Can the expansion of the Universe reduce the energy of a photon to 0?

[DaveBeal]

Is it theoretically possible for the expansion of the universe to red-shift the energy of a photon all the way to zero? If so, what happens to the photon? Or does the photon's energy only approach zero as an asymptote?

 

[Dale]

No. There is no inertial frame where the energy of a pulse of light is zero.

 

[PeterDonis]

Is it theoretically possible for the expansion of the universe to red-shift the energy of a photon all the way to zero?

No. The "energy" you refer to, which is redshifted by expansion, is the energy measured by a comoving observer who is co-located with the photon. This energy is the inner product of the comoving observer's 4-velocity with the photon's 4-momentum, and it is impossible for that to be zero. Mathematically, that is because it is impossible for the inner product of a timelike vector and a lightlike vector to be zero.

 

[Orodruin]

Mathematically, that is because it is impossible for the inner product of a timelike vector and a lightlike vector to be zero.

Mathematically, the zero vector would like to know where it belongs.

 

[Vanadium 50]

Mathematically, the zero vector would like to know where it belongs

Tell it we don't put up with its kind here.

A photon also has angular momentum. If it disappeared, where would that go? So "no" for a 2nd reason as well.

 

[PeterDonis]

Mathematically, the zero vector would like to know where it belongs.

It's not a valid timelike 4-velocity vector for an observer, and it's not a valid 4-momentum vector for a photon. So it doesn't belong anywhere in this discussion.

 

[Orodruin]

It's not a valid timelike 4-velocity vector for an observer, and it's not a valid 4-momentum vector for a photon. So it doesn't belong anywhere in this discussion.

In the physics discussion, no. In a purely mathematical statement about timelike and lightlike 4-vectors though.

 

[PeterDonis]

In a purely mathematical statement about timelike and lightlike 4-vectors though.

"Timelike" and "lightlike" are physical definitions. I don't think the zero vector meets the physical definition of a "lightlike" vector even though its norm is zero.

 

[Orodruin]

"Timelike" and "lightlike" are physical definitions. I don't think the zero vector meets the physical definition of a "lightlike" vector even though its norm is zero.

Not sure I agree with this. The nomenclature is obviously derived from physics, but that doesn’t make the definition that a timelike vector is a vector with positive square (or negative depending on convention) any less mathematical.

 

[PeterDonis]

@Orodruin, the definition of "timelike" isn't the issue, since the zero vector is obviously ruled out because its squared norm is zero, not positive or negative.

The definition of "lightlike" is the issue. I do not think the definition of a "lightlike" vector is meant to include the zero vector as lightlike, even though the zero vector does technically have a zero squared norm. I think the intended definition of "lightlike" is a nonzero vector that has zero squared norm.

 

[PAllen]

Ok, but the statement remains that if a null vector is not a zero vector per some basis at some event, then for any choice of basis, and any sequence of parallel transports, it remains nonzero. This precludes the possibility mentioned in the OP.

 

[martinbn]

I think the definition of light-like vector is a non-zero vector with zero norm.

 

[PeterDonis]

the statement remains that if a null vector is not a zero vector per some basis at some event, then for any choice of basis, and any sequence of parallel transports, it remains nonzero. This precludes the possibility mentioned in the OP.

Yes, agreed, but you still need to show that the antecedent is true. If @Orodruin is correct that we need to include the zero vector as "lightlike" on mathematical grounds, then it would be possible in principle to have a null vector that was a zero vector at some event. My point is that that's impossible: the definition of "lightlike" does not include the zero vector. Yes, ultimately that's based on the physical claim that the zero vector can't represent any physical thing, but it's still a mathematical definition of "lightlike" that excludes the zero vector.

 

[PAllen]

Yes, agreed, but you still need to show that the antecedent is true. If @Orodruin is correct that we need to include the zero vector as "lightlike" on mathematical grounds, then it would be possible in principle to have a null vector that was a zero vector at some event. My point is that that's impossible: the definition of "lightlike" does not include the zero vector. Yes, ultimately that's based on the physical claim that the zero vector can't represent any physical thing, but it's still a mathematical definition of "lightlike" that excludes the zero vector.

But then you would have light that had no energy or momentum ever ( at any event on its world line , or any basis choice). I guess that motivates non calling it light like. That is, a zero vector is zero in any basis and retains this property on any parallel transport.

 

[PeterDonis]

But then you would have light that had no energy or momentum ever ( at any event on its world line , or any basis choice). I guess that motivates non calling it light like.

Yes.

That is, a zero vector is zero in any basis and retains this property on any parallel transport.

Yes.

 

[Orodruin]

Yes, agreed, but you still need to show that the antecedent is true. If @Orodruin is correct that we need to include the zero vector as "lightlike" on mathematical grounds, then it would be possible in principle to have a null vector that was a zero vector at some event. My point is that that's impossible: the definition of "lightlike" does not include the zero vector. Yes, ultimately that's based on the physical claim that the zero vector can't represent any physical thing, but it's still a mathematical definition of "lightlike" that excludes the zero vector.

