My question is simple.
A photon is emitted by a distant galaxy and captured by human telescope. Because the distance is great, it undergoes redshift due to universe expansion. But this means the photon is captured at lower energy than at which it was emitted. Where does the energy difference go?
Where does the energy go?
Re: Where does the energy go?
The lost energy goes into the gravitational field
Re: Where does the energy go?
That sounds strange. What is the mechanism?
Re: Where does the energy go?
I often feel like lots of my energy goes into the gravitational field...more every day as I get older...dammit
Re: Where does the energy go?
It's not so strange if you think about it. The photons are losing energy because they are interacting with the gravitational field. Every action has a reaction so the gravitational field must be changed too. All fields in physics contain energy and the gravitational field is no exception, so the loss of energy to the photon must be balanced by a gain of energy in the gravitational field.
It is not very different from any other physical process where energy is transfromed from one form to another. Of course the gravitational field is related to the curvature of spacetime so the photons are having a small effect on the curvature and that includes the rate of expansion of the universe.
It is not very different from any other physical process where energy is transfromed from one form to another. Of course the gravitational field is related to the curvature of spacetime so the photons are having a small effect on the curvature and that includes the rate of expansion of the universe.
Re: Where does the energy go?
I always thought photon redshift is driven by expansion of the universe, not by gravitational field. Of course I know there is also gravitationally induced redshift but in general case gravity has negligible effect on photon redshift in universe - or at least I always heard that's so.
Doppler effect is similar to photon redshift (or at least it's always explained as such), but in this case the sound from escaping source is reaching me with lower energy but for longer time so the total energy is conserved (I think). Photon is just a particle, it doesn't interact "longer time". A chance for more photons being received than transmitted comes into mind but it looks a bit counterintuitive in scope of particle physics.
Plus, photon has no rest mass so how can it interact with gravity? Yes I know its trajectory is bent by spacetime curvature but - does it "straighten" it or what is actually happening then?
Doppler effect is similar to photon redshift (or at least it's always explained as such), but in this case the sound from escaping source is reaching me with lower energy but for longer time so the total energy is conserved (I think). Photon is just a particle, it doesn't interact "longer time". A chance for more photons being received than transmitted comes into mind but it looks a bit counterintuitive in scope of particle physics.
Plus, photon has no rest mass so how can it interact with gravity? Yes I know its trajectory is bent by spacetime curvature but - does it "straighten" it or what is actually happening then?
Re: Where does the energy go?
Photons have an energy proportional to their frequency. By E=mc^2, this means they have a mass (though no rest mass - so they cannot be at rest). Therefore, they source a gravitational field and bend spacetime, just like any other mass.
However, this does not explain the apparent violation of energy conservation.
Generally speaking, in General Relativity energy conservation only holds locally! This is a consequence of spacetime being curved. To go beyond this - to compare any quantities like velocities or energies at two different points in spacetime (e.g. at two different times) - you have to describe by what procedure you get from one point to the other. If you do this by going along the path of a photon that is being redshifted, you find you have just derived the redshift all over again - and in this sense the energies do "match".
Going beyond local energy conservation is possible, but that introduces all kinds of subtleties. Usually you will be talking about the energy of an isolated system, as measured by an observer sitting well outside it ("at infinity") where the spacetime is approximately flat and unchanging with time.
In cosmological settings, i.e. to deal with the expansion of the universe, you have to be even more careful. This is because energy conservation laws are a consequence of a symmetry, time translation invariance (roughly speaking, things dont change when you move forward or backwards in time). Clearly you have to be careful when extending this to the case of an expanding universe, which is not static or even stationary...
I fear I may not have answered you question exactly, but I hope this helps somewhat!
However, this does not explain the apparent violation of energy conservation.
