# Noether's theorem and Time invariance?

raid517
Hi, I know I'm probably going to get shot down in flames. I'm a total amateur to all of this. But I do try to read things and I do try to understand them - so I hope you guys will at least be patient with me.

But in any case I have been reading around about Noether's theorem and about the time invariant nature of general relativity. From what I can make out Noether's theorem appears to suggest (among other things) that the first law of thermodynamics agrees with Einstein's theory of General relativity in that it predicts that the first law can be considered as time invariant?

So this has led me to wonder somewhat if the first law of thermodynamics would apply equally well in reverse as it does when time is considered to be running forward?

I ask this because I wondered what would happen if you ran the entire process of the expansion of the Universe in reverse? Would the first law of the conservation of energy still hold - and if so what would it tell us? I wondered further what would happen if you ran the entire process back to the Planck epoch and perhaps even beyond? Would the first law continue to apply? (First let me make it clear that I am aware that it is not mathematically useful to try to describe events before the Plank epoch - I am merely curious as to what the first law of thermodynamics could imply if it were run back to that point?) I wondered this because I have a very hard job comprehending a Universe that could just have sprung into existence out of nothing.

So I pondered if perhaps the first law, due to it's time invariant nature (again I could be wrong on this) might actually be suggesting that even though we cannot know anything particularly useful beyond the Plank epoch, that everything that was needed to bring the Universe into being at time 0, was in some sense always already in place? I wondered if in fact the Universe did not have to emerge from nothing - because all it had to do was emerge from some previous 'pre-existing state?' (As is given by the statement that matter and energy cannot be created or destroyed, they can only converted). That is to say that although we cannot know anything useful beyond the Planck epoch, could we not say that whatever state existed prior to the this period merely had to be converted from one form into another in order to bring the Universe into being?

I suspect that I might be interpreting this rule too literally (as I usually do with such matters) and that there will be some special rule that would say that the first law of thermodynamics only applies up to a certain point. But as I said I am struggling to understand or come to terms with the idea that the Universe could ever really have had a beginning - because for me this implies a cause without an effect - which genuinely seems to me to be counter intuitive.

Also is it accurate to say that from our perspective the period of time leading prior to the Plank epoch leading up to time 0 would to us appear to last infinitely long? Thus implying perhaps that whatever existed before this period, could have existed for an infinitely long period of time? (Either this or the Universe would have needed to have always started out at a definite size - since before this period there would effectively be no measurable time). (I have some trouble with this concept too though, because it seems to suggest a universe that again spontaneously came into being out of nothing with a definite size - which in my own primitive imagination just does not seem plausible). Is the Planck time simply the shortest amount of time we are able to mathematically measure - even though some sense of something else may have existed beyond this time?

I know I am probably severely deluded, but really my ideal scenario would be a Universe that was born through natural and understandable laws and causes and which did not require that it was somehow just spontaneously brought into being - because in reality it had always existed - as had all of the rules and all of the material needed to bring it into being already in place.

So in other words, if the first law of thermodynamics can push our vision as far back as 10-43 seconds - why can't it push our vision even further back (if even purely through inference) to say that whatever state the universe existed in prior to the big bang - it still almost undoubtedly had a direct and real physical cause - or that indeed the universe was born through the conversion of one (unknowable) form of energy into another? Could the Universe in other words (through the first law of themodynamics) in some sense predict it's own birth?

Again I am sorry to trouble you guys with these dumb questions - but I do often find myself wondering about such things. I am often wrong and my musings are probably of not much significance to anyone other than myself - but I would just like to feel that I lived in a Universe that was at in some senses both definable and understandable - or that I could make at least some sense of.

Your input would therefore be greatly appreciated.

GJ

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## Answers and Replies

Staff Emeritus
Gold Member
Dearly Missed
I believe the first law of thermodynamics, conservation of energy, is reversable. At least we have Hamiltonian dynamics of conservative systems, and that is reversable.