Based on a brief check of the literature (selection based on what I had at home + Carroll's arXiv lecture notes):

• Rindler: Defines lightlike/null as having zero norm. There is no exclusion of the zero vector. To the opposite effect, there is a problem that explicitly refers to "S, T, N, V" as a spacelike, timelike, null, and general vector which are all real and non-zero.
• Schutz: Defines lightlike/null as having zero norm, then makes a comment that a null vector is not a zero vector. This may indicate that he is excluding the zero vector from the definition, but the way it is written gives the impression that the main takeaway should be that a null vector need not be the zero vector.
• Guidry: I can't find the definition easily from the index and cursory browsing ... weird, this should be in any relativity text.
• Carroll: "if $\eta_{\mu\nu} V^\mu V^\nu = 0$, $V^\mu$ is light-like or null" (p 15, at the bottom). He then makes a particular comment that a null vector does not need to be a zero vector, but does not exclude the zero vector from the null/light-like vectors.

Edit: Oh, I almost forgot another book I have at home ...

• Ohlsson and collaborator : Defines a vector $x$ "as ... lightlike if $x^2 = 0$"

Edit 2:

• MTW: p53: 'That one can classify directions and vectors in spacetime into ... and "null" or "lightlike" (zero squared length)'. There is no particular mention of excluding the zero vector from the null vectors.

 

[Orodruin]

then it would be possible in principle to have a null vector that was a zero vector at some event.

Just to add, this is not correct. Parallel transport needs to conserve not only the norm of vectors, but also the inner product between different parallel transported vectors. This makes it impossible for a non-zero null vector to be parallel transported into a zero vector.

 

[PeterDonis]

this is not correct

What I intended to state is correct, but I stated it ambiguously. Let me try again:

I agree that it is impossible to turn a nonzero vector into a zero vector by parallel transport. I was not claiming the contrary.

What I was saying is that, if we adopt the definition of "lightlike vector" that you have now given several sources for (I'll make comments on those references in a separate post, here I'm assuming for the sake of argument that they intended what you say they intended), then it would be possible in principle, physically, for the zero vector to represent some actual physical system. Such a system would have to have zero invariant mass, so it would be a "lightlike" object--but it would have zero energy and zero momentum, forever, because if it's the zero vector at one event on its worldline, it would have to be the zero vector at every event on its worldline.

That seems physically unreasonable to me, and the question I would ask the authors of each of the sources you give is, does it not seem physically unreasonable to them? And if they agree it does, then why would they not make sure that their definition of a lightlike vector mathematically excludes a possibility that they agree can't represent anything physical? Physicists are supposed to use math as a tool, not let math dictate things that make no physical sense.

 

[Orodruin]

As a description of something like the 4-momentum of a massless particle, yes, the zero vector is not really relevant. But this is not the only possible physical use of vectors with zero norm. And why, mathematically, would I want to exclude the zero vector from the categorisation into timelike, spacelike, and null? Sure, it is a very particular vector in itself, but isn't it enough to have three categories without having to explicitly categorise the zero vector? Physically, it is (or should be) always clear whether or not the zero vector is relevant.

Physicists are supposed to use math as a tool, not let math dictate things that make no physical sense.

I don't think defining a lightlike vector as one with zero norm dictates anything about the actual usage of the vectors. It is a definition and as far as all definitions go, you may like or dislike it for various reasons, but it does not inhibit your ability to use the maths in any way or form. Nothing states that you are forced to use a zero vector as a 4-momentum just because it may be classified as lightlike.

Edit: Compare to the Euclidean case, where typically the zero vector is considered orthogonal to all vectors, including itself. You can do what you will with that information.

 

[PeterDonis]

Schutz: Defines lightlike/null as having zero norm, then makes a comment that a null vector is not a zero vector. This may indicate that he is excluding the zero vector from the definition, but the way it is written gives the impression that the main takeaway should be that a null vector need not be the zero vector.

I disagree. His actual statement is: "It is particularly important to understand that a null vector is not a zero vector." The emphasis on "not" is in Schutz. That indicates to me that is specifically intends the zero vector to not be included in his definition of a null vector. His other remarks, I think, are with the same intent that I describe below with reference to Carroll--to emphasize to new students that the metric of spacetime is not Euclidean, but Minkowski, so that non-zero vectors can still have zero norm.

[Carroll] then makes a particular comment that a null vector does not need to be a zero vector

His actual statement is: "A vector can have zero norm without being the zero vector." That statement is ambiguous. It could mean that the zero vector is included in the definition of a null vector, but also that other vectors that aren't the zero vectors are; or it could be a way of pointing out to a new student who isn't familiar with the Minkowski metric that the definition of "null vector" just given is not vacuous: that it actually means those vectors which aren't the zero vector but still have zero norm, because the metric is not the Euclidean metric that the student is used to.

