The Friedmann-Lemaitre-Robertson-Walker spacetime describing the expanding Universe, on the other hand, has a slew of symmetries it does obey, but time-translation isn't one of them: an expanding Universe is different from one moment in time to the next. If you had a static spacetime that weren't changing, energy conservation would be guaranteed. But if the fabric of space changes as the objects you're interested in move through them, there is no longer an energy conservation law under the laws of General Relativity. In general, these symmetries are profoundly important to our understanding of the Universe, and have enormous additional implications for reality.

You see, there's a brilliant theorem at the intersection of physics and mathematics that states the following: every unique mathematical symmetry exhibited by a physical theory necessarily implies an associated conserved quantity. A time-translation symmetry leads to the conservation of energy, which explains why energy is not conserved in an expanding Universe.

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## The Definition of Density in General Relativity

Quantum corrections to classical gravity are visualized as loop diagrams, as the one shown here in white. Whether space or time itself is discrete or continuous is not yet decided, as is the question of whether gravity is quantized at all, or particles, as we know them today, are fundamental or not. But if we hope for a fundamental theory of everything, it must include quantized fields. But these two descriptions of the Universe are both incomplete. There are many questions we can ask about reality that require us to understand what's happening where gravity is important or where the curvature of spacetime is extremely strong where we need GR , but also when distance scales are very small or where individual quantum effects are at play where we need QFT.

These include questions such as the following :. We need something more: an understanding of gravity at the quantum level. A hologram is a 2-dimensional surface that has information about the entire 3-dimensional object displayed encoded in it. The idea of the holographic principle is that our Universe and the quantum field theoretical laws that describe it is the surface of a higher-dimensional spacetime that includes quantum gravity.

We don't have a working theory of quantum gravity, of course, or we'd be able to understand what symmetries it does and doesn't exhibit. But even without a full theory, we have a tremendous clue: the holographic principle.

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The AdS stands for anti-de Sitter space, which is frequently used to describe quantum gravity in the context of string theory, while the CFT stands for conformal field theory, such as the QFTs we use to describe three of the four fundamental interactions. While no one is certain whether this is applicable to our Universe, there are many good reasons to think it does.

In the Standard Model, the neutron's electric dipole moment is predicted to be a factor of ten billion larger than our observational limits show. The only explanation is that somehow, something beyond the Standard Model is protecting this CP symmetry in the strong interactions. We can demonstrate a lot of things in science, but proving that CP is conserved in the strong interactions can never be done.

Which is too bad; we need more CP-violation to explain the matter-antimatter asymmetry present in our Universe. Different frames of reference, including different positions and motions, would see different laws of physics and would disagree on reality if a theory is not relativistically invariant.

The fact that we have a symmetry under 'boosts,' or velocity transformations, tells us we have a conserved quantity: linear momentum.

## Energy Is Not Conserved

This is much more difficult to comprehend when momentum isn't simply a quantity associated with a particle, but is rather a quantum mechanical operator. This symmetry, if the holographic principle is correct, cannot exist globally. The classic arguments for all of them, in fact, are rooted in black hole physics and are known to require certain assumptions that, if violated, admit various loopholes.

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The first one means that there are no conservation laws that always necessarily hold. Even CPT and Lorentz invariance can be violated.

In , an experiment running under the leadership of Blas Cabrera, one with eight turns of wire, detected a flux change of eight magnetons: indications of a magnetic monopole. Unfortunately, no one was present at the time of detection, and no one has ever reproduced this result or found a second monopole. Still, if string theory and this new result are correct, magnetic monopoles, being not forbidden by any law, must exist at some level. The three quantum gravity conjectures that are demonstrated to hold for a holographic Universe have been around, in some form, since , but they were only conjectures until now.

There are no global symmetries; nothing in the Universe is always conserved under all imaginable circumstances even if you need to reach the Planck scale to see violations , and all non-forbidden charges must exist. It would be revolutionary for our understanding of the quantum Universe. Despite the results and implications of this study, it's still limited. We don't know whether the holographic principle is true or not, or whether these assumptions about quantum gravity are correct.

Paradoxically, if string theory is right, our expectations about hidden symmetries revealing themselves at a more fundamental level are not only wrong, but nature has no global symmetries at all. Update : First author of the paper, Daniel Harlow, has reached out to clarify a point that was not sufficiently appreciated by the author.

He relates the following:. I wanted to point out that there is one technical problem in your description The reason is that they are all actually gauge symmetries, not global symmetries. For electric charge I guess you are familiar with that, but in gravitational theory such as general relativity then translations, Lorentz transformations, CPT, etc are also gauge symmetries: they are just diffeomorphisms.

The difference between a gauge symmetry and a global symmetry is that the presence of gauge charge can be measured from far away, while the presence of a global charge cannot. For example in elecromagnetism if we want to know the total charge in a region, we just have to measure the electric flux through its boundary. Similarly in gravity if we want to know the energy of something, we can measure the fall-off of the metric far away basically looking for the M in the Schwarzschild metric. It isn't widely appreciated, but in the standard model of particle physics coupled to gravity there is actually only one global symmetry: the one described by the conservation of B-L baryon number minus lepton number.

