Sun Bending Light: Gravity Warps Space-Time?

[Garth]

Is this the key to figuring out the additional time dilation angle to be added to the GR curve in space (only 1/2 the needed angle)? Similar to the 90⁰ Doppler effect of red shift ( I don’t recall - what’s the correct term for this?) when a high speed light emitting source passes at it’s closest point to an observer.
While object (light) follows the curve space path of GR as it descends deeper into the gravity well and experiences a Blue Shift . Can we use that apparent increase in energy as an “apparent increase” in mass (that goes away as it departs the sun) to calculate an additional displacement for the higher mass to account for the angular change caused by the time dilation?

I think you are following a false trail here.

The photon travels along a straight line - a geodesic - and as it is light traveling at the speed of light that geodesic is a null-geodesic because the photons's proper time along its path is zero.

That straight line appears 'curved', that is it is deflected, because the space-time hyper-surface along which it is traveling is curved by the presence of the Sun's mass and this curvature is that which gives the Sun its gravitational field.

The photon has no rest mass, the change of its energy does not alter its path as all objects or photons fall at the same rate in GR, whatever their mass or energy - that is the essence of the Equivalence Principle and verified in experiments by Galileo onwards. (Cannon balls and feathers fall at the same rate in a vacuum.)

In order to calculate the amount of deflection you have to solve the equation of motion of the photon across the curved space-time surface, it is an involved calculation that however may be broken down into two components.

One component is due to the photon 'falling' towards the Sun, which is half the total, and this can be identified with the effect of time dilation at a series of momentarily stationary frames of reference along the path through which the light ray passes. The other half component is due to the curvature of space on its own.

I hope this helps.


Garth

 

[lightgrav]

I was wondering when I read this, light does not follow normal Newtonian physics (it has constant velocity.) So when light is pulled towards the sun it can not accelerate as a normal particle would. So would this effect the equations for the space part of GR's equation? Or does this effect have no significant on lights arc?
I'm also curious about the derivation of the time factor.

Using the corpuscular theory of light that was fashionable in Newton's day,
theorists would have expected light to speed up as it "fell" toward the Sun.
Unlike SpaceTiger, who pulled the "v" outside the integral in his impulse approximation, Newton would've treated it as a ponderous object which would travel faster near the Sun, and so be deflected less than you might expect based on the constant-speed straight-line path.
It is remarkable that the actual trajectory of a massive object takes it just enough closer than the straight-line path, so that the angular deflection infinitely far from the Sun is the same as derived in the impulse approximation (for small deflections). This is a coincidence, only being true for 1/r potentials.

The EASY way to get the time-factor is to consider the (scalar) Gravitational Potential (/c^2) as an increase in the index of refraction.

 

[SpaceTiger]

Unlike SpaceTiger, who pulled the "v" outside the integral in his impulse approximation, Newton would've treated it as a ponderous object which would travel faster near the Sun, and so be deflected less than you might expect based on the constant-speed straight-line path.

These results are only valid in the small-angle (nearly constant velocity) limit. The deflection angle at infinity is not equal to $\frac{2GM}{rv^2}$ for a body whose kinetic energy is comparable to its potential energy at closest approach.

so that the angular deflection infinitely far from the Sun is the same as derived in the impulse approximation (for small deflections). This is a coincidence, only being true for 1/r potentials.

I don't feel like solving the necessary differential equations to check this, but I'm pretty sure that's incorrect. For the 1/r^2 case, as one approaches the limit I mentioned above, the total impulse given to a passing body will be dominated by the time during which it's closest to the central mass. If the force law were steeper (such as 1/r^3), this would be even more true and I would expect the above approximation to be even more accurate. Clearly, it will break down for some force laws (like F~const.), but the result almost certainly isn't a coincidence of 1/r^2.

 

[RandallB]

I think you are following a false trail here.
…. ….. …. to calculate the amount of deflection .. to solve the equation of motion …. may be broken down into two components.
One component is due to the photon 'falling' towards the Sun, which is half the total, and this can be identified with the effect of time dilation at a series of momentarily stationary frames of reference along the path through which the light ray passes.

The other half component is due to the curvature of space on its own.
I hope this helps.
Garth

It’s your trail from an earlier post I’m trying to pick up.

So far this one thread has given more about Einstein’s bending starlight prediction then anything I’ve ever found on the net. Most do little better than recite a news headline about Einstein being the predictor of star locations.

