It's time translation, which is a continuous symmetry-and, by Noether's theorem, implies the existence of a conserved quantity-in this case the energy. Time reversal is a discrete symmetry, therefore doesn't lead to conserved quantities.

This is explained in any textbook, usually.

Manuel Urueña Palomo > "And time translation is also violated in an expanding universe."

This is not a violation of time translation invariance of the equations of motion (the Friedman equations), but something introduced by each specific solution to these equations (in this case the one which describes our universe to some accuracy). For models where the matter part of the Friedman type of universe models can be described by a Lagrangian, the Noether theorem can be applied to find a conserved energy for the system (i.e. universe).

Just (try to) work it out for yourself!

Kåre Olaussen But if you consider dark energy or an energy of the vaccum in an expanding universe, total energy is not conserved over time right?

No, that is not right.

As I have already said, there is a total energy expression, which is conserved over time, due to time translation invariance. It is insufficient to consider only partial contributions to this expression.

I think I have discussed this in more explicit mathematical detail in some earlier RG answer(s).

Here I am only referring to the Friedman type of universe models, where the analysis is simple. But I believe this is a general property of cosmological models based on general relativity.

I will assume with my interpretation of your confusion about this as I was confused about it for a long time. I can be wrong in assessing your confusion but still I will try my best.

1) First let's get the basics out of the way that Noether's theorem is about the continuous change in a system, i.e. whatever changes you can think a system can have that change (technically it means change in its configuration e.g. to its wave function) will be a continuous function and will be generated by some operator/ "Generator of that change". And if that system is symmetric under that change then there exist some observable of that system which is unchanged/conserved and mathematically that "generator" is that conserved observable.

2) When it comes to "change over time" or you may call it time translation, if there exist some observable of your system that remains unchanged then that will be generator of time translation which we call as "Energy" and thus the conserved quantity of the system. See.. how this is about finding the generator of a known "change", not the other way around and that is the genius of this idea.

3) Now if there exist a time translation symmetry in your system that means yes your system's energy would not care about at what particular instant of time the system is and its energy will be conserved and that system can be anything e.g. a pool game or a universe.

But if you are confusing this symmetry with time reversal in the sense that your system does not care about the arrow of time/ 2nd law of thermodynamics (thinking of reversing the time). Which I also used to get confused about , which I then realized it is slightly wrong way of thinking because ARROW OF TIME is a statistical phenomenon. E.g. You can have the same energy of a system over time but its entropy will still increase. Entropy increases means system is loosing information (not in absolute sense of the word) about the distribution of energy into it's constituents, Energy is simply getting redistributed such that number of ways to describe the same system are increasing (It is like getting more noisy, information is still there but it is harder to crack, which is bound to happen because there exist more number of ways to do it then to become "easier" to crack). But the TOTAL ENERGY is still the same. So arrow of time is still there even if there exist a time-symmetry.

If that was your confusion about, I hope it helps, if not then I will be happy to discuss more.

I say: Both.

Abhishek .. What about the fact that due to the metric expansion of space, and the energy of the vaccum/dark energy associated to it, there is more total energy every new time instant?

when you say there is more energy each instant , well less. because the expansion of the universe requires it.. but the fact is curved means there is unconserved energy , and it can be exploited.

or we are running in reverse but we would not notice that. see the Einstein paper hes starts it that way.

it losing kinetic energy and surface tension is decreasing, and that's rotational energy.. its free energy.. Noether saved GR , the lost of energy from gravity waves is offset by the gain in space or degrees of freedom.. so that wil keep bodies in orbit longer. or they will fall into the sun without gaining KE energy they cant to to a lower orbital.. i dont thik you should expand the metric because you need more planx units to put stuff in and things dont stretch as it expands. i mean the ricci tensor.. a the masses start to form the optimally small area to contain that pressure or energy. the potential energy is going toward the max.. .. i dont think 4d is a dynamic.. its a kinematic or ontic, a product of recording.

1. Noether and d'Hooft are among the few careful to separatee observables from generators.. the ADs /Cft and holographic "principle " or is that the generators lie on a shell with 2 degrees of freedom. there are no singularities.. See Susskind on dimensional reduction. We would think the universe is expanding if it were contracting became encoding memory requires heat dissipation. Hawking, arrow of time.. reversible computing.

2. We remember a time of less entory. it takes time and space and room for chaos and we self organize feeding on chaos.. see non reciprocal dyanmics in groups

3. the metric ( the coordinate systems, don't stretch that pleases) you can but .. we want to known how the dynamics go and not warp the description. Extrinsic embeddings are fine. Neother says a 4d box with a shehre in it is a good description.. put all the mass on shell. use a spherical symmetry The metric expansion of space is one way to look at ti but that isn't very agreeable. its better to consider a density function of ( x, y , ct) or Ricci tenson on a Ds brane .. The invariante is timelike. If tis reversible its not a 4d theory its 3d.. a f( a, b, R)..

