclidean space and not real space for this example.
So same object, same Euclidean spacetime, yes vastly different velocities (rate of change of proper separation) due to different ways of drawing abstract lines of simultaneity in that Euclidean space.

I mean let's be fair the Hubble law is more properly called the   "Hubble-LeMaitre  law" since Georges LeMaitre was predicting Hubble's law from General Relativity years before Hubble made made those observations.

No better evidence for a theory than a prediction like that being made before there was evidence that necessitated it.

The Milne model seems like a desperate attempt to show that special relativity and not general relativity might be enough to explain some things.

That's pretty much what it tried to do: GR used to describe local wiggles in a Minkowskian universe on the largest scales, rather than SR being used to describe local flat space in an expanding universe at the largest scales. It rightly fell flat, and I only use it 'locally' where SR still reasonably applies.

Example: I have two objects bolted to a long stick (several light years long) way out in deep space between galaxy clusters. There are super sensitive force sensors that measure force of the objects on the stick. At some point in the universe history, there might be zero tension or compression on the stick, but if it's sensitive enough it will detect tension now, and compression a long time ago. Milne model doesn't allow that. That's the most local test I can think of.
Interesting exercise to work out exactly when the stick has zero force on it, which is a function of all kinds of things like the mass of the objects, length of stick, and the local gravitational repulsion due to tidal effects from the nearest galaxy clusters. Yes, gravity can repel objects, something that can be illustrated with the rubber sheet analogy.

I have to shut up now. I'm well into just posting thoughts that go through my head and I've digressed.

[Zer0]

Hello Mark.
🙏

Hiya Sally!
🙋

If the Universe is a Closed ended system with no way out.
Then clearly everything just dissolves & dilutes but Remains!

Ps - Wonder from where do Virtual Particles appear.
🤔

[Eternal Student]

Hi Halc.

   Formal definitions of Euclidean and/or Flat:   I had written something here but I've deleted it.  It's boring and you can look up the official definitions whenever you need them.    It won't help much anyway,  most of us mess it up slightly and someone else will always have a different idea of the terms.

But ds/dt is still frame dependent. Is an expanding metric applied over Euclidean spacetime (a transformation done by Milne) make it no longer Euclidean?

   This is the more interesting bit.   Technically, a Euclidean metric has all positive eigenvalues    (or it just adds all the components and never takes one away, if you want it in plain language).   One of the nice things about it is that this "metric" is actually a bona fide, real, honest, straight-up,  satisfies the formal mathematical definitions of........     a metric.     In particular,    the distance  between two points is zero      if and only if    the points are the same.
    Meanwhile the main alternative is called a Lorentzian metric and has signature  (- + + +)   or   (+ - - -) if you prefer.   Although the physicists call it a "metric" it isn't really.   It's a pseudo-metric or semi-metric and you can see why:   Two points can be different points but still have 0 metric distance between them.   This can help to remind you that Minkowski space or anything with a Lorentzian metric isn't formally a Euclidean space,  it isn't even a proper metric space.
    Yet, some people will still say that Minkowski space is Euclidean - but that's people for you.  It's because they are talking about the purely spatial dimensions in Minkowski space.  Minkowski space does have all the properties of a Euclidean space provided we keep time quite separate and don't try to mess up the metric by including some time differences in it.
   Let's just consider a photon travelling through Minkowski space.   It travels on a null path.   The school definition of it's speed is as follows:
       Speed =    (Distance travelled)   /   (Time taken).
    If we use the full extent of the Lorentzian metric then that distance is actually what we normally call a spacetime interval.   So a photon covers 0 metric distance  in  1 unit of time,  so it has a speed of 0.   This is technically correct,  the rate of change of that Lorentzian measure of distance travelled with time is zero.   However, this is not what we have in mind for the speed of a photon, it is certainly not it's speed through 3-D space.   We would normally restrict our definition of the distance travelled to a purely spatial measure of distance,  if we do this then the photon covers a spatial distance of   ct   in  time  t   and so it has a speed of c.     Provided we keep time and space separate like this, then Minkowski space has Euclidean properties (and you can understand why people might say that it is Euclidean,  I mean they are wrong - but you can understand why they are saying it).

