Deriving Relativity

[geordief]

Apparently Feynman had his own method

 

https://books.google.ie/books?id=G0paDwAAQBAJ&pg=PA87&lpg=PA87&dq="Einstein+himself,+of+course,+arrived+at+the+same+Lagrangian**+but+without+the+help+of+a+developed+field+theory,+and+I+must+admit+that+I+have+no+idea+of+how+he+ever+guessed+at+the+final+result.+We+have+had+troubles+enough+in+arriving+at+the+theory+--+but+I+feel+as+though+he+had+done+it+while+swimming+underwater,+blindfolded,+and+with+his+hands+tied+behind+his+back!&source=bl&ots=IGoZy4MI0z&sig=ACfU3U1pAweHTDVX73wpqtGr5ucyNNda5w&hl=en&sa=X&ved=2ahUKEwicjrzO0JXsAhUnSBUIHTTsDJsQ6AEwCHoECAkQAg#v=onepage&q=blindfolded&f=false

 

""Einstein himself, of course, arrived at the same Lagrangian** but without the help of a developed field theory, and I must admit that I have no idea of how he ever guessed at the final result. We have had troubles enough in arriving at the theory -- but I feel as though he had done it while swimming underwater, blindfolded, and with his hands tied behind his back!"

How many distinct (not distinguished by methodology but really distinct)  methods are there to derive relativity?

 

 

 

 

[joigus]

Feynman arrived to that formulation of GR from a quantum elementary-particle batch of arguments.

He also arrived at the homogeneous Maxwell equations from Heisenberg's uncertainty principle. He made astonishing connections.

The problem with Feynman is there are big jumps in the logical reasoning supported by an incredibly far-sighted intuition and very deep knowledge of physics. So it's very difficult to know whether he's making a compelling logical argument or somehow incorporating a very well-digested idea from his toolkit, using analogies across the communicating vessels of mathematically-similar theories...

Trying to answer your question, I'm not aware of any other way that's been practiced to arrive at GR.

[geordief]

1 hour ago, joigus said:

Feynman arrived to that formulation of GR from a quantum elementary-particle batch of arguments.

He also arrived at the homogeneous Maxwell equations from Heisenberg's uncertainty principle. He made astonishing connections.

The problem with Feynman is there are big jumps in the logical reasoning supported by an incredibly far-sighted intuition and very deep knowledge of physics. So it's very difficult to know whether he's making a compelling logical argument or somehow incorporating a very well-digested idea from his toolkit, using analogies across the communicating vessels of mathematically-similar theories...

Trying to answer your question, I'm not aware of any other way that's been practiced to arrive at GR.

Could it be derived simply from assuming that the spacetime interval is the same for all observers?

Perhaps there are two observers and A flashes a light pulse to B so that both A and B have 2 (identical events)

 

Then perform the same operation with A accelerating towards B .

Then repeat with the acceleration increasing  linearly

 

Then repeat with the last operation accelerating similarly wrt the last operation 

 

And so on ..(the distance between A and B the same on each occasion) 

 

Is  the spacetime interval invariance a stand in for Einsteins's "physical laws are the same everywhere" postulate?

[dimreepr]

4 minutes ago, geordief said:

Perhaps there are two observers and A flashes a light pulse to B so that both A and B have 2 (identical events)

Nope, A and B must be in a different location and therefore have 2 different event's... 

[geordief]

20 minutes ago, dimreepr said:

Nope, A and B must be in a different location and therefore have 2 different event's... 

But the same spacetime interval ,no?

 

The first event is the departure of the light from A and the second is its capture by B

 

But only one spacetime interval,surely....

[joigus]

4 hours ago, geordief said:

Could it be derived simply from assuming that the spacetime interval is the same for all observers?

You also need general covariance, and the assumption that the field equations are a 2-rank tensorial, instead of higher rank, I think. They sound quite innocent assumptions, but they are really meaty. From there, the form of the Einstein tensor is almost forced (except for cosmological term) if you want the equations to be consistent with covariant conservation of the matter tensor.

Personally, I find Einstein's reasoning quite transparent. Absolutely brilliant, but transparent. Feynman's, in general, I find more difficult to see the compelling character of some steps.

I also need refreshing part of this material, I must confess.

I wish @Markus Hanke or @Mordred expressed their opinion.

[Markus Hanke]

17 hours ago, geordief said:

Could it be derived simply from assuming that the spacetime interval is the same for all observers?

No, because this doesn’t allow you to derive what that interval actually is. It will give you most of SR within small, local patches, but it won’t give you the field equations which are necessary to derive the metric from given distributions of energy-momentum.

20 hours ago, geordief said:

How many distinct (not distinguished by methodology but really distinct)  methods are there to derive relativity?

It depends on what you consider ‘distinct’. In addition to the ones already mentioned, what comes to mind is the geometric reasoning put forward by Misner/Thorne/Wheeler in their book, the reasoning of String Theory where GR emerges as a necessary consistency condition on the background spacetime, and possibly a thermodynamic approach from quantum information theory (entropic gravity). 

[geordief]

3 hours ago, Markus Hanke said:

No, because this doesn’t allow you to derive what that interval actually is. It will give you most of SR within small, local patches, but it won’t give you the field equations which are necessary to derive the metric from given distributions of energy-momentum.

 

So it would  give most of SR ? Can you give me an example of what it would fail to show in SR ?(I sound argumentative but am just curious)

What additional requirement  might  allow it to completely derive SR?

[Markus Hanke]

20 hours ago, geordief said:

So it would  give most of SR ? Can you give me an example of what it would fail to show in SR ?(I sound argumentative but am just curious)

What additional requirement  might  allow it to completely derive SR?

I think just demanding the interval to be invariant is not enough to uniquely determine the geometry of the underlying manifold. For example, you can write out this interval in ordinary 3D Euclidean space, the kind you learn about in school - which just gives you the Pythagorean theorem. The theorem is obviously true no matter what coordinate system you use, so there is also a notion of invariance there - but it won’t give you SR (e.g. vectors always add linearly, which they don’t in SR). Or you could have a curved spacetime - again, the interval is covariant, but it’s not SR.

I think in addition to invariance, you also need to give the full metric, which has to be of a specific form in order to give you SR.

[geordief]

5 hours ago, Markus Hanke said:

I think just demanding the interval to be invariant is not enough to uniquely determine the geometry of the underlying manifold. For example, you can write out this interval in ordinary 3D Euclidean space, the kind you learn about in school - which just gives you the Pythagorean theorem. The theorem is obviously true no matter what coordinate system you use, so there is also a notion of invariance there - but it won’t give you SR (e.g. vectors always add linearly, which they don’t in SR). Or you could have a curved spacetime - again, the interval is covariant, but it’s not SR.

I think in addition to invariance, you also need to give the full metric, which has to be of a specific form in order to give you SR.

Thanks. Appreciate your help and patience.