Generally speaking, in General Relativity energy conservation only holds locally! This is a consequence of spacetime being curved. To go beyond this - to compare any quantities like velocities or energies at two different points in spacetime (e.g. at two different times) - you have to describe by what procedure you get from one point to the other. If you do this by going along the path of a photon that is being redshifted, you find you have just derived the redshift all over again - and in this sense the energies do "match".
Going beyond local energy conservation is possible, but that introduces all kinds of subtleties. Usually you will be talking about the energy of an isolated system, as measured by an observer sitting well outside it ("at infinity") where the spacetime is approximately flat and unchanging with time.
In cosmological settings, i.e. to deal with the expansion of the universe, you have to be even more careful. This is because energy conservation laws are a consequence of a symmetry, time translation invariance (roughly speaking, things dont change when you move forward or backwards in time). Clearly you have to be careful when extending this to the case of an expanding universe, which is not static or even stationary...
I fear I may not have answered you question exactly, but I hope this helps somewhat!
Last edited by photino on Tue May 04, 2010 10:49 pm, edited 2 times in total.
Re: Where does the energy go?
Expansion of the universe is more than just galaxies moving apart. It is a feature of the metric of spacetime itself. The metric is also the gravitational field. So saying that photon redshift is due to expansion of universe implies that it is also an interaction with the gravitational field. This is not the gravitational pull of the photons on each other. It is a background field.
Energy is a relative concept that differs from one reference frame to another. To fix our understanding of energy we must fix a specific standard of time. When we talk of the energy of the photons in an expanding universe it is normally understood that we take time as the age of the universe and energy is defined in relation to that. If you take a different definition of time (which you are entitled to do) then the energy will be different and this difference can be interpreted as a Dopler shift.
Gravity does not just interact with rest mass, it interacts with energy and momentum which a photon still has with no rest mass.
Energy is a relative concept that differs from one reference frame to another. To fix our understanding of energy we must fix a specific standard of time. When we talk of the energy of the photons in an expanding universe it is normally understood that we take time as the age of the universe and energy is defined in relation to that. If you take a different definition of time (which you are entitled to do) then the energy will be different and this difference can be interpreted as a Dopler shift.
Gravity does not just interact with rest mass, it interacts with energy and momentum which a photon still has with no rest mass.
Re: Where does the energy go?
There are a lot of well known cosmologists and relativity experts who like to say things such as "energy conservation is only local in GR" or "energy conservation is only non-local in GR" or "energy conservation only works in special cases such as static spacetimes or asymptotically flat spacetimes" or "energy is quasi-local" or even "energy is not really conserved in GR" or "Energy Conservation is an empty trivial statement in GR"
Dont believe them, they are not only contradicting each other, they are talking nonsense. Most of their claims can be put to rest by showing that the same argument applies to energy conservation for an electron in an electric field or other similar circumstances where conservation is normally accepted. Energy conservation works perfectly well in GR for all circumstances and it is not trivial or approximate. It follows from Noether's theorem when applied to the general covariance of all fields with gravity included.
Dont believe them, they are not only contradicting each other, they are talking nonsense. Most of their claims can be put to rest by showing that the same argument applies to energy conservation for an electron in an electric field or other similar circumstances where conservation is normally accepted. Energy conservation works perfectly well in GR for all circumstances and it is not trivial or approximate. It follows from Noether's theorem when applied to the general covariance of all fields with gravity included.
Re: Where does the energy go?
Thank you I think I understand where I did the mistake. The photon has high energy only in the frame of reference of the source galaxy and that doesn't change - and it has low energy in the frame of reference of my galaxy and that also doesn't change all the time. Mixing these two was the problem.
It all looks so obvious now I can't understand how comes I didn't see it
It all looks so obvious now I can't understand how comes I didn't see it
Re: Where does the energy go?
