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Why save energy if physics says energy is conserved anyway? Did Einstein really say that energy is not conserved? And what does energy have to do with time? This is what we will talk about today.

I looked up “energy” in the Encyclopedia Britannica and it told me that energy is “the capacity for doing work”. Which brings up the question, what is work? The Encyclopedia says work is “a measure of energy transfer.” That seems a little circular. And as if that wasn’t enough, the Encyclopedia goes on to say, well, actually not all types of energy do work, and also energy is always associated with motion, which actually it is not because E equals m c squared. I hope you are sufficiently confused now to hear how to make sense of this.A good illustration for energy conservation is a roller-coaster. At the starting point, it has only potential energy, that comes from gravity. As it rolls down, the gravitational potential energy is converted into kinetic energy, meaning that the roller-coaster speeds up. At the lowest point it moves the fastest. And as it climbs up again, it slows down because the kinetic energy is converted back into potential energy. If you neglect friction, energy conservation means the roller-coaster should have just exactly the right total energy to climb back up to the top where it started. In reality of course, friction cannot be neglected. This means the roller-coaster loses some energy into heating the rails or creating wind. But this energy is not destroyed. It is just no longer useful to move the roller coaster.

This simple example tells us two things right away. First, there are different types of energy, and they can be converted into each other. What is conserved is only the total of these energies. Second, some types of energy are more, others less useful to move things around.

But what really is this energy we are talking about? There was indeed a lot of confusion about this among physicists in the 19th century, but it was cleared up beautifully by Emmy Noether in 1915. Noether proved that if you have a system whose equations do no change in time then this system has a conserved quantity. Physicists would say, such a system has time-translation invariance. Energy is then by definition the quantity that is conserved in a system with time-translation invariance.

What does this mean? Time-translation invariance does not mean the system itself does not change in time. Even if the equations do not change in time, the solutions to these equations, which are what describe the system, usually will depend on time. Time-translation invariance just means that the change of the system depends only on the amount of time that passed since you started an experiment, but you could have started it at any moment and gotten the same result. Whether you fall off a roof at noon or a midnight, it will take the same time for you to hit the ground. That’s what “time-translation invariance” means.

So, energy is conserved by definition, and Noether’s theorem gives you a concrete mathematical procedure to derive what energy is. Okay, I admit it is a little more complicated, because if you have some quantity that is conserved, then any function of that quantity is also conserved. The missing ingredient is that energy times time has to have the dimension of Pla()nck’s constant. Basically, it has to have the right units.

I know this sounds rather abstract and mathematical, but the relevant point is just that physicists have a way to define what energy is, and it’s by definition conserved, which means it does not change in time. If you look at a simple system, for example that roller coaster, then the conserved energy is as usual the kinetic energy plus the potential energy. And if you add air molecules and the rails to the system, then their temperature would also add to the total, and so on.

But. If you look at a system with many small constituents, like air, then you will find that not all configurations of such a system are equally good at causing a macroscopic change, even if they have the same energy. A typical example would be setting fire to coal. The chemical bonds of the coal-molecules store a lot of energy. If you set fire to it, this causes a chain reaction between the coal and the oxygen in the air. In this reaction, energy from the chemical bonds is converted into kinetic energy of air molecules. This just means the air is warm, and since it’s warm, it will rise. You can use this rising air to drive a turb(ain), which you can then use to, say, move a vehicle or feed it into the grid to create electricity.

But suppose you don’t do anything with this energy, you just sit there and burn coal. This does not change anything about the total energy in the system, because that is conserved. The chemical energy of the coal is converted into kinetic energy of air molecules which distributes into the atmosphere. Same total energy. But now the energy is useless. You can no longer drive any turbine with it. What’s the difference?

The difference between the two cases is entropy. In the first case, you have the energy packed into the coal and entropy is small. In the latter case, you have the energy distributed in the motion of air molecules, and in this case the entropy is large.

A system that has energy in a state of low entropy is one whose energy you can use to create macroscopic changes, for example driving that turbine. Physicists call this useful energy “free energy” and say it “does work”. If the energy in a system is instead at high entropy, the energy is useless. Physicists then call it “heat” and heat cannot “do work”. The important point is that while energy is conserved, free energy is not conserved.

So, if someone says you should “save energy” by switching off the light, they really mean you should “save free energy”, because if you let the light on when you do not need it you convert useful free energy, from whatever is your source of electricity, into useless heat, that just warms the air in your room.

Okay, so we have seen that the total energy is by definition conserved, but that free energy is not conserved. Now what about the claim that Einstein actually told us energy is not conserved. That is correct. I know this sounds like a contradiction, but it’s not. Here is why.

Remember that energy is defined by Noether’s theorem, which says that energy is that quantity which is conserved if the system has a time-translation invariance, meaning, it does not really matter just at which moment you start an experiment.

But now remember, that Einstein’s theory of general relativity tells us that the universe expands. And if the universe expands, it does matter when you start an experiment. And expanding universe is not time-translation invariant. So, Noether’s theorem does not apply. Now, strictly speaking this does not mean that energy is not conserved in the expanding universe, it means that energy cannot be defined. However, you can take the thing you called energy when you thought the universe did not expand and ask what happens to it now that you know the universe does expand. And the answer is, well, it’s just not conserved.

A good example for this is cosmological redshift. If you have light of a particular wavelength early in the universe, then the wave-length of this light will increase when the universe expands, because it stretches. But the wave-length of light is inversely proportional to the energy of the light. So if the wave-length of light increases with the expansion of the universe, then the energy decreases. Where does the energy go? It goes nowhere, it just is not conserved. No, it really isn’t.

However, this non-conservation of energy in Einstein’s theory of general relativity is a really tiny effect that for all practical purposes plays absolutely no role here on Earth. It is really something that becomes noticeable only if you look at the universe as a whole. So, it is technically correct that energy is not conserved in Einstein’s theory of General Relativity. But this does not affect our earthly affairs.

In summary: The total energy of a system is conserved as long as you can neglect the expansion of the universe. However, the amount of useful energy, which is what physicists call “free energy,” is in general not conserved because of entropy increase.

Thanks for watching, see you next week. And remember to switch off the light.