Would Einstein been able to "solve" his general relativity field equations without the earlier mathematical work or Riemann's metric tensor?
Or would he have "discovered" the metric tensor himself, like Newton "discovered" calculus in order to solve his equations?
I'll ask the same questions in math and physics sections to see what different responses I get, if any.
Or would he have "discovered" the metric tensor himself, like Newton "discovered" calculus in order to solve his equations?
I'll ask the same questions in math and physics sections to see what different responses I get, if any.
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Let me answer that in this way.
People did QCD for years...even Feynman, before he decided to come up with 'Feynman Diagrams'. They didn't necessarily facilitate discovery per se , but allowed it to be done easier and a little faster.
Most notation in math is for making something easier. Could be easier for understanding perhaps, but mostly for the sheer work involved. If you have ever tried solving a set of 3 simultaneous equations, it's very doable. Try it with 4 equations.....harder, go to 5 and it gets really tough...it CAN be done, but the work can be intractable. This is why tensors make it easier, but even the tensors have horrid notation, and there aren't very many decent books about it out there.
So, in my opinion, I think Einstein would have had the concept down, but would have had trouble 'proving' it without the easier notation to work with.
Even with the notation you still have to think down to the hardcore level of what's really going on conceptually.
Here's a quick example...in Quantum Mechanics we work with phase space...coordinates (p, q). 'q' is a generalized position coordinate, and 'p' is momentum, so while you are dealing with these kinds of 'points' in 'phase space' you still have to think down to the level of the quantities you're working with in some cases. I will give you one more example.....and I do this often.
If I were doing something with the force due to gravitation: F = -GmM/r², and the focus was the distance between the centers of the two masses, I would probably condense this down to: F = Q/r²
It's easier to handle and work with, and easier to see conceptually without the extra clutter, but.....you still have t account for what's going on....what's involved and the interactions taking place.
I know these are odd examples, but they do touch on what I'm talking about. Certainly we can't ever know how things would have turned out had Einstein not had the MT, but I tend to think he would have, but it may have taken him longer.
People did QCD for years...even Feynman, before he decided to come up with 'Feynman Diagrams'. They didn't necessarily facilitate discovery per se , but allowed it to be done easier and a little faster.
Most notation in math is for making something easier. Could be easier for understanding perhaps, but mostly for the sheer work involved. If you have ever tried solving a set of 3 simultaneous equations, it's very doable. Try it with 4 equations.....harder, go to 5 and it gets really tough...it CAN be done, but the work can be intractable. This is why tensors make it easier, but even the tensors have horrid notation, and there aren't very many decent books about it out there.
So, in my opinion, I think Einstein would have had the concept down, but would have had trouble 'proving' it without the easier notation to work with.
Even with the notation you still have to think down to the hardcore level of what's really going on conceptually.
Here's a quick example...in Quantum Mechanics we work with phase space...coordinates (p, q). 'q' is a generalized position coordinate, and 'p' is momentum, so while you are dealing with these kinds of 'points' in 'phase space' you still have to think down to the level of the quantities you're working with in some cases. I will give you one more example.....and I do this often.
If I were doing something with the force due to gravitation: F = -GmM/r², and the focus was the distance between the centers of the two masses, I would probably condense this down to: F = Q/r²
It's easier to handle and work with, and easier to see conceptually without the extra clutter, but.....you still have t account for what's going on....what's involved and the interactions taking place.
I know these are odd examples, but they do touch on what I'm talking about. Certainly we can't ever know how things would have turned out had Einstein not had the MT, but I tend to think he would have, but it may have taken him longer.