So this is the only known symmetry we are actually saying must be violated! Spatial translation symmetry leads to the conservation of momentum; rotational symmetry leads to the conservation of angular momentum. Then, the argument from above still follows, but that energy is a charge living on that surface. Since integrals over a surface can be converted to integrals over a bulk by Gauss's theorem, we can, in analogy with Gauss's Law, interpret these energies as the energy of the mass and energy inside the surface. If the surface is conformal spacelike infinity of an asymptotically flat spacetime, this is the ADM Energy.

If it is conformal null infinity of an asymptotically flat spacetime, it is the Bondi energy. You can associate similar charges with Isolated Horizons, as well, as they have null Killing vectors associated with them, and this is the basis of the quasi-local energies worked out by York and Brown amongst others. What you can't have is a tensor quantity that is globally defined that one can easily associate with 'energy density' of the gravitational field, or define one of these energies for a general spacetime.

The reason for this is that one needs a time with which to associate a conserved quantity conjugate to time. But if there is no unique way of specifying time, and especially no way to specify time in such a way that it generates some sort of symmetry, then there is no way to move forward with this procedure.

## Quantum Gravity in Everyday Life: General Relativity as an Effective Field Theory

For this reason, a great many general spacetimes have quite pathological features. Only a very small proprotion of known exact solutions to Einstein's Equation are believed to have much to do with physics. Energy conservation does work perfectly in general relativity. The overall Lagrangian is invariant under time translations and Noether's Theorem can be used to derive a non-trivial and exact conserved current for energy.

The only thing that makes general relativity a little different from electromagnetism is that the time translation symmetry is part of a larger gauge symmetry so time is not absolute and can be chosen in many ways. However there is no problem with the derivation of conserved energy with respect to any given choice of time translation. There is a long and interesting history to this problem. Einstein gave a valid formula for the energy in the gravitational field shortly after publishing general relativity. The mathematicians Hilbert and Klein did not like the coordinate dependence in Einstein's formulation and claimed it reduced to a trivial identity.

They enlisted Noether to work out a general formalism for conservation laws and claimed that her work supported their view. The debate continued for many years especially in the context of gravitational waves which some people claimed did not exist. They thought that the linearised solutions for gravitational waves were equivalent to flat space via co-ordinate transformations and that they carried no energy. At one point even Einstein doubted his own formalism, but later he returned to his original view that energy conservation holds up.

The issue was finally resolved when exact non-linear gravitational wave solutions were found and it was shown that they do carry energy. Since then this has even been verified empirically to very high precision with the observation of the slowing down of binary pulsars in exact agreement with the predicted radiation of gravitational energy from the system. Wikipedia has a good article on these and how they confirm energy conservation. Although pseudotensors are mathematically rigorous objects which can be understood as sections of jet bundles, some people don't like their apparent co-ordinate dependence.

Despite these general formulations of energy conservation in general relativity there are some cosmologists who still take the view that energy conservation is only approximate or that it only works in special cases or that it reduces to a trivial identity.

In each case these claims can be refuted either by studying the formulations I have referenced or by comparing the arguments given by these cosmologists with analogous situations in other gauge theories where conservation laws are accepted and follow analogous rules. One area of particular contention is energy conservation in a homogeneous cosmology with cosmic radiation and a cosmological constant. Despite all the contrary claims, a valid formula for energy conservation in this case can be derived from the general methods and is given by this equation. This always comes to zero in a perfectly homogeneous cosmology.

The first two terms describe the energy in matter and radiation with the matter energy not changing and the radiation decreasing as the universe expands. Both are positive. The final two terms represent the gravitational energy which is negative to balance the other terms. This equation holds as a consequence of the well-known Friedmann cosmological equations , that come from the Einstein field equations, so it is in no sense trivial as some people have claimed it must be. This then is a statement which can be interpreted as the constancy of an observable labeled energy.

This happens with the FLRW equation of cosmology. So we are not able to derive a conservation of energy from first principles. The cosmological constant is dependent on the vacuum energy density, plus pressure terms. Does this connects with something deeper than just a "detailed balance? It might, and I suspect that Phillip's analysis connects with this. The deSitter metric is a time dependent conformal theory of a flat metric.

Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Energy conservation in General Relativity Ask Question. Asked 8 years, 8 months ago. Active 8 years, 8 months ago. Viewed 9k times. Here are some examples, is there a convenient way to define energy in these scenarios? Just a system of gravitational waves. A point mass moving in a static but otherwise arbitrary space-time.

Equivalent if I'm not mistaken to a test mass moving in the field of a second much larger mass, the larger mass wouldn't move. Two rotating bodies of similar mass. Malabarba Malabarba 2, 3 3 gold badges 19 19 silver badges 28 28 bronze badges. Certain people hold that the conservation of energy is vacuous in GR while others think that concepts like ADM energy make the conservation non-trivial.