I'm reading Peter Atkins' book, Galileo's Finger, and in the chapter on energy, he makes the points that the conservation of momentum stems from the shape of space (that it's smooth and not lumpy) and that the conservation of energy stems from the shape of time (that it's smooth and not lumpy). I'm not totally clear on how the shape of spacetime leads to the conservation laws. Could someone elucidate the relationship, in layman's terms?
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John Reny's answer, is to the point. Shift symmetry of the space means that if a relation for a system is valid in $x$, is also valid if the system is shifted to $x+dx$, so having this symmetry and calculating the total momenutm of a system, it leads to $dp_{tot}/dt = 0$, which is the consnervation of momentum. Noethers theorem states that whenever a system has some symmetry another (associated) quantity is conserved. Whether this is a space property per se (and not a system property), is sth that can be debated. – Nikos M. Jun 13 '14 at 17:29
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You might want to have a look at the answers to the question Can Noether's theorem be understood intuitively?. This is almost a duplicate of yours, though I suspect it might be pitched at slightly too complex a level for a non-physicist.
The conservation laws are related to a symmetry through a theorem proved by the mathematician Emmy Noether in 1915, and widely referred to as Noether's theorem.
For example conservation of momentum is related to position shift symmetry. This symmetry means that if you do some experiment, then move your kit some random distance in space and do the experiment again you'll get the same results. There are obvious objections to this: for example an experiment done on Earth's surface may well give a different result to one done in the weightless conditions of the International Space Station. But the experiment includes all interacting systems, so that includes the Earth (and the Sun, etc). If you moved the whole Solar System some random distance in space and redid your experiment you'd get the same results.
In a similar way conservation of energy is related to time shift symmetry. This symmetry means that if you do some experiment, then wait for some some random time and do the experiment again you'll get the same results. The experiment will give the same results tomorrow as it does today.
It's a long time since I read Galileo's finger and I don't remember exactly what point Atkins is making. However it should be obvious that space shift symmetry only works if spacetime is (on average) the same everywhere, otherwise moving in spacetime would affect your experiment. I assume that this is what Atkins means by not too lumpy.
John Rennie
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