I tried to ask one of your questions using DeepSeek with deep thinking, it answered:

> Okay, the user is asking if the ADM formalism is strictly equivalent to General Relativity. Let me start by recalling what the ADM formalism is. ADM stands for Arnowitt, Deser, and Misner, who developed a way to reformulate Einstein's equations into a Hamiltonian framework. This is useful for things like numerical relativity and quantum gravity.

So, first, I need to confirm whether ADM is just a reformulation of GR or if it changes the theory. From what I remember, ADM splits spacetime into space and time, using a 3+1 decomposition. They introduce variables like the lapse function and shift vector to handle the foliation of spacetime into spatial hypersurfaces.

Wait, does this decomposition change the physics? Or is it just a different mathematical approach? I think it's the latter. The Einstein equations in ADM form should be equivalent to the original ones, just expressed in terms of different variables. But I should check if there are any subtleties, like constraints or boundary conditions that might affect equivalence.

Also, the user might be confused because sometimes when you change the formalism, especially in constrained systems, there can be issues. For example, in ADM, you have the Hamiltonian and momentum constraints. These need to be satisfied at all times, and numerical implementations have to ensure that. But theoretically, as long as the constraints are satisfied, the ADM equations should be equivalent.

Another point is the initial value problem. ADM is used for that, which suggests it's a way to set up initial data consistent with GR. If the equivalence wasn't strict, then initial data in ADM wouldn't correspond properly to GR solutions. But I think they do, as long as the constraints are met.

However, I should mention that while the formalism is equivalent, practical implementations (like in numerical relativity) might face issues like constraint violation over time, which aren't a problem with the theory itself but with the method of solving the equations.

Also, maybe the user is coming from a background where they heard about different formulations of GR, like tetrad formalism or others, and is wondering if ADM is another such equivalent approach. So, confirming that ADM is a Hamiltonian formulation that doesn't alter the physical content of GR, just the mathematical structure.

Need to make sure to explain that equivalence here means the solutions to the equations are the same, and any differences are in the mathematical handling, not the physical predictions. Also, perhaps touch upon the applications where ADM is useful, to highlight why this reformulation is valuable even though it's equivalent.

I should also check if there's any gauge freedom or choice in the ADM formalism that might lead to different interpretations. For example, the lapse and shift are gauge variables, but that's part of the coordinate freedom already present in GR. So, choosing different lapses and shifts corresponds to different coordinate choices in standard GR, which doesn't affect the physical equivalence.

In summary, the ADM formalism is a reformulation that breaks down Einstein's equations into a set of constrained evolutionary equations. The physical content remains the same as GR, so they are strictly equivalent in terms of physical predictions, but the mathematical approach is different, suited for certain types of problems like numerical simulations or canonical quantization.

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It's far far beyond my capability to validate if it's right, but I think it would be helpful if the LLM can explain how it thinks?

I agree LLM often being confidently incorrect, it often gives me code that completely un-compilable, users need to review its content very carefully, and we need to do that for content from other sources also

What I want to explain is that LLMs are helpful at many scenarios(not as hardcore as astrophysics, for example, there's an article about myths of seven sister https://news.ycombinator.com/item?id=42809925

And I want to know if there're other "sisters-like" myths, I can not get the right keyword to search, and then LLM helps, and I can go to do source check for the given answer by its keywords