I thing most will get the idea about the curve in space:
From your post #12

A ray of light passing close to the Sun on this surface follows a straight line, a (null) geodesic, along the surface. This ray is 'bent' because the surface it travels along is curved.
The deflection of a ray just grazing the Sun's surface at distance R from its centre, is determined by the curvature of 'space' to be
$$\alpha = \frac{2GM}{Rc^2}$$ radians as stated by ST.

Even without a detailed derivation here it compares well with Swartzchild formula AND SpaceTiger gave excellent Newtonian examples in a couple different ways.
Even Einstein only had this much before 1916. The key to GR is his additional part of the prediction to include time dilation.
Again from your post #12

However there is a further effect due to the dilation of time.
This extra effect is caused by space-time being curved rather than just space.
Instead of there being a 'curvature of time' as well as a 'curvature of space' the time component of space-time curvature reveals itself as a time dilation between clocks at different levels in the gravitational field as determined by the equivalence principle, or equivalently by the SR treatment of Newtonian gravity. The time dilation component adds an extra
$$\alpha = \frac{2GM}{Rc^2}$$ radians

Of the two options to “reveal” the time dilation component 1) as determined by the equivalence principle, or 2) equivalently by the SR treatment of Newtonian gravity.
It’s by using the “SR treatment of Newtonian gravity” to derive this last half of the total deflection that I believe will be the most understandable explanation.

That’s the only piece missing, to have this thread hold a complete explanation of Einstein’s GR calculation of starlight deflection by the sun.
Can you or someone show a derivation to build this last half, or reference that does? It would make this thread complete.

 

[RandallB]

Just following up on your post #12

the time component of space-time curvature reveals itself as a time dilation between clocks at different levels in the gravitational field as determined by the equivalence principle, or equivalently by the SR treatment of Newtonian gravity.

The time dilation component adds an extra
$$\alpha = \frac{2GM}{Rc^2}$$ radians
making a total of
$$\alpha = \frac{4GM}{Rc^2}$$ radians = 1.75" arc for the Sun

Do you remember how to derive these time dilation parts or a reference where it came from?
I was hoping the “equivalently by the SR treatment of Newtonian gravity” part would be even simpler than the impulse version of the first half (non-time dilation portion) of the answer.

 

[Garth]

You will find the splitting of the time dilation and space curvature components of light deflection in C. M. Will, Was Einstein Right?: Putting General Relativity to the Test, Basic Books (1993). (This is a popular account of tests of general relativity.)

The mathematics of the time dilation part as a "SR" treatment of Newtonian gravity has already been given by ST above.


The space-time deflection of light in terms of the Robertson parameters for a general metric is given by
$$\theta =\frac{4G_mM}R\left( \frac{1+\gamma }2\right)$$
where $\gamma$ is the amount of space (only) curvature per unit mass, and in GR $\gamma = 1$.

By elimination you can see that the other component (also 1) is that caused by the dilation of time.

I hope this helps.

Garth

 

[RandallB]

The mathematics of the time dilation part as a "SR" treatment of Newtonian gravity has already been given by ST above.

Sorry I don’t see that.
I assume by ST you mean Space Tiger not “Space-Time”
He did provide very valuable examples of the Newtonian equivalents for finding the sun’s bending of light.

One used the impulse approximation for any object applied to gravity in the Newtonian limit. Very effective and intuitive once you understand it.https://www.physicsforums.com/newreply.php?do=newreply&p=857454

Another using any in-falling mass following a hyperbolic orbit he diagramed the angle of the semilatus rectum to arrive at the same answer. A little harder to follow but it works.

As both of these approaches give the same answer for objects of any small mass, it is no stretch at all to apply the same conclusions to the “no-mass” photon, by considering no mass as included in “any small mass”.
And these results match perfectly with Einstein’s GR predictions of 1911 to 1915 and are all only half the total correct solution. (Not until 1916 did Einstein’s final adjustments to GR include the affect of time dilation in it for the total solution we accept today.)

BUT, I don’t see where any of these have derived anything to support that final half of the light bending solution for time-dilation. And I consider this thread incomplete without it. As I said earlier I find most web references to simple state the end result equation with no explanation as to how they get there. What is need is the explanation or derivation of how the time dilation was built into the final equation, not just that final equation shown in two parts.

The editorial I find on your referenced book says it will “explain the implications of this highly complex theory without using any mathematics beyond geometry”. Explaining implications doesn’t sound like it is focused on providing a derivation of the original. But I’ll try to get hold of a copy and see what’s there, if I can extract a reasonable explanation of the time dilation half of the solution is I will post it here. I just figured it would be better to come from someone that already knew and understood more about that time dilation calculation than I do.