@PhilG: I hesitate to say this because you are not quoting anyone or anything specific - but I think you are rather quick to call "nonsense" when referring to statements taken completely out of context. Quite often what happens is that a particular mathematical fact is expressed in words, and that can be misleading if taken on its own. For instance, a statement like "energy conservation is trivial in GR" might refer to the fact that the conservation of the energy-momentum tensor is automatic in GR, in the sense that it follows from the Bianchi identity together with the Einstein field equations. It's a matter of taste whether one prefers this to a derivation via general covariance.
Of course energy conservation always holds in GR, for all spacetimes, if by that you mean the conservation of the matter energy-momentum tensor (which is a field, and so this is a *local* statement). But I'm sure you'll agree one has to be careful when translating this back to questions like "where does the lost energy go?", because in GR you can't take an energy at time 1 and an energy at time 2, subtract one from the other and expect to obtain an meaningful result, outside of special circumstances or careful definitions. I think it's only fair to point this out when such questions come up, at the risk of complicating things a bit.
In a standard FRW cosmology, with standard comoving coordinates (like you suggest), the energy density part of the conservation of the (photon) energy-momentum tensor is precisely the equation that describes that the photons redshift along with the expansion of the universe. This is of course a consequence of the interaction of the photons with the spacetime metric, as you point out. But the integrated statement (for comparing the energy of a photon at two different times) is also a consequence of the symmetries of the FRW metric (there is a Killing tensor).
What puzzles me about your statement that "the lost energy goes into the gravitational field" is that it makes it sound like the gravitational field energy increases. As far as I know it is tricky (but not impossible) to define what is meant by "the energy of the gravitional field" in GR. So if this is what you were implying, I would be very interested to know the precise version of the statement!
Of course energy conservation always holds in GR, for all spacetimes, if by that you mean the conservation of the matter energy-momentum tensor (which is a field, and so this is a *local* statement). But I'm sure you'll agree one has to be careful when translating this back to questions like "where does the lost energy go?", because in GR you can't take an energy at time 1 and an energy at time 2, subtract one from the other and expect to obtain an meaningful result, outside of special circumstances or careful definitions. I think it's only fair to point this out when such questions come up, at the risk of complicating things a bit.
In a standard FRW cosmology, with standard comoving coordinates (like you suggest), the energy density part of the conservation of the (photon) energy-momentum tensor is precisely the equation that describes that the photons redshift along with the expansion of the universe. This is of course a consequence of the interaction of the photons with the spacetime metric, as you point out. But the integrated statement (for comparing the energy of a photon at two different times) is also a consequence of the symmetries of the FRW metric (there is a Killing tensor).
What puzzles me about your statement that "the lost energy goes into the gravitational field" is that it makes it sound like the gravitational field energy increases. As far as I know it is tricky (but not impossible) to define what is meant by "the energy of the gravitional field" in GR. So if this is what you were implying, I would be very interested to know the precise version of the statement!
Re: Where does the energy go?
Photino, Hi! The kind of stuff you are saying is exactly what many cosmologists and relativity experts say. I have looked at these things very hard and my conclusion is that they are nonsense. No offense intended to anyone but these things need to be put straight!
Let's look at some of the things you are syaing, or perhaps quoting. "Energy conservation in GR is trivial" I believe this was first said by Wheeler in his early book "Geometrodynamics" and people have been repeating it ever since. of course, if you are free to define what you mean by "trivial" then anything can be made trivial. That is why I like to compare these things with something else more basic that people have a better feel for. You say it is trivial because it follows from the bianchi identity and the field equations. OK, but this works exactly the same for any gauge theory. In elcetrodynamics for example, the electric charge is conserved because the electromagnetic field is divergence free from its definition (analogous to Bianchi identity in GR) When combined with Maxwells equations this makes the current have zero divergence. So you see that exactly the same argument works and if anything it is simpler in this case, yet noone claims that this makes charge conservation "trivial" in EM, so why do they say it makes energy conservation "trivial" in GR?
