The Perspective of Motion Through Space

[LS George]

So in this instance I'll talk about photons. As light passes through a gravitational field its path is bent/deflected (however you want to phrase it). I can accept that. What bugs me is the how the particle will view its motion through space.

 

From its perspective, is it travelling in a straight line? Or not?

 

If it is, does the same apply to non-zero mass particles? (Protons, electrons etc.)

 

All help is appreciated!

[ajb]

From its perspective, is it travelling in a straight line? Or not?

 

If it is, does the same apply to non-zero mass particles? (Protons, electrons etc.)

It is not so easy talking about the perspective of massless particles, so lets not. Your question is valid for massive particles also.

 

Locally we have inertial frames of reference, so over very small distances the observer will travel in straight lines, however over larger distances the observer will travel in a curved path and he will notice this.

[Strange]

As a concrete example, if you are in free fall you are travelling along a geodesic (the equivalent of a straight line in curved space-time).

 

When you are sitting in your chair, you are (I think) being accelerated along a curved path.

 

[Don't be surprised if someone more knowledgeable says I have got that wrong. I certainly won't be ... ]

[xyzt]

As a concrete example, if you are in free fall you are travelling along a geodesic (the equivalent of a straight line in curved space-time).

 

When you are sitting in your chair, you are (I think) being accelerated along a curved path.

 

[Don't be surprised if someone more knowledgeable says I have got that wrong. I certainly won't be ... ]

You have it right.

[studiot]

 

As a concrete example, if you are in free fall you are travelling along a geodesic (the equivalent of a straight line in curved space-time).

 

I can't accept that a geodesic is the 'equivalent' of a straight line, except in very loose terms.

 

You can have 'straight' lines in curved space and geodesics in the same space.

 

Are these equivalent?

 

You can have 'straight' lines in euclidean space and geodesics also in euclidian space.

 

Are these equivalent?

 

Remember also that Newton did not talk about straight lines, in relation to motion. His actual phrase was

 

"In its right line"

 

I think it would be best to explain the difference between the purely geometric statement of geodesic or straight line and the least energy formulation of physics.

Edited January 7, 2014 by studiot

[ajb]

You can have 'straight' lines in curved space and geodesics in the same space.

 

How do you define a stright line on a curved space?

[studiot]

How do you define a stright line on a curved space?

 

 

 

my point exactly but I actually said

 

 

in curved space

However do (some?) ruled surfaces not incorporate both straight lines and geodesics?

[Strange]

 

I can't accept that a geodesic is the 'equivalent' of a straight line, except in very loose terms.

 

The words are, inevitably, ambiguous.

 

On the surface of the Earth, a geodesic is equivalent to a straight line in that it is the shortest distance between two points; i.e. part of a great circle. Projected onto a map it may not appear straight. Considered in 3D space it is not straight.

 

The definition in GR is somewhat more complex.

[studiot]

it is the shortest distance between two points

 

Doesn't that depend upon your distance function?

[moth]

To predict a geodesic path in GR, is it enough to use the lagrangian idea of action (kinetic energy - potential energy) when it's minimized?

 

[studiot]

 

To predict a geodesic path in GR, is it enough to use the lagrangian idea of action (kinetic energy - potential energy) when it's minimized?

 

 

 

http://en.wikipedia.org/wiki/Geodesics_as_Hamiltonian_flows

 

I can also recommend the long didcussion on Geodesics in chapter 9 of

 

Dodson and Poston : Tensor Geometry.

 

They discuss the origin of the term geodesic in Ancient Greece and Persia along with the meaning of 'straight '

 

both in terms of geometry and of physics bring out their differences and physical interpretations (if any)

 

 

A curve, c in a manifold M is a geodesic if its tangent vector field is parallel, del c'(t)=0

If c is thought of as describing the motion of a particle c'(t) becomes the velocity at time t. del c'(t) becomes the rate of change of velocity or acceleration. So a geodesic is the path of a poarticle subject to no forces, constrained only by the geometry of the manifold (We give another interpretation in paragraph3)

Edited January 7, 2014 by studiot

[ajb]

Doesn't that depend upon your distance function?

Exactly, you need to pick a metric.

 

And what is your technical difference between on and in? I don't think there is much of a distinction in this case.

 

I say on as I think of curves as maps, but you may say in as we have a subset of the manifold. It now depends on a few technicalities if this is really a submanifold, but anyway.

 

To predict a geodesic path in GR, is it enough to use the lagrangian idea of action (kinetic energy - potential energy) when it's minimized?

You can formulate geodesics in that way quite independent of GR. You can also formulate them in terms of a Hamiltonian, which is also neat.

Edited January 8, 2014 by ajb

[decraig]

All of the particulars of this can be found in Sean Carrol's http://preposterousuniverse.com/grnotes/ chapter 3.

 

Of course this means doing all the hard work in chapters 1 through 3, but some will find it well worth it. There is no better author for a conversational approach to a topic that could be otherwise be very dry.

 

There is more than one way to define a geodesic on the manifold of spacetime, and more than one shortest-distance definition. All the complication arises in the ambiguity of choosing the definition of the derivative of a vector, and other objects where the manifold is not Minkowskian--that is, not flat.

 

The definition of a geodesic in general relativity reduces to the path of an unaccelerated particle where the spacetime manifold is sufficiently flat. This involves about 6 separate requirements to say how the derivatives themselves are to be distinguished from all other possible definitions.

Edited January 11, 2014 by decraig