Alright. To preface this, let me say that I'm sorry if this is a stupid issue that's been addressed elsewhere. I'm still working my way through the sequences.
But... the jump from discrete causality to continuous causality seems to be hiding a big issue for the argument against time travel. It's not an insoluble issue, but the only solution that I see does pose problems for the locality of this definition of "causal universe".
To start from the beginning: the argument in the discrete case relies heavily on the computability of the universe. In a causal universe, we can compute time t=8 based on complete knowledge about time t=7, and then we can compute time t=9 on our newly found knowledge about time t=8.
But as far as I can see, there's not similar computability once we move into a continuous universe. In particular, if we have complete knowledge about a subspace K of spacetime, even with infinite computing power we can only find out the state of the universe along what I'll call the "future boundary" of the space K: in particular, we can only compute the state of points P in space-time whose past light cone of height d lies entirely inside K, for some d. That means that we can never compute time t=8 based on knowledge that we can have at time t=7, in fact, we can't compute time t=8 unless we have complete knowledge about time t<8.
So computability doesn't seem to pose a problem for Time Turners, because even without Time Turners the (non-local) future is not computable. To put it another way, continuous causality has been defined in an entirely local way, which means that non-local cycles don't seem to be a problem. In fact, positing a Time-Turner jump from times t<9 to time t=8 merely requires redefining the topology of time in such a way that point P at times t<9 is in the past light cone of the same point P at time t=8.
The obvious reply is that (by definition) point P at time t=8 must be in the past light cone of the same point P at all times 8<t<9. So we have the past light cone of point P intersecting the future light cone of point P. So should our definition of a "causal universe" exclude that? That seems perfectly reasonable, but doing so seems to destroy the locality of our definition of a causal universe. Because the intersection of the light cones that we're objecting to is not a local phenomenon: for a point P at time t<9 arbitrarily close to 9 (i.e. in the local past time cone of P at t=8), point P at time t=8 is only its non-local past time cone.
Does the issue I'm trying to get at make any sense? I can rephrase if that would help, and I'd be happy to read anything that addresses this.
Alright. To preface this, let me say that I'm sorry if this is a stupid issue that's been addressed elsewhere. I'm still working my way through the sequences.
But... the jump from discrete causality to continuous causality seems to be hiding a big issue for the argument against time travel. It's not an insoluble issue, but the only solution that I see does pose problems for the locality of this definition of "causal universe".
To start from the beginning: the argument in the discrete case relies heavily on the computability of the universe. In a causal universe, we can compute time t=8 based on complete knowledge about time t=7, and then we can compute time t=9 on our newly found knowledge about time t=8.
But as far as I can see, there's not similar computability once we move into a continuous universe. In particular, if we have complete knowledge about a subspace K of spacetime, even with infinite computing power we can only find out the state of the universe along what I'll call the "future boundary" of the space K: in particular, we can only compute the state of points P in space-time whose past light cone of height d lies entirely inside K, for some d. That means that we can never compute time t=8 based on knowledge that we can have at time t=7, in fact, we can't compute time t=8 unless we have complete knowledge about time t<8.
So computability doesn't seem to pose a problem for Time Turners, because even without Time Turners the (non-local) future is not computable. To put it another way, continuous causality has been defined in an entirely local way, which means that non-local cycles don't seem to be a problem. In fact, positing a Time-Turner jump from times t<9 to time t=8 merely requires redefining the topology of time in such a way that point P at times t<9 is in the past light cone of the same point P at time t=8.
The obvious reply is that (by definition) point P at time t=8 must be in the past light cone of the same point P at all times 8<t<9. So we have the past light cone of point P intersecting the future light cone of point P. So should our definition of a "causal universe" exclude that? That seems perfectly reasonable, but doing so seems to destroy the locality of our definition of a causal universe. Because the intersection of the light cones that we're objecting to is not a local phenomenon: for a point P at time t<9 arbitrarily close to 9 (i.e. in the local past time cone of P at t=8), point P at time t=8 is only its non-local past time cone.
Does the issue I'm trying to get at make any sense? I can rephrase if that would help, and I'd be happy to read anything that addresses this.