Others have asserted that if the expansion ran backwards, thermodynamics would reverse, including the unsymmetrical second law (entropy increase). Some thinkers have identified the "arrow of time" with the expansion of the universe, so reversing that would reverse the other. Back in the days when cosmologists thought there would be a big crunch, there were statements that the contraction toward that would reverse the thermodynamic arrow. All of this is and has always been controversial because the true origin of the arrow of time is not agreed upon.

raid517
I believe the first law of thermodynamics, conservation of energy, is reversible. At least we have Hamiltonian dynamics of conservative systems, and that is reversible.

Others have asserted that if the expansion ran back wards, thermodynamics would reverse, including the unsymmetrical second law (entropy increase). Some thinkers have identified the "arrow of time" with the expansion of the universe, so reversing that would reverse the other. Back in the days when cosmologists thought there would be a big crunch, there were statements that the contraction toward that would reverse the thermodynamic arrow. All of this is and has always been controversial because the true origin of the arrow of time is not agreed upon.

Well I am only really concerned with the first law in this specific instance. It is rather like taking the first law and 'pretending' that the other laws do not exist. Clearly the second law would tell us that actually running time in reverse would be impossible - and at least in this sense General relativity wouldn't be able to tell us anything very useful. General relativity in this case (so far) is not really 'aware' of the second law of thermodynamics.

Really all I am asking is not if time really CAN be run in reverse - because it is no more than a thought experiment at this point. Rather I am employing the same method that Edwin Hubble used when he looked at the night sky and realized that all of the objects in the Universe were moving away from each other - and that this meant at some point in the past they must have been progressively closer to each other - so that if you ran the process in reverse you would end up back at a period where all of the the matter and all of the energy was tightly compacted together at a single point.

All I am curious about is if you ran the same process of expansion in reverse and applied the rules of General Relativity - and the time invariant nature of the first law of thermodynamics (which BTW can be shown to work equally well in a forward or backwards time direction) then as well as saying that matter and energy on a purely localized (and fairly small scale) cannot be created or destroyed, could you not use there rules to say that the Universe itself could not be created or destroyed? Or in other words whatever (unknowable) state the Universe existed in prior to the big bang it must always have contained all of the 'ingredients' needed in order to bring the Universe into existence - and that whatever these ingredients were the first law of thermodynamics would state that in order for us (and the Universe to exist) these ingredients could only have been converted from one form into another? Put yet another way, could the first law of thermodynamics (in conjunction with General relativity) be used to imply that the Universe, whatever it's beginnings may have been, must have had a definite (and to a degree definable) physical cause?

This I guess can be summed up in asking the question, at what point (if you run the entire expansion of the Universe in reverse) does the first law of thermodynamics (in isolation from the other laws) break down - if any - and can this be conclusively proved?

GJ

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raid517
Here is a quote I found on this subject that appears to suggest that first (and other laws of thermodynamics) can hold even right up to the very extreme initial conditions only a short period after the big bang.

In the macroscopic world, the domain of ‘classical’ physics, the laws of thermodynamics are, and have always been, true. On the quantum scale, it is a very different matter. Heisenberg’s uncertainty states that there will always be a level of uncertainty when you try to make measurements of particles and other quantum scale occurrences. You can never know everything about a particle’s position and motion at anyone time. This is an intrinsic uncertainty, it is not due to limitations on our measuring devices. This uncertainty of the energy of anything of the Planck scale is size allows some very bizarre phenomena to occur.

To us, vacuums appear to contain nothing at all. But, it you were to look closely, very, very closely (to the order of 10^-35m), space is actually a foaming mass of quantum activity. This quantum foam is made of particles and micro-black holes popping in and out of existence, apparently in contravention of the second law of thermodynamics, they appear out of nothing with energy, then disappear again just as quickly. The key to this is the uncertainty principle. The disturbance is permitted to ‘borrow’ a tiny amount of energy and exist for a very short length of time, and then it must return the energy and disappear again. But, the more energy it borrows, the less time it is allowed to exist. These ‘temporary’ particles, called virtual particles, are not just theoretical, they have been proven to have real effects on scientific experiment.

The only thing that prevents these virtual particles from coming into permanent existence is a lack of energy. However, it is possible to artificially supply energy to the particles therefore promoting them into reality. This could be done in a lab by creating very strong electric fields, but these fields are very difficult to create. On the other hand, intense gravitational fields could also do the job.

