Why does light bend during acceleration

[kavonkazem]

Im studying general relativity right now and I dont understand the equivalence principle. I understand it when relating to objects with mass. When in a sufficiently small space you cannot tell whether your are being accelerated in space far away from any gravitational force, or if you are in a gravitational field. How ever it states that there is no experiment you can do to tell the difference between the two, including a light experiment. This doesnt make any SENSE!!! I was under the impression light has no mass, which means when it bends, it is due to the curvature of space time or if you like a gravitational potential. Which means sitting on the earth light would bend downwards towards the earth. But why would it bend in a spacecraft accelerating through space? Why is acceleration bending light? please help me

[swansont]

But why would it bend in a spacecraft accelerating through space? Why is acceleration bending light? please help me

 

If you have a beam going across the spaceship, the wall of the spaceship will have moved an additional distance, relative to an inertial frame. That gives you a curved path from the rocket's perspective.

[IM Egdall]

The Equivalence Principle (EP) has its limits. The beam going across the spaceship will bend only half the amount of a beam in a equivalent graviational field. The EP takes into account only the warping of time, and in fact gives identical predictions as Newtonian gravity. General relativity gives the full effect due to both time and space warp -- which agrees with actual measurements of starlight passing the Sun.

Edited February 16, 2012 by IM Egdall

[elfmotat]

The Equivalence Principle (EP) has its limits. The beam going across the spaceship will bend only half the amount of a beam in a equivalent graviational field.

 

This is patently false. A spaceship with the appropriate amount of acceleration is indistinguishable from a local gravitational field.

 

I'm guessing this is coming from the factor of 2 difference between the Newtonian and GR predictions about the deflection of light around the sun. I assure you this has nothing to do with a failure of the equivalence principle. In fact, it gives more credence to it.

[DrRocket]

Im studying general relativity right now and I dont understand the equivalence principle. I understand it when relating to objects with mass. When in a sufficiently small space you cannot tell whether your are being accelerated in space far away from any gravitational force, or if you are in a gravitational field. How ever it states that there is no experiment you can do to tell the difference between the two, including a light experiment. This doesnt make any SENSE!!! I was under the impression light has no mass, which means when it bends, it is due to the curvature of space time or if you like a gravitational potential. Which means sitting on the earth light would bend downwards towards the earth. But why would it bend in a spacecraft accelerating through space? Why is acceleration bending light? please help me

 

The equivalence principle was philosophically useful to Einstein in the discovery and development of general relativity. However, unlike the axioms of special relativity, it is not central to the logical structure of general relativity, which is based on (pseudo) Riemannian geometry and the relatioship between the stress-energy tensor and the Einstein curvature tensor. It is still useful in some situations, but one must be careful not to confuse onself with it.

 

That said, light follows a geodesic in spacetime.

 

So does any object influenced by no force other than gravity.

 

Consider a ball floating in your spacecraft in free fall (eg. in orbit around a massive body). That ball also follows a geodesic in spacetime, as does the spacecraft until thrust is appplied. Now fire your rockets and accelerate the spacecraft. The ball will continue to follow a spacetime geodesic until it hits something. But from the perspective of a passenger in the spacecraft, that ball will follow some more or less arbitrary path determined by the acceleration of the ship around the ball.

 

The only significant difference between light and the ball is that that light follows a null geodesic in spacetime, while the ball follows a timelike geodesic. From the perspective of an observer with a coordinate system referenced to the body of the spacecraft that path can appear to be almost anything.

[IM Egdall]

This is patently false. A spaceship with the appropriate amount of acceleration is indistinguishable from a local gravitational field.

 

I'm guessing this is coming from the factor of 2 difference between the Newtonian and GR predictions about the deflection of light around the sun. I assure you this has nothing to do with a failure of the equivalence principle. In fact, it gives more credence to it.

 

Sorry, but I do not think you are right here.

 

Einstein came up with the EP in 1907. Based on the EP and his free-falling elevator thought-experiment, he calculated the bending of starlight as is passes very close to the Sun's surface. He got a value of 0.875 arcseconds. This is the same (incorrect) value predicted by Newtonian gravity. And it takes into account only the warping of time.

 

Then in 1915, with his new field equations of general relativity, Einstein revised his prediction to twice the amount: 1.75 arseconds. Experiments since have confirmed this value to extreme accuracy. (This takes into account the warping of both time and space).

[elfmotat]

Based on the EP and his free-falling elevator thought-experiment, he calculated the bending of starlight as is passes very close to the Sun's surface. He got a value of 0.875 arcseconds.

 

No, he used the accelerating elevator though-experiment to demonstrate that light would, in fact, fall due to gravity (something that wasn't known at the time). He made no such calculations.