4. in a cyclic universe, expansion and contraction, there is conservation of energy and angular momentum , and energy, else only locally.it requires a full contraction and expansion.. Einstein like to start with it stretched out.

https://www.jstor.org/stable/1968714?read-now=1&seq=36#page_scan_tab_contents citiing ..

now i cite Chernitzky. quantum gravity, need peer review.. no dark energy, no forces, no particels, 2 pages,.

if i made a mistake please correc t im just a puppet maker .. im make simulations.. uisng physics engines.. for decades.. i know that there is theory that has a possible point and some that is just why or wow.. and i and a n engineer and now i need to get better theory. i wnat to learn about if we can tunnel under this brane .. its not string theory at all.. just a bit like it..

Manuel Urueña Palomo , like the others have said concisely, which isnt my talent, i'm glad that there is agreement, but Noether currents may explain something about the apparent change in energy over time, the dark energy you mention here.. seems that something would have to be in the field to have a vector field.. Chernitzky has a field theory that quantized by the circular orbits like a bohr lowest potential energy state. the Soliton's and repel they can act over the entire cosmos "like a tsunami" does not mention shocks in the timelike direction but the surface tension can make the geodesic bumps and orbits. Sure two solitons can appear to bounce, not interact, push or pull and extend as solitary , long period but waves with vorticity, but what is the field of? I don't know. So but this is a thin shell cosmos like a black hole, lots of questions but lets assumed its not real. its math. a Cohomology, She states (lost in some translations) the Lie groups tell less that the relations in the spherical symmetry, in fact if we use guesses we cant observe we might finally get some new observables , up close. So if you take the sphere thats hollow for the universe, you take 10^14 r^3 as r^2. no need for this darkmatter construct that doesn't interact with anything mass or light. its would be very dark if light is dispersing as in 3d on a hypersurface. its doesnt fit the Webb data very well,

if all. A black holes mass ( observed indirectly When you take Noether relations as real generalors, predictors, or physical laws. You have one less dimension, and cant be transformed to observables, but these are rear view mirror distorted. So we cant say for every orbiting body in an expanding, or apparent so universe there is energy added to compensate for the gravity waves shedding over time... To make a curve or an acceleration , you need that expansion in this case. Just having a curve can give you non reciprocity. that feeds on chaos.. So without the bulk its possible for surface pushing and pulling to expand or contract the balloon in an anisotropic way. So all these usually counter rotating currents create a tension that can give a hooks law like wave propagation fabric , that has killing vectors orthogonal to against time tike directions. The dispersion relation and the type of wave on that maybe be faraday like , or matter waves, im speculating. maybe someone can read this , its too complex for me but it think thes tackling the question here and we need this harmonic relation .. the CpT symmetry you mentioned , well over the whole cycle it wil become a sphere, and stop spinning before contracting again. Then its conserved. Its that's al true and reversible the FFT is unitary. And its a 2D universe , basically . a factions of ( x, y, an cT) t= 0 being the center mass of the universe.. . This goes against the holy GR and SR and other, but Einstein later papers we seriously going there.. Few read them. Hope if helps i work on 2D toy universes and have spend 7 months on research on this with some phDs.. tis time to drop the Standard Model because, its iots true, we are never going to leave the solar system. So i don't care about any cosmic measurements , of places we wil never go to. I do make assumptions, that that to me is the assumptions because we don't have time to just wonder about the big bang..

You are asking the right questions. Asl on Quora there is Roy Kerr has answered many and it you want to know about spinning black holes with give clues, hes the authority at 90 years old still snapping out answers.. not to everything the Kerr metric .. that's his.. frame dragging, bumpy, and very much a microcosm in some ways of what our universe could be like.

Einstein starts his toy universe backwards ( at the bounds) because you can't start a calculation or simulation with crazy hot numbers. runs it

from this linked above:

The Gravitational Equations and the Problem of Motion

A. Einstein, L. Infeld and B. Hoffmann there others..

here's a nice one.. a few pages.

Article Gravitation in Unified Scalar Field Theory

this is too technical for me and maybe a simpler one is in Chernitzky ( whos ideas are applied in chemistry and practically applications and just now 2023 ones are being cited. he started in 2005 i think deserves more attention) .. for the simplification of this Standard Model , but that bores me to tears, while staying focused on Gravity and reduction, it explains the spiral arms of galaxies.

Article Noether currents for the Teukolsky Master Equation