Recent content by vis_insita

It depends on how the standard library implementers organize their header files. gcc apparently declares std::negate in a header used by std::string, which is included via iostream. You can get this information from the compiler output on onlinegdb: The latest gcc version also pulls...

BTW, whatever compiler version onlinegdb uses, it also compiles if you don't include iostream. I think the lessons here are 1) avoid "using namespace std;" and 2) don't include what you don't use.

There is also a std::negate, which is probably found via argument dependent lookup? Try replacing the "using namespace std" by "using std::vector".

Yes, my understanding is that some of these proofs are relevant to Hawking radiation/Black hole thermodynamics. That is why I assumed Wald's definition would be the most relevant one in this discussion too. It also follows from that physically meaningful definition that the future and past of...

Wald, General Relativity, defines ##J^+(S)=\bigcup_{p\in S}J^+(p)##, which looks exactly like the definition @Elias1960 is using. I have never seen a different definition of ##J^+(S)## (MTW, and Hawking and Ellis seem to use the same) and as far as I understand the discussion (which is...

Yes, I know, which is why I called both solutions "different". It doesn't matter why it differs, only that it differs.

I tried to answer that question above. I think you can prove consistency by constructing simple toy models of your theory. If such models exist then the theory must be consistent. Because no model can satisfy both statements X and not-X. The easiest way to accomplish this may be to just find...

I think the solution to the approximate equation is different from the solution to the exact equation, if this is what you are asking. No, but it is also not my definition. I think you misunderstood something. I just claimed that applying both Newtonian Physics and GR to the same particular...

As I tried to make clear above, the relevant relation is that of logical implication. In that sense neither theory is a subset of the other. To derive Newtonian Mechanics from SR, e.g, you need an additional assumption, like ##v \ll c##, which is not itself implied by SR (nor Newtonian...

I did not say or imply that GR contradicts itself. The Newtonian limit of GR is not GR itself.

Whether there is a contradiction or not is a purely logical question. If GR implies "The perihelion precession of Mercury is 574″ ± 5'' per century", while Newtonian gravity implies that this is not the case, then there is as clear a contradition between both theories as can be. Of course...

@MathematicalPhysicist gave one above: In this sense inconsistent theories are pretty much worthless, since everything follows from them.

I understand that. Again, I claim that "A approximates B" doesn't show that there is no contradiction between A and B. On the contrary. It only means that A and B agree on some facts, namely those which are stated vaguely enough that the difference A-B doesn't matter. On the other hand they...

No, I'm not. I claim that the Newtonian prediction is accurate but wrong, not true but inaccurate.

True, of course. But the fact that GR and Newtonian physics agree on some facts in some situations doesn't show that they don't contradict each other. They are in contradiction if they disagree on some facts, which they do.