I think what you are saying about "local" conservation refers to the fact that the covariant divergence of the energy momentum tensor is zero. As you know this does not in itself lead to a conservation law. You can only integerate up to get a proper conservation law when you have a vector (or antisymmetric form) which has zero covariant divergence. The Energy-momentum tensor is a symmetric tensor so its divergence does not lead to a proper conservation law unless it can be contracted with a killing vector. Someitmes people use this to justify statements like "energy conservation is only local in GR" or even "energy conservation is only approximate". In this case an analogy would be a non-abelian gauge theory like QCD if you look only at the colour charge on the quarks and ignore the fact that the gluons have colour too. Of course you need to include all the color charges to get the colour charge conservation law. In GR the gravitational field has energy and momnetum too. If you just look at the energy-momentum tensor for matter fields which are included in the EM tensor then you can not expect a conservation law unless the gravitational field is static so that its energy does not change. To get the full conservation law you need to do a more complete calculation of the energy by applying Noether's Theorem to the full system of equations. This has been understood for many years, in fact Noether developed her theroem(s) because of the need to do this for GR. The result is completely general and exact, no ifs or buts required.
I think it was Peebles in one of his textbooks who made the errornous statement that redshift of photons contradicts conservation of energy. He was badly wrong, no disrespect for his great achievments intended, but sometimes the textbooks have errors and this is one of them. For some reason it is often repeated by people who should know it is wrong.
Yes you can do something special for the FRW case because it has a conformal symmetry giving a "killing tensor" but you dont need that. The conservation of energy always holds in GR, even with a cosmological constant included.
What I mean by that statement is that given any choice for "time translation" as specified by a timelike vector field N, you can then apply Noether's theorem to the full set of fields including gravity. The symmetry is the infinitesimal diffeomorphism generated by N. You dont need this to be a killing vector, any diffeomorphism is a symmetry of the Lagrangian and the full system of equations. From this you can derive a vector current that has a dependence on N. The covariant divergence of this current is zero. This can be integrated over a finite region of space with a boundary over time to show that the change in the total enegy in that region is equal to the total flux of the current leaving the region. This is what you need for a conservation law.
There are technicalities in applying Noether's Theorem due to the second derivatives of the metric tensor in the langrangian. In the past people used to remove these terms by adding a term that is always divergence free. This leads to the many Pseudo-tensor formulations that we all loathe (apparently you can make these more acceptable by calling them sections of jet bundles.) Another approach is to keep the second derivatives in the Langrangian but use a more general form of NT which can deal with second derivatives. It is easy to generalise in this way. There is also a little known second Noether's Theorem for gauge theories that makes the calulation easier to do. (Noether understood many things about this that most cosmologists today have apparently not learnt) The result is a covariant formulation for the conserved energy current but the price is that its dependence on N is more complicated than you might have hoped for. Whatever your taste these are all perfectly good energy conservation laws that work.
There are many more ifs and buts that people have tried to use to justify their claims that energy conservation is somehow not right in GR, but every one of them falls down on closer analysis.
Did someone open this thread with "My question is simple"? LOL
Let's look at some of the things you are syaing, or perhaps quoting. "Energy conservation in GR is trivial" I believe this was first said by Wheeler in his early book "Geometrodynamics" and people have been repeating it ever since. of course, if you are free to define what you mean by "trivial" then anything can be made trivial. That is why I like to compare these things with something else more basic that people have a better feel for. You say it is trivial because it follows from the bianchi identity and the field equations. OK, but this works exactly the same for any gauge theory. In elcetrodynamics for example, the electric charge is conserved because the electromagnetic field is divergence free from its definition (analogous to Bianchi identity in GR) When combined with Maxwells equations this makes the current have zero divergence. So you see that exactly the same argument works and if anything it is simpler in this case, yet noone claims that this makes charge conservation "trivial" in EM, so why do they say it makes energy conservation "trivial" in GR?
