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Pentashagon comments on Open Thread, January 16-31, 2013 - Less Wrong
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I have a question about timeless physics. If the future state of the universe is only based on the current state with no reference to time then what determines how much the universe changes from state to state? Removing time seems to reintroduce Zeno's paradox. Either the universe changes in discrete steps or something else has to keep track of how much the universe changes at each step and the only way I can think of to measure how much it changes is a derivative with respect to "time".
Any better insights?
Do you think continuous spatial + temporal dimensions have problems continuous spatial dimensions lack? If so, what and why?
It may be that I don't have a good understanding of quantum mechanics. In Newtonian mechanics the state of the universe is dependent on the prior position and velocity of and forces on all the particles. The velocity and forces are both expressed in terms of the derivative of time so if time was removed from the equations Zeno's paradox would imply that either nothing could ever move or that motion was discontinuous whenever the next state of the universe was calculated.
From browsing wikiepdia it looks like that there are time-dependent as well as time-independent Schrödinger equations used for moving and stationary states, respectively. Is it actually possible to express the entire universe as a single time-independent equation? If so, does that mean that what we actually experience at any "time" is just a random sample from the steady-state probability distribution? Does that mean we should always expect the universe to tend toward some specific distribution (maybe just the heat death)?
Yes.
If your original question, "what determines how much the universe changes from state to state?", is meant to refer to spacelike "states", then the answer (which requires only general relativity) is the geometry of spacetime. But the "states" in the Wheeler--DeWitt equation are spacetimes, so in that context "the universe" differs "from state to state", but it doesn't "change."