      This is essentially where the issues appear in the Milne model.   The underlying space never was Euclidean anyway.  It had a Lorentzian metric (so that the time component was subtracted).  More than this, there was no attempt to keep time separate from the spatial dimensions.   Quite the opposite, the whole thing rests entirely upon the ability to mix up some time with some space and apply Lorentz transformations.   So that's it straight from the word go.   Minkowski space used this way was never going to be Euclidean or show properties we would expect in a Euclidean space.    On top of this they introduce a second metric which also has a non-Euclidean signature and it's just a mess.   Nothing in the the model is Euclidean by the end.   However, it wasn't really the introduction of the second metric that was the problem.  Minkowski space isn't Euclidean to start with and they (Milne et.al.) were determined to exploit the Lorentzian nature of spacetime.

    I can barely remember how the original use of  ds/dt   came up....

Best Wishes.

[Halc]

   This is the more interesting bit.   Technically, a Euclidean metric has all positive eigenvalues    (or it just adds all the components and never takes one away, if you want it in plain language).   One of the nice things about it is that this "metric" is actually a bona fide, real, honest, straight-up,  satisfies the formal mathematical definitions of........     a metric.     In particular,    the distance  between two points is zero      if and only if    the points are the same.

So I suspected. Thanks for that. I've always hesitated to use the term Euclidean when I just mean flat.

Meanwhile the main alternative is called a Lorentzian metric and has signature  (- + + +)   or   (+ - - -) if you prefer.

Always the latter I think since timelike intervals are expressed as positive/real and spacelike intervals as negative/imaginary. I can't find a site that uses the -+++ standard, but maybe I didn't look hard enough.

This is essentially where the issues appear in the Milne model.   The underlying space never was Euclidean anyway.  It had a Lorentzian metric (so that the time component was subtracted).  More than this, there was no attempt to keep time separate from the spatial dimensions.   Quite the opposite, the whole thing rests entirely upon the ability to mix up some time with some space and apply Lorentz transformations.

The transformation I did was not a Lorentz transformation, but rather one from an inertial frame to a hyperbolic frame (a frame anchored on an event instead of a time axis). Energy is not conserved in the latter sort of frame, but any arbitrary event can be used to define the frame.

[Eternal Student]

Hi again Halc.

The transformation I did was not a Lorentz transformation, but rather one from an inertial frame to a hyperbolic frame.........  (and stuff)....

    It's actually quite difficult to find a lot of good texts about the Milne model anymore but I think I've got an overview of the ideas.   I'm mostly onboard with what was done in the transformation.   The Milne universe seems to be using something like Rindler co-ordinates.
   
   Your original question was something like this:

......... Is an expanding metric applied over Euclidean spacetime (a transformation done by Milne) make it no longer Euclidean?

    To which the main thing to note is that it wasn't an honest Euclidean space to start with.   The underlying spacetime was Minkowski which isn't Euclidean and won't have Euclidean properties unless you are extremely careful to try and keep time and space separate.   Applying a second metric made it worse but it just wasn't Euclidean to start with.  As far as I can see, the entire methodology relies upon exploiting the non-Euclidean nature of the Minkowski spacetime right from the start.

Late addition:   The  (- + + +) convention  is used in the texbook I have.    Sean Carroll,  An introduction to spacetime and Geometry, page 9.     
He defines   (Δτ)²  =  -(Δs)²,    and  states  "the interval is defined to be (Δs)² not the square root of this quantity".    So he avoids imaginary numbers all the time anyway.
    I'm sure that many other sources use the other convention and also define the interval to be the square root.


Best Wishes.

[Halc]

Hi again Halc.

The transformation I did was not a Lorentz transformation, but rather one from an inertial frame to a hyperbolic frame.........  (and stuff)....

    It's actually quite difficult to find a lot of good texts about the Milne model anymore but I think I've got an overview of the ideas.   I'm mostly onboard with what was done in the transformation.   The Milne universe seems to be using something like Rindler co-ordinates.

Hyperbolic coordinates, not Rindler. The latter is for references with constant proper acceleration, which is not true of hyperbolic coordinates. The Milne universe is only one special case of the general hyperbolic coordinates. One can assign such coordinates to any event in spacetime, especially say one where grenade explodes in a vacuum.
   

Is an expanding metric applied over Euclidean spacetime (a transformation done by Milne) make it no longer Euclidean?

