In 1918, the great mathematician Amalie Emmy Noether proved two theorems which have since proved to be amongst our greatest insights into nature. In this part I'll talk about the first of these theorems, which shows that there is a deep link between symmetries and conservation laws. In particular, I'll talk about the conservation of energy and the conservation of momentum, because these two quantities play an important part in the Einstein field equations.

A symmetry is a change that you can perform on a system that leaves it the same. For example, a square has lots of symmetries. Firstly, if you take the square and rotate it through 90 degrees, 180 degrees or 270 degrees then it's just the same as how it started (we aren't considering the corners as being distinguishable from each other). In addition to these rotational symmetries, a square has a number of reflection symmetries: it's unchanged by reflection through any line that joins two opposite corners, or through through any line that joins the midpoints of opposite edges. One final, more subtle symmetry is the identity symmetry: just do nothing at all to the square. Any pair of symmetries can be combined by "composition" to give another symmetry (it may, though, be the same as one or both of the symmetries being combined). The way composition works is that you do the first operation and then the second (you have to keep track of the corners of the square to work out what the composite symmetry is, despite the fact that the corners are really all identical to each other). For example, doing a rotation through 90 degrees and then another one through 90 degrees is equivalent to doing a single rotation through 180 degrees.

The symmetries of the square form a mathematical structure called the "symmetry group" of the square. The four requirements for being a group are that two elements (here symmetries) combine to give an element of the group; that there is an identity element that leaves all the other elements unchanged regardless of whether you do the identity first or second; that each element has an "inverse" element such that if you combine it with its inverse you get the identity (for example, the reflections are their own inverses, the rotation through 180 degrees is its own inverse too, and the rotations through 90 degrees and 270 degrees are inverses of each other); and finally that if you're combining three elements then it doesn't matter if you combine the first and second first and then combine the result of that with the third, or if you combine the second and third first and then combine that with the first (although in general the order matters, so that you can't combine the first and third first and then combine that with the second like you can with addition). (I'd render all that in maths for clarity, but nobody would speak to me again...)

The symmetries that we'll be concerned with here are symmetries of the laws of physics governing the system under consideration (or, more precisely, "continuous variational symmetries"; I'll explain the "continuous" bit later, and the "variational" part just means that the particular physical theory has to be based on a variational principle as discussed way back in Part 5). In other words, if we have a history of the system then a symmetry is a way to push or pull or stretch the parts of the system (and change the values of fields and stuff like that) which gives another possible history of the system which also obeys the laws of physics. For example, suppose our system consists of consists of a planet in orbit around a star (and that the system is a long way away from anything else so that it's effectively isolated). One obvious symmetry is translation in space: the history that's just the same as the original history except that everything is shifted a million kilometres in some direction will also be a physically possible history (it won't have the same "boundary conditions", but if you saw a film of the planet-star system in both locations then you wouldn't be able to find any way in which one case obeys the laws of physics and the other one doesn't). A second example is rotation: if you move to a new history in which the star system is tipped over by some angle then that history works just fine too. The final important example is movement in time: suppose the star system was artifically created in the year 3000 and then left to its own devices; then a history in which it was created in the year 3010 in the same state as the state it was created in during the first case would be a time translation of the original history. Notice that all of these symmetries are "continuous" in the sense that all the histories given by intermediate translations or rotations are also possible.

Another important type of symmetries are "gauge symmetries" which consist of simultaneously changing the values of "matter fields" and "gauge fields" in certain ways. For example, the laws of electromagnetism have a gauge symmetry which consists of changing the value of a particle's phase by different amounts at each event and changing the values of the electromagnetic field in ways that compensate for these phase changes. In fact it's possible to derive the properties of electromagnetism entirely from the requirement that local changes in phase be a symmetry. All the other fundamental laws of force are derived from similar (but more complicated) gauge symmetries.

In all these cases you have to move or modify the whole system, which includes everything that influences its behaviour. You might then claim that this is a silly idea, because by moving enough stuff you could always ensure that the two systems behave in the same way. However, this need not be the case - for example, different directions in the universe could in principle have different properties (as they do in crystals) and then continuous rotations wouldn't be a symmetry no matter how much other stuff you tip over too. Similarly time translation wouldn't be a symmetry if the properties of space are changing with time (as they are in the real universe, so all of this stuff only applies on a scale small enough that these changes can be ignored). Furthermore, not all changes made to a system are symmetries, or else the laws of physics would be pretty useless things. Given any law of physics there's a systematic way to find all of its continuous symmetries.

Now I'm going to have to talk for a bit about conservation laws. The idea here is that there is some quantity that you can work out from the state of the system at a given time, and that if you work out the quantity at different times you always get the same value; we say that the quantity is conserved. For example, the law of the conservation of momentum says that if you have a system of particles and you add up the mass of each times its velocity (*not* its speed), then this quantity, the total momentum, is conserved. Other examples of conserved quantities include energy, angular momentum and electric charge. However, there's more to conservation laws than this. In all cases of physical interest, the conservation is local, which means that the change in the amount of the quantity in some region between two times is given by the amount flowing into the region minus the amount flowing out. For example, the conservation of energy is a local conservation law: you can't create energy in one place and balance the books by destroying it somewhere else. Indeed, from what I've said about the relativity of simultaneity you can see that conservation laws in relativistic theories must be local, because if the creation and destruction that add up to overall conservation happen at the same time but different places in one frame, then in other frames they will happen at different times and so there will be an interval during which energy isn't conserved.

Finally, I can tell you about the special case of Noether's theorem used by physicists. This says that for every continuous variational symmetry of the laws of physics governing a system there is a corresponding local conservation law. This seems to me (and lots of other people!) to be a most remarkable result. As we can systematically find all the variational symmetries of a system, we can find lots of useful conservation laws (we can't find them all, because some correspond to "generalised symmetries" and are covered by the general case of Noether's theorem, which I have no intention of trying to explain here). Each of the conservation laws that have been known for hundreds of years is turns out to be linked to a symmetry. If translation in time is a symmetry then energy is conserved; we say that energy is the "generator" of time translation. Similarly, if translation in space is a symmetry then momentum is conserved. Combining these into invariance under translation in spacetime gives the conservation of energy-momentum (sometimes called 4-momentum). (This correspondance means that the fact that people in different frames slice spacetime into space and time in different ways implies that they will also divide 4-momentum into energy and momentum in different ways.) Rotational symmetry gives conservation of angular momentum. The symmetry of electromagnetism under gauge changes gives the conservation of charge.