George Musser sent me and David Hogg an email with some questions about the paper Hogg and I wrote about the interpretation of the redshift (which I’ve written about before). The discussion may help to clarify a bit what Hogg and I are and are not claiming, so here it is (with Musser and Hogg’s permission, of course).

Musser’s original question:

I’m still absorbing your paper from a couple of years ago on the cosmological redshift, being one of those people who has made the distinction with Doppler shift and, more generally, between “expansion of space” of “motion through space”.

If these are equivalent and, in fact, the latter is preferred, then should I think of the big bang as spraying out galaxies through space like a conventional explosion — i.e. the very picture cosmologists have been telling us is wrong all these years? If the rubber-sheet model of space is so problematic, then what picture

shouldI keep in my head?Also, if the photon only ever sees locally flat spacetime, is that why the cosmological redshift does not entail a loss of energy?

Hogg’s reply:

The only cosmologist saying that that the “explosion” picture is

wrongis Harrison (who himself is very wrong), although others think it is uncomfortable, like Sean Carroll (who is not wrong). Empirically, there is no difference; what is definitely wrong is the idea that the space is “rubber” or has dynamics of its own. There is no absolute space–the investigator has coordinate freedom, and the empty space has no dynamics, so this rubber sheet picture is very misleading. And no, the photon does not “lose energy” in any sense. It is just has different energies for different observers, and we are all different observers on different galaxies.

Musser:

That is helpful, but I am still confused. An explosion goes off at a certain position in space and matter shoots outward in every direction. Is that really a valid picture of the big bang? What do I make of the presence of horizons?

And finally me:

It’s still true that there is no spatial center to the expansion. That is, there is no point in space that is “really” at rest with everything moving away from it. Space is homogeneous, which means that whatever point you pick looks as much like the center as any other point.

One thing that can be said for the expanding-rubber-sheet picture: at least in its form as an expanding balloon, it conveys this idea of homogeneity tolerably well. (Well, except that it’s hard for people to remember that only the surface of the balloon counts as “space” in this metaphor. People always want to think of the center of the balloon’s volume as “where” the Big Bang happened.)

So I’d rather you not think of the Big Bang as an explosion “at a certain position in space”. It’s still true that it happened everywhere rather than somewhere. There’s no preexisting space into which stuff expands. For instance, if we imagine a closed Universe (i.e., one that has a finite volume today), its volume was smaller in the past, approaching zero as you get closer to the Big Bang. So in that sense space really is expanding.

[A bit of fine print: All of the above is true as applied to the standard model of the Universe, in which homogeneity is assumed. Whether it’s true of our actual Universe is of course an empirical question. The answer is yes, as far as we can tell so far. But there’s no way to tell — and there probably never will be any way to tell — what space is like outside of our horizon. But anyway, this point is independent of the question of interpretation that we’re discussing at the moment. So it’s safe to ignore this point for the present discussion.]

As Hogg says, the main thing we object to is the idea that the rubber sheet has its own dynamics and interacts with the stuff in the Universe — that is, that the stretching of the rubber sheet tends to pull things apart, or that it “stretches” the wavelengths of light. As far as I’m concerned, the main reason for objecting to this language is not because it gives the wrong idea about cosmology, but because it gives the wrong idea about relativity. The most important point about relativity is that space doesn’t have any such powers and abilities. If you’re a small particle whizzing through space, at every moment space looks to you just like ordinary, gravity-free, non-expanding space.

So if you’re going to abandon the heresy of the rubber sheet, what should you replace it with? I don’t have anything as catchy as the rubber sheet, unfortunately. What I visualize when I visualize the expanding Universe is just a bunch of small neighborhoods, each one of which is completely ordinary gravity-free space, but each of which is moving away from its neighbors.

In this picture, the redshift is easy to understand. If a guy in one neighborhood tosses a ball to his neighbor, the speed of the ball as measured by the catcher will be less than the speed as measured by the thrower. That is, the two measure different energies for the ball, not because there’s some phenomenon taking energy away, but just because they’re in different reference frames. If the catcher then turns around and throws again to his neighbor, the same thing happens again, and so on. That’s all the redshift is. It’s not some mysterious “stretching.”

“The only cosmologist saying that that the "explosion" picture is wrong is Harrison (who himself is very wrong), although others think it is uncomfortable, like Sean Carroll (who is not wrong).”