    To which the main thing to note is that it wasn't an honest Euclidean space to start with. The underlying spacetime was Minkowski which isn't Euclidean and won't have Euclidean properties unless you are extremely careful to try and keep time and space separate.

Agree to all, but the question still seems to stand:
Is an expanding metric applied over Minkowskian spacetime (a transformation done by Milne) make it no longer Minkowskian?
That's what I meant to ask, and I'm not actually sure what I meant be the equation. The arbitrary coordinate system I choose to slice up the spacetime has no physical effect on the spacetime, but it isn't the Minkowskian way of assigning the coordinates, so I guess the answer is no if you take the question as an abstract one, and yes if you take the question as a physical one.

Late addition:   The  (- + + +) convention  is used in the texbook I have.    Sean Carroll,  An introduction to spacetime and Geometry, page 9.     
He defines   (Δτ)²  =  -(Δs)²,    and  states  "the interval is defined to be (Δs)² not the square root of this quantity".    So he avoids imaginary numbers all the time anyway.

Yes, all the texts I have also show the interval to be the square and not just 's', but I don't usually see the -+++ part. So a 10 second duration has an interval of 100, or -100 if you're Carroll.
If I parse his line correctly, the interval is (Δs)² and it is the negation of (Δτ)², which seems only true for an interval between two events at the same location in space, but by assigning it as the negation like that, he uses the -+++ convention. Functionally it matters not so long as we know which convention is being used.

[Eternal Student]

Hi.

.....but the question still seems to stand:
Is an expanding metric applied over Minkowskian spacetime (a transformation done by Milne) make it no longer Minkowskian?
That's what I meant to ask, and I'm not actually sure what I meant be the equation. The arbitrary coordinate system I choose to slice up the spacetime has no physical effect on the spacetime, but it isn't the Minkowskian way of assigning the coordinates, so I guess the answer is no if you take the question as an abstract one, and yes if you take the question as a physical one.

   It's hard to find texts on the Milne model as I mentioned earlier. 

Page 341,  Sean Carroll,  Spacetime and Geometry  indicates the following:
    The Riemann curvature tensor of the Milne Universe = 0  (has all components =0).   Therefore, it is locally equivalent to flat space  (Minkowski).    In this case, it can be shown to be quivalent to a small patch of Minkowski space -  the interior of a future light cone of some fixed point of Minkowski space, foliated by negatively curved hyperboloids.

   This is the treatment of the Milne Universe in General Relativity.  It doesn't seem that the Sean Carroll's book cares at all for the TWO metrics that Milne used in his original treatment which only required special relativity.  There is one Milne metric as far as that textbook is concerned.  This seems to be the one you describe as "the expanding metric applied over Minkowski spacetime".
    Carroll considers this metric:      which is equivalent to the Milne metric stated in Wikipedia when the curvature, k is normalised (set k=-1 and adjust r and a(t))  and then identifying new co-ordinates   

   Anyway, it seems that using that metric, the space (with the Milne metric as described above) is equivalent to a small patch of Minkowski space.

    I don't doubt that Milne developed his model without GR and using two metrics,  there's even this section written about it in Wikipedia:
Milne developed this model independent of general relativity but with awareness of special relativity. As he initially described it, the model has no expansion of space, so all of the redshift (except that caused by peculiar velocities) is explained by a recessional velocity associated with the hypothetical "explosion".
   So, he may have had two metrics and potentially treated them as separate things.   However, we already know that in the first metric the space was assumed to be Minkowski space.   So I think a fair and reasonable answer to your question is as follows:   Yes, both metrics describe a space with 0 curvature and are locally equivalent to flat Minkowski space.   That's all you (a Physicist) could physically determine if they were in those spaces anyway, they have 0 curvature everywhere and local co-ordinates exist which would seem natural to you and make the space behave like Minkowski space.
    (I can't sensibly use the two different metrics simultaneously - anything with two metrics defined on it can't be Minkowski space by definition, it's some new mathematical object that I'm not sure we have a name for).

Best Wishes.

[EvaH]

Asked and answered here!

https://www.thenakedscientists.com/podcasts/question-week/what-happens-lights-lost-energy

[hamdani yusuf]

A space ship is moving away from us at 0.1c. It sends a 1 Watt 600 nm laser pulse, one second long in its frame of reference.
What's the power of the laser that we will receive? what's the wavelength? how long is the pulse?
What if the space ship is moving toward us at the same speed?