 

 

This is the same (incorrect) value predicted by Newtonian gravity.

 

After demonstrating that light should fall due to gravity, he then calculated by how much using Newtonian gravity.

 

 

And it takes into account only the warping of time.

 

I've seen Newtonian gravity formulated geometrically using a time dilation factor in the tt component of the metric, but I wouldn't go so far as to say that Newtonian gravity "takes into account the warping of time."

 

 

Then in 1915, with his new field equations of general relativity, Einstein revised his prediction to twice the amount: 1.75 arseconds. Experiments since have confirmed this value to extreme accuracy. (This takes into account the warping of both time and space).

 

The reason he obtained an incorrect value the first time 'round was because he was using an incorrect theory of gravitation. This says nothing about the validity of the EP. The EP is actually one of the axioms of GR, so if it weren't true then GR wouldn't be either. There have been numerous tests of the EP over the years, and they have all confirmed it to high accuracy - some even up to 13 decimal places. A few of them are listed on wikipedia: en.wikipedia.org/wiki/Equivalence_principle .

Edited February 17, 2012 by elfmotat

[IM Egdall]

No, he used the accelerating elevator though-experiment to demonstrate that light would, in fact, fall due to gravity (something that wasn't known at the time). He made no such calculations.

 

From Hans Ohanian's book, Einstein's Mistakes - The Human Failings of Genius, p. 226:

 

"(Einstein's) 1911 calculation of the bending of rays of light, which was based on the Principle of Equivalence, yielded a result half as large as the new calculation (in 1915) based on his new theory of gravitation. Einstein understood that the reason for this discrepancy was the warping of space, whereas the 1911 calculation had effectively included only the warping of time . . . (Thus) the bending of a ray of light in an accelerated box is half as large as the bending in a box at rest in a gravitational field.

 

" . . . in 1913, Einstein's friend and colleague Ehrenfest . . . had published a short paper presenting a general proof about the failure of the Equivalence Principle for the propagation of light."

 

As I understand it, Einstein deduced gravitational time dilation from the EP. From this, in 1911 he calculated the wrong-by-half value for the bending of light. By considering only the warping of time, Einstein's mathematics gave identical predictions as Newton's. (I do not believe he realized this at the time.)

Edited February 17, 2012 by IM Egdall

[DrRocket]

 

The reason he obtained an incorrect value the first time 'round was because he was using an incorrect theory of gravitation. This says nothing about the validity of the EP. The EP is actually one of the axioms of GR, so if it weren't true then GR wouldn't be either. There have been numerous tests of the EP over the years, and they have all confirmed it to high accuracy - some even up to 13 decimal places. A few of them are listed on wikipedia: en.wikipedia.org/wiki/Equivalence_principle .

 

General relativity is not an axiomatic theory in the sense that special relativity it. Modern presentations of general relativity are not logically reliant on the equivalence principle. It served a purpose in the discovery process of Einstein, but it is largely irrelevant to the logical foundations of the subject.

 

It is absolutely untrue that "The EP is actually one of the axioms of GR, so if it weren't true then GR wouldn't be either."

[elfmotat]

As I understand it, Einstein deduced gravitational time dilation from the EP. From this, in 1911 he calculated the wrong-by-half value for the bending of light. By considering only the warping of time, Einstein's mathematics gave identical predictions as Newton's. (I do not believe he realized this at the time.)

 

The error comes from the fact that the equivalence principle applies only locally, whereas Einstein used it in coordination with Newtonian gravity to predict an incorrect value for the deflection of light. If you think the equivalence principle is in error, by all means come up with a explanation for why experiment agrees with it so well.

 

 

General relativity is not an axiomatic theory in the sense that special relativity it. Modern presentations of general relativity are not logically reliant on the equivalence principle. It served a purpose in the discovery process of Einstein, but it is largely irrelevant to the logical foundations of the subject.

 

It is absolutely untrue that "The EP is actually one of the axioms of GR, so if it weren't true then GR wouldn't be either."

 

Would you agree with the following(?): "Any physical law which can be expressed in tensor notation in SR has exactly the same form in a locally inertial frame of a curved spacetime." That is the strong equivalence principle, also called the "comma-goes-to-semicolon rule" because of the switch from partial to covariant derivatives. If this doesn't count as an axiom, I honestly don't know what does.

Edited February 18, 2012 by elfmotat

[elfmotat]

The error comes from the fact that the equivalence principle applies only locally, whereas Einstein used it in coordination with Newtonian gravity to predict an incorrect value for the deflection of light. If you think the equivalence principle is in error, by all means come up with a explanation for why experiment agrees with it so well.