It is possible that during the big bang, black holes the size of a nucleus popped into existence due to the quantum foam. The interesting thing is that the smaller a black hole is, the more strongly space-time is distorted around it and distortions in space-time imply the existence of very strong gravitational fields. Stephen Hawking has shown that the gravitational field around such a hole would give enough energy to the quantum foam to promote the particles into real existence. Calculations show that in the big bang the initial extreme conditions would also have been enough to create real particles out of the gravitational energy of the rapidly expanding universe.

And as for how the universe actually came into being itself, it is believed that also in the quantum foam, virtual space-time bubbles also continually pop in and out of existence, like virtual particles, only to disappear again. However, it is possible that one of these space-time bubbles, which is actually an unimaginably small universe, could avoid rapidly disappearing again and be promoted to a full size universe, such as ours. However, for this to work some sort of repulsive force is needed, a sort of anti-gravity. Many scientists believe in the existence of such a force at the time of the creation of the universe, but as I’ve answered your question and that’s a whole other topic, I think I’ll stop before I go off on too much of a tangent.

To summarise, due to the uncertainty principle, particles and space-time bubbles continually pop in and out of existence for short times depending on their energy, without breaking the law of conservation of energy as they disappear again. Think of it like an accountant (the universe) who balances the books at the end of every month. If someone (a virtual particle) was to borrow some money on the 4th day of the month (pop into existence)then put it back on the 8th day,(disappear again) then as far as the bookkeeper would know, nothing had gone amiss and no rules (or laws) had been broken. If a particle is to come into complete and real existence, it must take its energy from somewhere, such as a gravitational field.

I can grasp this to an extent and do appreciate what it is saying. But this only says that the first (and second) laws of thermodynamics are not broken by this process. So I was merely curious of at 'what point', if any did the first law of thermodynamics break down? I guess probably in a sense I can answer my own question in that really the answer to this question might be unknowable. It might be unknowable simply because we cannot really see beyond the Plank epoch. So while we might assume or imply whatever we may wish - there is no direct way to arrive at a definite answer?

So really it might just be a philosophical question - as opposed to a scientific one - because it all comes down to a certain degree to what we might prefer to believe. It seems to simply be an arbitrary choice universe that had a definite cause - or one that had no cause whatsoever? (Although I don't think even Stephen Hawkins quite implies this). All I wondered was if at least you could us logic and the reverse argument to an argument through ignorance, to imply that given the rules of nature as we currently understand them, it is much more likely that they are indicating to us that the Universe in fact did have a direct physical cause?

On another possibly related note, it seems to me that the first law of thermodynamics and Einstein's special theory of relativity seem to be saying almost exactly the same thing. (Although special relativity can say a lot more about a variety of different conditions). The first law of thermodynamics states that energy cannot be created or destroyed and Einstein's theory of special relativity says that energy and matter are essentially different aspects of the same thing. I find it very hard to tell the difference between these two statements. So I was curious if Einsteins equations were in any sense derived directly from a study of the first (or any of the subsequent laws) and if so to what extent. The statement that energy cannot be created or destroyed simply seems to me like quite a profound and significant thing to say and it almost seems purely like a matter of semantics to go from saying energy and matter cannot be destroyed and energy and matter are different aspects of the same thing.

Again, I would just like to clear this stuff up in my own mind.

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Staff Emeritus
The existence of Poincarre recurrence

http://www.math.umd.edu/~lvrmr/History/Recurrence.html

convinces me that extreme caution is needed when thinking about the second law of thermodynamics.

But you seem to be mainly interested in the conservation of energy, so let's address that topic.

One of the easiest places to start, IMO, is the sci.physics.faq

Is energy conserved in General Relativity

In special cases, yes. In general -- it depends on what you mean by "energy", and what you mean by "conserved".

Specifically, if you ask do FRW cosmologies (Friedmann-Robertrson-Walker cosmologies, the standard isotropic cosmologies usually used to model the big bang) conserve energy, the answer is no.

They (FRW metrics) do not have the associated timelike symmetry needed to apply Noether's theorem. Nor can one apply other standard notions of energy based on asymptotic flatness of space-time.

As a consequence, there is not a constant number that represents the "total energy" or "total mass" of the universe.