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> It is stored in the geometry of spacetime And because changing the relative position of objects changes the geometry and the conservation of energy and momen

by acqq 6y ago

> It is stored in the geometry of spacetime And because changing the relative position of objects changes the geometry and the conservation of energy and momentum exists, can't we say that we don't even have to think about the geometry as the "store", but that it's enough to know that it is a property of the whole system we observe and that when it changes the conservation is not violated?

pdonis 6y ago
> changing the relative position of objects changes the geometry Yes, although the change due to the spaceship moving is far too small to measure. (Technically, the global spacetime geometry doesn't "change", since spacetime is a 4-dimensional geometry that already contains all the information about "changes" in it, so spacetime itself doesn't change at all. But if we think of the local spacetime geometry changing along the worldline of an object like the spaceship, that concept makes sense.) > conservation of energy and momentum exists You have to be careful with this. There are actually three different senses in which this can be taken in GR, a local one and two global ones. The local sense is that the covariant divergence of the stress-energy tensor is zero. This basically says that matter and energy in tangible form (which does not include "energy stored in the gravitational field" or "energy stored in spacetime geometry") can't be created or destroyed. The first global sense is that in certain kinds of spacetimes, we can do integrals that evaluate the "total energy" of the spacetime, and these integrals will obey certain conservation laws. But those laws don't necessarily correspond to what we normally think of as "conservation of energy" in a global sense. The second global sense is that, again in certain kinds of spacetimes, there is a constant of free-fall motion that can be interpreted as the kinetic plus potential energy of the free-falling object, such as a spaceship on a free-fall trajectory between two planets. This is the sense that has implicitly been used in this discussion. But this constant of free-fall motion only exists in stationary spacetimes, and in stationary spacetimes, the relative positions of gravitating masses cannot change with time. If the relative positions of gravitating masses do change with time, the spacetime is not stationary and the constant of free-fall motion that is interpreted as kinetic plus potential energy does not exist. So there is no such thing as a spacetime which has changing relative positions of objects and conservation of energy in the sense we have been using that term in this discussion.

by acqq 6y ago

pdonis 6y ago

> changing the relative position of objects changes the geometry Yes, although the change due to the spaceship moving is far too small to measure. (Technically, the global spacetime geometry doesn't "change", since spacetime is a 4-dimensional geometry that already contains all the information about "changes" in it, so spacetime itself doesn't change at all. But if we think of the local spacetime geometry changing along the worldline of an object like the spaceship, that concept makes sense.) > conservation of energy and momentum exists You have to be careful with this. There are actually three different senses in which this can be taken in GR, a local one and two global ones. The local sense is that the covariant divergence of the stress-energy tensor is zero. This basically says that matter and energy in tangible form (which does not include "energy stored in the gravitational field" or "energy stored in spacetime geometry") can't be created or destroyed. The first global sense is that in certain kinds of spacetimes, we can do integrals that evaluate the "total energy" of the spacetime, and these integrals will obey certain conservation laws. But those laws don't necessarily correspond to what we normally think of as "conservation of energy" in a global sense. The second global sense is that, again in certain kinds of spacetimes, there is a constant of free-fall motion that can be interpreted as the kinetic plus potential energy of the free-falling object, such as a spaceship on a free-fall trajectory between two planets. This is the sense that has implicitly been used in this discussion. But this constant of free-fall motion only exists in stationary spacetimes, and in stationary spacetimes, the relative positions of gravitating masses cannot change with time. If the relative positions of gravitating masses do change with time, the spacetime is not stationary and the constant of free-fall motion that is interpreted as kinetic plus potential energy does not exist. So there is no such thing as a spacetime which has changing relative positions of objects and conservation of energy in the sense we have been using that term in this discussion.