Harrison is dead and so can’t defend himself. I think it is going a little far to call him “wrong” on this point, though. Harrison was very careful about his terminology and many others weren’t or aren’t. In particular, only those who have never stated that the relativistic Doppler formula should be used at high redshifts should be allowed to even discuss this matter. 😐

“Empirically, there is no difference; what is definitely wrong is the idea that the space is "rubber" or has dynamics of its own. There is no absolute space–the investigator has coordinate freedom, and the empty space has no dynamics, so this rubber sheet picture is very misleading.”

If there is no difference empirically, and the situation is in some sense analogous to which interpretation of quantum mechanics one prefers, then calling Harrison “wrong” seems even worse.

What about the de Sitter universe, or in general any universe with no matter content? These are limiting cases of expanding universes with less and less matter. In the limiting case where one can think of the matter as just “test particles”, i.e. tracing the expansion but not influencing it, then it seems clear that space expands. Yes, maybe it doesn’t in some choice of coordinates, but it DOES in the natural limit of a universe with non-zero matter content.

The good thing about the rubber-sheet analogy is that it avoids the Doppler shift as an explanation for the cosmological redshift, which, even if it can be made to work in some sense (as in your paper) is probably more trouble than it’s worth (especially if it leads people to think that the relativistic Doppler formula has any place in the discussion of the cosmological redshift).

“And no, the photon does not "lose energy" in any sense. It is just has different energies for different observers, and we are all different observers on different galaxies.”

This is definitely wrong. Photons are redshifted while their number remains constant. The energy of the universe decreases. Harrison has also discussed this at length, and stating that it just ain’t so isn’t enough. If Harrison is wrong on this point, I want to see a refereed-journal paper explicitly showing where he is wrong.

Think of the CMB and an observer who sees it having the same temperature in all directions. The temperature drops with time. This corresponds to the photons losing energy.

Of course, for a PARTICULAR PHOTON one can give it an arbitrary energy via a suitable choice of reference frame. But the sum of the energies of all photons drops with time.

“In this picture, the redshift is easy to understand. If a guy in one neighborhood tosses a ball to his neighbor, the speed of the ball as measured by the catcher will be less than the speed as measured by the thrower. That is, the two measure different energies for the ball, not because there's some phenomenon taking energy away, but just because they're in different reference frames. If the catcher then turns around and throws again to his neighbor, the same thing happens again, and so on. That's all the redshift is. It's not some mysterious "stretching."”

However, when we measure a redshift of z, then 1+z is the ratio of the scale factor of the universe now to the time when the light was emitted. If one thinks of the wavelength of the photon as stretching with the expansion of the universe (not necessarily “being stretched” by the expansion of the universe), then this is a natural result. If it is an infinite series of infinitesimal Doppler shifts, it is not immediately obvious that knowing z gives us knowledge of the change in scale factor during the time the light was on its way to us.

Harrison’s book has a lot to recommend it — it’s one of the main things I give my research students to read. But Hogg is right that when he discusses the relation between relativity and cosmology, he’s flat-out wrong. I’m away from home and don’t have the book handy, so I can’t give you chapter and verse at the moment, but I can in a few weeks if you like.

(Incidentally, the fact that he’s dead is irrelevant. Presumably it’s OK to discuss frankly the merits of the ideas in a widely influential book, even after the author’s death.)

In general, Phillip and I simply disagree, and I’m quite content to leave it at that: neither of us is going to convince the other. I do have to make one comment, though. Phillip says: “Photons are redshifted while their number remains constant. The energy of the universe decreases.” The second statement is wrong, or to be more precise it is meaningless. There is no sensible way to define the “energy of the universe” in such a way that this is true.

The quantity Phillip is talking about is apparently obtained by adding up the energies of a bunch of photons, with each of these energies defined in a different frame. You’re not allowed to do this. That’s not something esoteric about relativity — you’re not even allowed to do it in Newtonian physics. Suppose you measure the kinetic energy of one baseball in your backyard, and I measure the kinetic energy of another baseball while traveling in a 747. Nothing meaningful results from adding the two. Similarly, nothing meaningful results from adding the energies of two CMB photons separated by a cosmological distance.

We’re not the only ones who are fans of Harrison’s book; I’m aware of several others. When you give this to your students, do you say “By the way, he’s totally wrong in chapter so and so”?

Of course, his death is irrelevant except that it is useless to ask whether he really believed such and such or was simply misunderstood, since we can’t ask him.

