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Could you please help me to understand two statements in physics:

1) Universe by definition is an isolated system. But universe expands, space pours into universe, space (vacuum) has zero-point energy, so does expansion means more energy in the universe? If yes, how can it be an isolated system?

2) There is time arrow, but no space arrow. If universe can only expand, isn't it that space goes only in one direction: from smaller to larger, that is the space arrow? Aren't time and space arrows similar?

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Energy is not conserved in General Relativity. See for a discussion of this, or Google for lots more info available on the web. – John Rennie May 23 '13 at 7:20

As John pointed out in his comment the concept of conservation of energy, which you are using to state that an expanding universe is not isolated, does not hold in General Relativity, but only in subspaces that are flat and governed by the Lorenz transformation. Space and time are not uniform in GR, the coordinates are curved.

The two dimensional example of the surface of an expanding balloon is a good analogue for the way all matter distances itself from all other matter. Two points drawn on the expanding balloon will recede from each other without an "arrow" for the direction of recession, so you cannot define an arrow of space, except by the statement : it is expanding. The same for the arrow of time , it just means "it is growing". It is not an arrow in the sense of a vector in any dimension. In this sense they are similar.

Two dimensional people on the surface of the balloon would not be able to determine a center of expansion of the two dimensional surface. Conceptually as analogue to the Big Bang , the beginning of expansion would have all the balloon surface at one point. In a similar manner we are all at the center of the original expansion, because when the BB happened all spacetime points were at the singularity. Our astronomical observations are from the neighborhood of our original expansion, the way that two points on a balloon see each other receding if they were neighbors in the original point before the balloon is pressurized.

Hope this helps.

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Thank you, John and Anna! I read some of the articles, and got more questions of course. Could you please help me with this one: Does the statement "gravitational field has negative energy" means that energy can become (transform into) spacetime? – Tatiana May 26 '13 at 2:00
this is a good answer for energy conservation in GR… . As for your statement, please give a link for it because out of context it has little meaning . – anna v May 26 '13 at 3:34
Thank you Anna! Here is the link: – Tatiana May 29 '13 at 6:02
Thank you Anna! Here is the link: […. Citation: "in general relativity spacetime can give energy to matter, or absorb it from matter". Does it mean that energy can become spacetime, and spacetime can become energy? – Tatiana May 29 '13 at 6:13
No, it means that the concept of energy and matter are ill defined when one is workingin a general relativity framework. In local frameworks lorenz invariance hold and E^2-p^2=m^2 is the tie of matter and energy. But these variables need a space time to be defined, and if the rug is being pulled out from under them, i.e. no longer Lorenz invariant but GR rules, the concept of energy is not well defined. All the rest is confusing, though masses are defined by spacetime distortions in the GR framework. – anna v May 29 '13 at 6:49

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