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Is there any way of obtaining the exact same theory of Special/General Relativity but using a completely different framework? That is, without manifolds, without vectors, etc...
"Any way" covers a lot of ground. I knew someone who wrote a complete recursive descent parser generator (BNF in, ALGOL code out) as a TECO macro. (I chose a more traditional programming language for this exercise).kent davidge said:Is there any way of obtaining the exact same theory of Special/General Relativity but using a completely different framework? That is, without manifolds, without vectors, etc...
Why?kent davidge said:Is there any way of obtaining the exact same theory of Special/General Relativity but using a completely different framework? That is, without manifolds, without vectors, etc...
With math it is always possible to do something in a more complicated way.kent davidge said:Is there any way of obtaining the exact same theory of Special/General Relativity but using a completely different framework? That is, without manifolds, without vectors, etc...
Because I'm almost giving up! Trying hard since end of last year and can't understand some fundamental stuff regarding manifoldsmartinbn said:Why?
What source have you been using?kent davidge said:Because I'm almost giving up! Trying hard since end of last year and can't understand some fundamental stuff regarding manifolds
Special relativity was done before Minkowski got ahold of it and introduced Minkowski space, so I don’t see why you couldn’t get pretty much all of SR without all that.kent davidge said:Is there any way of obtaining the exact same theory of Special/General Relativity but using a completely different framework? That is, without manifolds, without vectors, etc...
Yes, but without vectors?Sorcerer said:Special relativity was done before Minkowski got ahold of it and introduced Minkowski space, so I don’t see why you couldn’t get pretty much all of SR without all that.
Is it any easier that way than using Minkowski space? The required math background is the same either way - you don't even need any calculus if the examples are properly chosen so we're comfortably in the realm of high-school math. For example: spaceship flies away from Earth at .5c for a year, turns around and returns to Earth two years after it left. How much time passed on the ship? Try solving this problem using the methods Einstein used in his classic paper (top of page 11). Now compare with the Minkowski approach: spacetime interval out is ##\sqrt{3}/2##, spacetime interval back is the same, total time is ##\sqrt{3}## years.A big chunk of it is here, by Einstein, and it’s all undergraduate math, and not even the hardest undergraduate math.
http://hermes.ffn.ub.es/luisnavarro/nuevo_maletin/Einstein_1905_relativity.pdf
Oh, I’m not saying Minkowski didn’t make things cleaner or easier. Just saying you can get the theory just using algebra and calculus (and to be honest I find calculus easier than linear algebra, tensors and all the rest). As for vectors, I don’t see how you can escape vectors in physics unless you work only in terms of energy, and correctly if I’m wrong, in SR you’re STILL working with vectors (4-vectors) when talking about energy.Nugatory said:Yes, but without vectors?
Is it any easier that way than using Minkowski space? The required math background is the same either way - you don't even need any calculus if the examples are properly chosen so we're comfortably in the realm of high-school math. For example: spaceship flies away from Earth at .5c for a year, turns around and returns to Earth two years after it left. How much time passed on the ship? Try solving this problem using the methods Einstein used in his classic paper (top of page 11). Now compare with the Minkowski approach: spacetime interval out is ##\sqrt{3}/2##, spacetime interval back is the same, total time is ##\sqrt{3}## years.
The question is where do you want your complexity? You can hide it behind a standardised notation with standardised tools for manipulating it, or you can put it out front and handle it yourself. Sure, there's a learning curve for the tools (still climbing it myself), but it more than repays itself by saving you from errors trying to do the book keeping manually.kent davidge said:Because I'm almost giving up! Trying hard since end of last year and can't understand some fundamental stuff regarding manifolds