I wouldn’t rule out being convinced. In fact, I would welcome it, since that would mean I learn something new. I admit I might be biased to some extent. I think we all must agree on the following statement: Using the relativistic Doppler formula to calculate the velocity of an object at high redshift does not yield a meaningful answer in the the velocity so derived is not the temporal derivative of ANY distance used for other purposes in cosmology. (Easy way to see this: the relativistic Doppler formula doesn’t contain any cosmological parameters, so the result must be the same for all cosmological models.) Perhaps seeing so many people goof up on this point led me to distrust any mention of the Doppler shift in a cosmological context. I think it’s more trouble than it’s worth, even though your article and an old article by (IIRC) Narlikar show that it is possible in some sense (though whether the enlightenment outweighs the confusion is something we could agree to disagree on).

I agree that the idea of energy in GR is a thorny issue, and thus the “total energy of the universe” is a difficult concept. But consider the following: At a given instant of cosmic time (this is a well defined concept), all observers in the rest frame of the CMB will measure the same temperature. Imagine a network of such observers covering the entire universe, in the limit where each point has an observer. Each observer measures the energy of the CMB photons at his location at the given instant. The sum of these might not be the “total energy of the universe”, but I think it is clear that this is a quantity which decreases with time. Harrison’s point is that the energy lost due to the decrease in the CMB temperature with time doesn’t go anywhere, but simply disappears (not a problem for the conservation of energy since the relevant symmetry doesn’t exist in the cosmological context). Does Hogg dispute the fact that it decreases with time, or does he claim that it goes somewhere else?

To answer the first question: Yes, I do, when I remember to, especially if the student in question has a strong interest in and grasp of relativity.

You say, “I think we all must agree on the following statement: Using the relativistic Doppler formula to calculate the velocity of an object at high redshift does not yield a meaningful answer in the the velocity so derived is not the temporal derivative of ANY distance used for other purposes in cosmology.”

I do not agree with the “does not yield a meaningful answer” clause, but I do agree that the velocity is not the temporal derivative of a distance. That’s not a requirement for something called a velocity to be meaningful, as far as I’m concerned. The velocity of a spectroscopic binary star, as measured from a terrestrial telescope, is also not the derivative of a distance, at least not in any sense of distance that’s useful for any other purpose. The point here is exactly the same as in the cosmological context: the velocity in question is comparing things at two different times, whereas “distance” in the sense in which you seem to want to use the term is defined at an instant (in some inertial frame).

You say, “The sum of these might not be the "total energy of the universe", but I think it is clear that this is a quantity which decreases with time.”

Sure, but since it’s not the total energy of the Universe, I can’t work up any interest in this quantity. It’s simply not a physically meaningful quantity. When this quantity changes, it’s meaningless to ask “where it goes,” since there’s no reason we should have expected this quantity to behave in any particular way. And I strongly object to the use of language suggesting that the non-conservation of this quantity means that “energy is not conserved,” because this quantity is NOT energy.

In general, I don’t speak for Hogg, but I can confidently answer your last question: like me, he agrees that this quantity decreases with cosmic time, and like me he doesn’t think it’s useful or interesting to ask “where it goes,” since it’s not a physically meaningful quantity.

“the velocity is not the temporal derivative of a distance” But this is the standard definition of a velocity. Why the confusing terminology?

“The velocity of a spectroscopic binary star, as measured from a terrestrial telescope, is also not the derivative of a distance, at least not in any sense of distance that's useful for any other purpose.” I claim that what is measured is the derivative of the proper distance. Whether that is useful for any other purpose is a different question, but it is the derivative of a standard distance.

“whereas "distance" in the sense in which you seem to want to use the term is defined at an instant (in some inertial frame)” That’s true for some distances. Others, like the angular-size or luminosity distance, are defined through some measurement prescription.

I don’t think I was very clear about the binary star. Remember that (a) the Earth is going around the Sun, in addition to the motion of the star, and (b) the velocity measured connects two different moments in time. There is no sense, at least none I can think of, in which the measured velocity is the rate of change of the distance of anything. Explain to me the function D(t) such that the measured velocity is dD/dt.

“I don't think I was very clear about the binary star. Remember that (a) the Earth is going around the Sun, in addition to the motion of the star, and (b) the velocity measured connects two different moments in time.”

It seems I can’t win. If I define a velocity as something like v=HD in the velocity-distance relation, you say it is not a distance one can directly measure. If I define a distance by a measurement prescription, you say it connects two different moments in time.

