Photon Propagation : Straightline or Helix ?

Thanks rpenner. Farsight seems to want a Nerf guns at dawn - subject to available time I'll go for that...
If you (rpenner) would be kind enough to grit your teeth while I try for concepts - that would be appreciated - unless I go horribly wrong - in which case your intervention would be even more appreciated.

Answering where did Einstein say...

From wikipedia
https://en.wikipedia.org/wiki/Equivalence_principle

The equivalence principle was properly introduced by Albert Einstein in 1907, when he observed that the acceleration of bodies towards the center of the Earth at a rate of 1g (g = 9.81 m/s2 being a standard reference of gravitational acceleration at the Earth's surface) is equivalent to the acceleration of an inertially moving body that would be observed on a rocket in free space being accelerated at a rate of 1g. Einstein stated it thus:

we [...] assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system.
— Einstein, 1907


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For the purposes of thought experiments I adopt the convention that everything that is to be considered will be included in the opening description. If this leaves the experiment fatally flawed then I'll go away and come back later with (hopefully) the flaw fixed.

So we have a carriage (of mass m) containing Alice and a Nerf Gun. Alice fires a Nerf (from point A on the floor) vertically up at the ceiling. The Nerf travels (up) a distance x at muzzle velocity v and hits the ceiling at point B. Alice marks point B on ceiling with a delible marker pen. The Nerf take time t to travel AB where t=x/v.

A locomotive is attached and the carriage accelerated at a m/(s^2). We work out the acceleration by knowing the mass of the coach (m) and the force (f) exerted by the coupling (f) to the locomotive. We find a=f/m.
Alice fires another Nerf in the same direction as before. The Nerf again takes x/v seconds (travelling in a straight line) to reach the ceiling but while it is is in flight the coach has advanced a distance (1/2)at^2.

Now for some serious pulling power... we accelerate the coach at g m/(s^2) where g is 'the acceleration due to gravity'. Alice fires the Nerf gun again the Nerf travels up in a straight line and Alice makes another delible mark on the ceiling at point C where the Nerf hit.
The distance BC is (of course) (1/2)gt^2.

Now we get the crane out and hang the coach vertically from its coupling. We can tell the carriage is being accelerated by the tension in the coupling (mg).

Alice fires the Nerf gun (now horizontally), the Nerf travels in a straight line and (to within a gnatscock) hits point C after travelling for time t.

Einstein says:-
"[there is] complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system."

More significantly Reality says the same thing.

-C2.

 

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What's even more basic is that Einstein does not say that the material point moves in a straight line.

 

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explain then why the definition is required.

 

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Yeah, well what you and The God and paddoboy seem to be missing is

1. Even in Euclidean space, it is necessary to define exactly what is meant by a "straight line". Usually this is taken to be the shortest distance between 2 points. The proof that it is the shortest distance (and is unique) is given by a technique (known for close on 200 yrs) known as the calculus of variations

2. The term that seems to have been used a lot here to refer to geodesics - namely the "straightest possible line in non-Euclidean space" - is without content. If it is supposed to refer to the shortest distance between 2 points in non-Euclidean space, this requires some work, especially to show it is unique. Nowhere has the meaning been made clear in context

I am afraid that the variational calculus may not help you here very much - this would require a constant metric in the region that separates the 2 points in question. You will need the sort of calculation that rpenner gave - note that he explicitly uses the Levi-Civita connection in his exposition, precisely because the metric is not constant

 

Hey - I'm missing things too - loads of 'em.

 

Hmmm, Funny since I have argued [with appropriate references] against the pseudoscience of both the god and Farsight continually, that you then lump me in with them.....
The gist of this thread if you havn't already guessed, is the fact that light and all bodies will tend to travel in a straight line unless acted on by a force [Newtonian] or influenced by the curvature/warping of spacetime [GR]
I don't believe I have deviated from that position, despite the constant obfuscation from our two friends,

 

For the sake of variation let's try dropping the carriage vertically from a considerable height - high enough for Alice to fire the Nerf gun and bale out successfully.

Under these circumstances does the bullet hit point B (the same point as when horizontal with no acceleration) or point C (the same point as when horizontal with acceleration=g) or some other point. If 'other point' then where?

 

https://en.wikipedia.org/wiki/Introduction_to_the_mathematics_of_general_relativity

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High-precision test of general relativity by the Cassini space probe (artist's impression): radio signals sent between the Earth and the probe (green wave) are delayed by the warping of space and time(blue lines) due to the Sun's mass. That is, the Sun's mass causes the regular grid coordinate system (in blue) to distort and have curvature. The radio wave then follows this curvature and moves toward the Sun.


A good example of this is the surface of the Earth. While maps frequently portray north, south, east and west as a simple square grid, that is not in fact the case. Instead, the longitude lines running north and south are curved and meet at the north pole. This is because the Earth is not flat, but instead round.