 

I wasn't very clear here. I'll elaborate: The equivalence principle is a local principle. A gravitational field is locally indistinguishable from an accelerated reference frame. Einstein used Newtonian gravity along with the EP to calculate the deflection of light relative to local straight lines, lines that could be established by rigid rulers. This is fine, but only locally. When applied to a global system, like it was, it fails. Straight lines are bent near sources of gravitation, and by just enough to yield the factor of two difference between the general relativistic and Newtonian predictions. This doesn't indicate a failure of the equivalence principle, it's just a misapplication of it. Like I said, there is tons of experimental support for the EP.

[elfmotat]

I've seen Newtonian gravity formulated geometrically using a time dilation factor in the tt component of the metric, but I wouldn't go so far as to say that Newtonian gravity "takes into account the warping of time."

 

Also, for anyone unfamiliar with this, I think it's interesting enough to warrant some explanation. If you have a metric given by (in units where c=1, Ω=constant for simplicity):

 

[math]ds^2=-(1-\frac{2GM}{r})dt^2+dr^2[/math]

 

it's easy to show that geodesics reduce to Newtonian gravity. Using the principle of extremal proper time:

 

[math]\delta \int d\tau =\delta \int \sqrt{-g_{\mu \nu}\dot{x}^\mu \dot{x}^\nu}dt=0[/math]

 

and the appropriately chosen Lagrangian:

 

[math]L=(1-\frac{2GM}{r})-\dot{r}^2[/math]

 

, then:

 

[math]\frac{\partial L}{\partial r}=\frac{\partial }{\partial t}\frac{\partial L}{\partial \dot{r}}\Rightarrow \ddot{r}=-\frac{GM}{r^2}[/math]

 

 

Incidentally, [math]\Delta \tau =\Delta t \sqrt{1-\frac{2GM}{r}}[/math] is the gravitational time dilation predicted by the Schwarzschild metric. Geometric Newtonian gravity can therefore be viewed as an approximation to GR which neglects spacial curvature.

Edited February 19, 2012 by elfmotat

[DrRocket]

 

Would you agree with the following(?): "Any physical law which can be expressed in tensor notation in SR has exactly the same form in a locally inertial frame of a curved spacetime." That is the strong equivalence principle, also called the "comma-goes-to-semicolon rule" because of the switch from partial to covariant derivatives. If this doesn't count as an axiom, I honestly don't know what does.

 

What that says is the special relativity is the localization of general relativity.

 

I don't know if I would call that an axiom, or just something necessitated by globalizing special relativity and thereby formuating a theory of gravitation.

 

Given the success of special relativity to predict non-gravitational phenomena, this is pretty much unavoidable.

 

As I said earlier, I don't find general relativity to be axiomatically derived in any meaningful sense. To me it is just (pseudo) Riemannian geometry and the physics comes in via the field equations. I guess you could call the field equations axioms, but I think that to be not in keeping with the usual spirit of a set of axioms.

 

On the other hand I generally do not find physics to be axiomatic anyway. The axiomatization of physics was one of the original Hilbert Problems from 1900 and remains open. I suspect that physics may never be rigorously formulated in terms of axioms. After all, physics is not mathematics.

[elfmotat]

What that says is the special relativity is the localization of general relativity.

 

I don't know if I would call that an axiom, or just something necessitated by globalizing special relativity and thereby formuating a theory of gravitation.

 

Given the success of special relativity to predict non-gravitational phenomena, this is pretty much unavoidable.

 

As I said earlier, I don't find general relativity to be axiomatically derived in any meaningful sense. To me it is just (pseudo) Riemannian geometry and the physics comes in via the field equations. I guess you could call the field equations axioms, but I think that to be not in keeping with the usual spirit of a set of axioms.

 

On the other hand I generally do not find physics to be axiomatic anyway. The axiomatization of physics was one of the original Hilbert Problems from 1900 and remains open. I suspect that physics may never be rigorously formulated in terms of axioms. After all, physics is not mathematics.

 

Fair enough. I suppose "axiom" is a bit of a strong word to use. Even so, the SEP was definitely still a core assumption in the formulation of GR.

[IM Egdall]

The error comes from the fact that the equivalence principle applies only locally, whereas Einstein used it in coordination with Newtonian gravity to predict an incorrect value for the deflection of light. If you think the equivalence principle is in error, by all means come up with a explanation for why experiment agrees with it so well.

 

Well said. I think this agrees with my earlier point. A beam of light across an accelerating elevator will bend. But by half the amount we would see in an equivalent gravitional field. This is because, as you say say, the EP applies only locally. In that sense, the EP is limited (I did not say it was in error). It does not take into account the geometry of space See link:

 

http://www.einstein-online.info/spotlights/equivalence_deflection