I’m missing your point here. You made the argument that something can’t be called a v unless it’s a dx/dt. I’m saying that that’s not a required criterion, and giving an example to show it.

I don’t remember all of the details of our previous conversations, but here’s what I think is true regarding v=HD. I guess I’m reasonably happy for you to call HD “a velocity” if you want. But the fact that that velocity doesn’t match the observed redshift is not evidence that the redshift is not a velocity. Even if v=HD is “a velocity”, it’s not the velocity that’s relevant in explaining the redshift, since HD is evaluated at a moment of cosmic time, whereas the quantity that’s relevant to the redshift is the velocity of us now relative to the galaxy then.

You say “the quantity that's relevant to the redshift is the velocity of us now relative to the galaxy then”. I take it this is a general definition, and this is what ANY redshift (other than a gravitational redshift) measures. Is this just a visualisation or can one actually define a velocity this way (at two different times, which you otherwise indicate is something of a problem)? For large cosmological redshifts, is there a simple formula which relates this velocity to the redshift? Or does one have to think of it as an integral and integrate?

No need to answer the last question; I just re-read your paper. I’m now in total agreement. 🙂

Let me talk briefly about the tethered galaxy. I don’t think it is obvious that, in the expanding-space picture, one would necessarily expect it to “get caught up in the expansion of the universe” and hence move away. One can think of a peculiar velocity (“relative to the rubber sheet”) exactly cancelling the cosmological expansion. Since in general the latter varies with time, so must the former. When the tether is cut, this relationship is no longer enforced, and the former stays fixed while the latter evolves depending on the cosmological parameters. Which one wins out depends on the cosmological parameters and the initial distance. Again, we all agree on what the calculations show; I’m just questioning the fact that the expanding-space paradigm intuitively leads one to a wrong answer.

I don’t think that I am the only one who thinks of a velocity as the change in a(n otherwise useful) distance with time. Again, I think this is the source of the confusion. I’ve become accustomed to thinking about “unobservable” distances such as the proper distance at a given instant of cosmic time, but have been more conservative with regard to velocities. Once one gives up the idea that a velocity is defined as the change in (an otherwise useful) distance with time, the whole issue vanishes. The fact remains though that IF one thinks of the velocity given by the relativistic Doppler formula for large redshifts as the change in some distance (perhaps

definedby this equation) with time, then it is important to remember that this distance is not directly related to any of the other “compression of distances” used in cosmology.Maybe expanding space should be viewed similarly to the membrane paradigm in black-hole theory; it isn’t what is really going on, but can be a useful heuristic in some cases if one understands the limitations.

I’m so glad you agree! In many discussions with many people on this topic, I think this is the first time that’s ever happened. (By the way, that’s my lame excuse for not understanding when you were agreeing with me over on the other thread!)

I agree with you on the tethered galaxy: there’s no necessary connection between the misconception and the expanding sheet paradigm. I’ve guessed that there is a connection in practice, because many people who have been taught the expanding sheet end up with the misconception. But that’s just anecdotal (I haven’t done a formal survey), and anyway we all know that correlation is not the same as correlation (although it is correlated with it!).

Thanks for all the discussion!

“correlation is not the same as correlation”

I guess you mean that correlation is not the same as causation. True, of course, and something many people miss, though there are numerous examples showing the difference (most people who die are married, for instance).

I think the hurdle is thinking of a velocity as dx/dt where x is some distance one is familiar with. If that assumption goes (and the binary star is a good example), then things become much more clear.

"the quantity that's relevant to the redshift is the velocity of us now relative to the galaxy then"

Think of a hypothetical universe which is static when the light is emitted, then expands, then is static again when the light is detected. All agree that 1+z is the ratio of the scale factors now (detection) and then (emission). Does the quote above still hold in this case?

I hate when I mess up my own jokes! Yes, it should be “correlation is not the same as causation.”

To answer your last question: Yes, it does. The point is that to compare the two velocities you need to parallel transport one to the other. When you do that, you’ll cross over the period of expansion, with the result that the relative velocity agrees with the redshift. That’s what Figure 3 of our paper was meant to illustrate.

(If you don’t like to talk in terms of parallel transport, you can instead just imagine that the two velocities are compared by a succession of observers along the path, each of whom signals to the one next to him. It’s the same thing.)

I have a blog against the expansion of the universe, with arguments that show this is impossible.

I write also over the evidences for the Big Bang.

I have arguments and Hypotheses in: http://bigbangno.wordpress.com

Thanks.