In general relativity, gravity has curvature effects on the four dimensions of the Universe.

 

https://www.google.com.au/search?q=...UKEwj1pN7vnvHMAhWnXaYKHeJnBMAQsAQIQQ&dpr=2.25

 

rpenner:

Please correct me if I am wrong in my understanding, but the 'path of least action' can be any shape, depending on the balance of forces (or absence of forces) which may be involved (or not) in any situation under study. Being 'straight line' is not a definitional necessity in this case, is it? If it is, can you please explain why, so that I may better understand what you and other posters are claiming about 'straight line' paths? Thanks.

 

It's really easy, all people are saying is that everything will tend to travel in a straight line unless acted on by forces [Newtonian] or who's path is governed by curved/warped space, in which case it travels in geodesics.[GR]

 

What makes the situation different from the least action principle are:
1) We are talking about geometry of a 4-dimensional manifold, not extrema of a 7 dimensional functional. Thus the geodesics have a natural meaning, independent of whether they are traversed.
2) General covariance -- lots of choices how to describe space and time
3) Different units -- typically length-squared, not action.
4) Coordinate Time is not an independent variable -- indeed some valid geodesics are necessarily paths that no particle of zero or finite mass may traverse.
5) Principle of equivalence, I -- the geodesics can be described, in the limit of small excursions relative to the tidal forces, as the straight lines where every corresponding to those described by Minkowski geometry. There is always a local coordinate system about a point where the affine connections, \(\Gamma_{\mu\nu}^{\rho}\) (but not necessarily derivatives, are all zero.
6) Principle of equivalence, II -- The geodesics are dictated by space-time geometry, not specific properties of the particle like electric charge. So the motions are as natural as Newton's First Law of Motion.
7) Crossings -- While trajectories don't cross in phase space ; geodesics in 4-dimensional space cross at every point in an infinite number of ways.
8) Closed curves -- While a conservative system may have closed curves in phase space; it's an open question if closed time-like curved exist in realizable spacetimes in GR.

Finally, there has been, since 1915, a least-action way of describing gravity, but it describes variation in spacetime metric and curvature, not trajectories of geodesics.
https://en.wikipedia.org/wiki/Einstein–Hilbert_action
Again, this is emphasizing that this is a geometric theory.

 

I feel the definition of straightline is non intuitive in curved spacetime.......

1. In case of Euclidean Flat spacetime, the least distance between 2 points, can be connected through a straightline.

2. The discatnce between these two points can be traversed by a particle even if acceleration is not zero, that means even in presence of non-zero force, linear motion is possible. Why other posters are pushing that motion in absence of force can be decribed as straightline, stems from the fact that in GR gravity is not considered as force.

3. IMO it is not right to call a curved path between two points as straightline, just because no force is present. For example the orbital path of any celestial object under Gravity (No force) is weirdly curved but since it is on geodesic and without external force and hence it is termed as straightline. This in my opinion is quite funny.

4. I give you one more example, take projectile motion, is it straightline or curved ?

I have expressed earlier also, and IMO the correct definition of straightline segment between two points should be the least time required to cover the same by light. That is instead of Geodesic, the null Geodesic between two points shall be termed as straightline. This is sensible also as between two points A and B, even in curved spacetime, there will be only one null geodesic but there can be many geodesics, depending on the initial conditions.

 

A geodesic can be a straight line. In curved spacetime it is still a geodesic.

Bingo! and a geodesic!

You are obfuscating because the premise of your OP was found wanting.
For the umpteenth time, everything tends to travel in a straight line unless acted on by an opposing force [Newtonian]. or influenced by the curvature/warpage of spacetime.

No they are all geodesics and are called orbits.

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It's a geodesic.

That's nice but not quite correct, while a straight is certainly the shortest distance between two points on a flat surface, a geodsic is the shortest distance between two points on a curved surface: In effect, a straight line is a geodesic.

 

I disagree, Geodesic is not the shortest between two points in curved spacetime, null geodesic is. Hope it is clear.

And IMO even in curved spacetime ideally null geodesic should be the straightline, not any other geodesic.

 

Shifting the goal posts again?

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We are talking about any straight line motion in curved spacetime not just a photon. And both are the shortest path between two points.

Again, geodesics are straight lines and we were talking about all bodies in curved spacetime including light..
But we are getting there!

 

Is it straightline or non-straightline ?

 

It's a geodesic path.

 

Finding it difficult to commit ?

Is it straightline or non straightline ?? You have no third choice, here...

 

Good that you attempted some content contribution, but unfortunately it is not correct in the context. Rpenner has softly nudged you but I will be blunt here...

Any path straight or curved, under acceleration or in absence of acceleration, as long as it is continuous, shall have infinitessimal small segment dx, as Euclidean straightline. This is true for both straightlines as well as curved lines (arcs), so you are offering nothing here, your attempted statement cannot distinguish between straight and curved....Good nonetheless, you wrote something technical without shrug and shake.