So I don't think we can say that Caroll intended the zero vector to be a null vector based on that statement. It is perfectly possible that he simply didn't want to go into such technical details in a book that was intended as an introduction to the subject. Similar remarks would apply to Schutz as discussed above.

 

[PeterDonis]

this is not the only possible physical use of vectors with zero norm

Can you give some others?

 

[Orodruin]

Can you give some others that are relevant in this context? Remember we are talking about the energy and momentum of a "photon" ("light pulse" would be a better term here).

No, you are talking about a light pulse, but the usage of the terminology "lightlike" does not have this restriction.

Just as an example, we could imagine talking about two inertial observers in Minkowski space and ask ourselves when a light signal from observer A would reach observer B. The reply would be "when the separation of the emission event and the event on the worldline of B is lightlike". If you do not include the zero vector in lightlike, then you would have to include "or the zero vector" in this statement.

 

[Orodruin]

So I don't think we can say that Caroll intended the zero vector to be a null vector based on that statement.

Agreed, but as far as the definition goes, we can only read what he has actually written, which is that a lightlike vector is one with zero norm.

 

[Orodruin]

is perfectly possible that he simply didn't want to go into such technical details in a book that was intended as an introduction to the subject

This is personal inference. Is MTW also to be considered as intended as an introduction to the subject? It is a rather extensive introduction in that case.

 

[PeterDonis]

we could imagine talking about two inertial observers in Minkowski space and ask ourselves when a light signal from observer A would reach observer B. The reply would be "when the separation of the emission event and the event on the worldline of B is lightlike". If you do not include the zero vector in lightlike, then you would have to include "or the zero vector" in this statement.

Why? Unless A and B are the same--which makes no sense since then you would not even talk about "the separation of the emission event and the event on the worldline of B", or even about a "light signal" at all, since there aren't two events, only one--the vector that connects the two events in question will not be the zero vector.

 

[PeterDonis]

This is personal inference.

Sure. One has to do that when reading any text.

Is MTW also to be considered as intended as an introduction to the subject? It is a rather extensive introduction in that case.

The entire book is of course not just an introduction. But the definitions of "timelike", "spacelike", and "null or lightlike" are given in Chapter 2, which is intended as an introduction to SR.

 

[Orodruin]

The entire book is of course not just an introduction. But the definitions of "timelike", "spacelike", and "null or lightlike" are given in Chapter 2, which is intended as an introduction to SR.

That's just being unreasonable. If it was that important to single out the zero vector from the set of lightlike vectors, they would have done so at some point in the book. If not, we have to conclude that they are perfectly fine with keeping the zero vector as "lightlike". As I have said, it changes absolutely nothing when discussing physics.

Why? Unless A and B are the same--which makes no sense since then you would not even talk about "the separation of the emission event and the event on the worldline of B", or even about a "light signal" at all, since there aren't two events, only one--the vector that connects the two events in question will not be the zero vector.

Physics is full of approximations. In this case the approximation is that A and B can actually be colocated.

You can also have a series of Lorentz transformations whereby a lightlike vector has the zero vector as a limit, which is not the case for a timelike or spacelike vector. This would be the case, e.g., for a series of consecutive observers, each with relative velocity $v$ to the previous one. Integration over loop momenta in Feynman diagrams includes integrating over the zero vector (admittedly with zero measure, but still, explicitly removing it seems weird).

 

[PeterDonis]

In this case the approximation is that A and B can actually be colocated.

If A and B are colocated (which means that there is just one worldline, not two) then you can't send a light signal between them, which was your example.

You can also have a series of Lorentz transformations whereby a lightlike vector has the zero vector as a limit

Integration over loop momenta in Feynman diagrams includes integrating over the zero vector

These seem more relevant, since, as you say, excluding the zero vector would be a complication for such cases.

 

[Orodruin]

If A and B are colocated (which means that there is just one worldline, not two)

This is an incorrect inference. That they are colocated at a particular time does not imply that they must be colocated at all later times.

then you can't send a light signal between them, which was your example.

A sends continuous signals. B receives them. This includes the signal emitted as they pass each other to the approximation that A and B can be colocated.

These seem more relevant, since, as you say, excluding the zero vector would be a complication for such cases.

To add to this, consider soft photons in the out states. In order to get convergent amplitudes, you need some IR cutoff. This includes excluding photons having the zero vector as 4-momentum as well as photons having small, but non-zero, 4-momentum. In that sense, the zero vector is not different from other "small" light like vectors.

In the end, the terminology does not matter much. What matters is what we do with the maths, not what things are called. I could call timelike vectors "green vectors". It would complicate understanding and be severely confusing, but physics would not change if this change was driven through consistently. What matters in the end is the model itself, not what we choose to call things.

 

[PeterDonis]

A sends continuous signals. B receives them.

This was not at all clear from your previous post about this. I thought you were only talking about a single light signal.