I think what you are saying about "local" conservation refers to the fact that the covariant divergence of the energy momentum tensor is zero. As you know this does not in itself lead to a conservation law. You can only integerate up to get a proper conservation law when you have a vector (or antisymmetric form) which has zero covariant divergence. The Energy-momentum tensor is a symmetric tensor so its divergence does not lead to a proper conservation law unless it can be contracted with a killing vector. Someitmes people use this to justify statements like "energy conservation is only local in GR" or even "energy conservation is only approximate". In this case an analogy would be a non-abelian gauge theory like QCD if you look only at the colour charge on the quarks and ignore the fact that the gluons have colour too. Of course you need to include all the color charges to get the colour charge conservation law. In GR the gravitational field has energy and momnetum too. If you just look at the energy-momentum tensor for matter fields which are included in the EM tensor then you can not expect a conservation law unless the gravitational field is static so that its energy does not change. To get the full conservation law you need to do a more complete calculation of the energy by applying Noether's Theorem to the full system of equations. This has been understood for many years, in fact Noether developed her theroem(s) because of the need to do this for GR. The result is completely general and exact, no ifs or buts required.
I think it was Peebles in one of his textbooks who made the errornous statement that redshift of photons contradicts conservation of energy. He was badly wrong, no disrespect for his great achievments intended, but sometimes the textbooks have errors and this is one of them. For some reason it is often repeated by people who should know it is wrong.
Yes you can do something special for the FRW case because it has a conformal symmetry giving a "killing tensor" but you dont need that. The conservation of energy always holds in GR, even with a cosmological constant included.
What I mean by that statement is that given any choice for "time translation" as specified by a timelike vector field N, you can then apply Noether's theorem to the full set of fields including gravity. The symmetry is the infinitesimal diffeomorphism generated by N. You dont need this to be a killing vector, any diffeomorphism is a symmetry of the Lagrangian and the full system of equations. From this you can derive a vector current that has a dependence on N. The covariant divergence of this current is zero. This can be integrated over a finite region of space with a boundary over time to show that the change in the total enegy in that region is equal to the total flux of the current leaving the region. This is what you need for a conservation law.
There are technicalities in applying Noether's Theorem due to the second derivatives of the metric tensor in the langrangian. In the past people used to remove these terms by adding a term that is always divergence free. This leads to the many Pseudo-tensor formulations that we all loathe (apparently you can make these more acceptable by calling them sections of jet bundles.) Another approach is to keep the second derivatives in the Langrangian but use a more general form of NT which can deal with second derivatives. It is easy to generalise in this way. There is also a little known second Noether's Theorem for gauge theories that makes the calulation easier to do. (Noether understood many things about this that most cosmologists today have apparently not learnt) The result is a covariant formulation for the conserved energy current but the price is that its dependence on N is more complicated than you might have hoped for. Whatever your taste these are all perfectly good energy conservation laws that work.
There are many more ifs and buts that people have tried to use to justify their claims that energy conservation is somehow not right in GR, but every one of them falls down on closer analysis.
Did someone open this thread with "My question is simple"? LOL
Re: Where does the energy go?
PhilG, thanks for your long reply! Really sorry I have not responded yet - I hope to have the time to do so properly soon.
Re: Where does the energy go?
Wait and all will be revealed ...
Scientific American, July 2010, "Is the Universe Leaking Energy", by Tamara M. Davis
She presents a very different interpretation of "Where did the energy go from the red shift?"
And if I might quote the last paragraph of her article (and I hope it peaks your curiosity to find the article) ... "Thus, the universe does not violate the conservation of energy; rather it lies outside that law's jurisdiction."
Scientific American, July 2010, "Is the Universe Leaking Energy", by Tamara M. Davis
She presents a very different interpretation of "Where did the energy go from the red shift?"
And if I might quote the last paragraph of her article (and I hope it peaks your curiosity to find the article) ... "Thus, the universe does not violate the conservation of energy; rather it lies outside that law's jurisdiction."