 

[Garth]

First there is only one effect in GR, the deflection of the light ray as a consequence of space-time curvature. To do it properly you have to solve the equation of motion of a null-geodesic grazing the Sun. There is no easy alternative, see
Weinberg "Gravitation & Cosmology" page 188 ff. or
MTW "Gravitation" page 184-5 (a simpler derivation) or
Wald "General Relativity" pages 144-8.

The result using the generalised metric in the Eddington & Robertson expansion is as above:
$$\theta =\frac{4G_mM}R\left( \frac{1+\gamma }2\right)$$
where the Robertson parameter $\gamma$ is the amount of space (only) curvature per unit mass, and in GR $\gamma = 1$.

So in GR half the effect is due to the curvature of space alone. But what is the other half due to? It must be the non-space part of space-time curvature i.e. time dilation.

That is my argument.

By the Equivalence Principle time dilation is the measure of the relative acceleration of two momentarily stationary freely falling particles at different altitudes in a gravitational well.

This acceleration is the normal Newtonian gravitational acceleration, therefore the calculation of a photon 'falling towards' the Sun is equivalent to the calculation of the time dilation effect of the metric at different points along the light path.

Furthermore, the Newtonian calculation of a photon 'falling towards' the Sun makes no allowance for the curvature of space (alone) as space is Euclidean in Newtonian physics.

Garth

 

[RandallB]

in GR half the effect is due to the curvature of space alone. But what is the other half due to? It must be the non-space part of space-time curvature i.e. time dilation.
That is my argument.
This acceleration is the normal Newtonian gravitational acceleration, therefore the calculation of a photon 'falling towards' the Sun is equivalent to the calculation of the time dilation effect ...
Furthermore, the Newtonian calculation of a photon ... makes no allowance for the curvature of space

Whoa – I think I see the flaw in your argument that has you running up the wrong trail!

If I understand, your arguing that Space Tigers explanations match up to the GR time-dilation components and not the Space Curve part of GR.
That sounds absurd (To borrow from J. Bell), STiger never used anything involving time changes.
And of course what he has shown us relates to the GR curve in space not in time. First of all we are not dealing with a photon that is “falling”! That implies accelerating into the sun due to it’s gravity. These are “traveling by” the sun well above escape speed and being affected by the suns gravity (gravity well) and crossing altitudes of changing time.

Take a good look at the formulas Tiger has given us, not based on time they are based on any mass following normal Newtonian orbits, including escape orbits.
Now reduce those speeds but keep the same fundament formulas; reduced till you get a circular orbit based on the perigee we are investigating. With this orbit there is no time levels being crossed but the Newtonian is giving us an orbit. How could GR ever correlate this with a curve in time! It cannot, the altitudes are not changing! The GR explanation for a circular orbit is the curve in space, the gravity well not GR time dilation. You can not just take those same Newtonian fundamental formulas that tie directly to the GR curve in space at low speeds and somehow claim they transform into time dilation at high speeds. They are comparable to the curve in space not time. So of course “the Newtonian calculation makes no allowance for the curvature of space” it is the equivalent of it. How could a Newtonian calc use a GR curve anyway?

I have no doubt the additional bend is due to light traveling by and crossing different altitudes of time. And the correct number for that part of it is:
$$\alpha = \frac{2GM}{Rc^2}$$ radians

But we have yet to see how arriving at that part of the solution can be described or was derived.

Maybe someone else has some ideas or knows.

 

[Garth]

If I understand, your arguing that Space Tigers explanations match up to the GR time-dilation components and not the Space Curve part of GR.
That sounds absurd (To borrow from J. Bell), STiger never used anything involving time changes.
And of course what he has shown us relates to the GR curve in space not in time.

You are confusing the orbit curving in space (Newtonian) and the curvature of space. GR explains the curvature of an orbit, such as the circular orbit of the Earth around the Sun as a 'straight line' - that is a geodesic - along a hyper-surface of space-time that itself is "curved". For such curvature you do not have to embed the hypersurface in a higher, fifth dimension, although it might help to visualise it, curvature is described by its intrinsic geometrical properties.

If the Earth's elliptical orbit around the Sun does not look much like a straight line that is because you are not viewing it in 4D space-time. To scale the Earth follows a spiral through space time where the radius is 1 AU and the pitch is I light year, about 10⁵ AUs.

First of all we are not dealing with a photon that is “falling”! That implies accelerating into the sun due to it’s gravity.

In the Newtonian/SR treatment yes we are - the photon is traveling so fast that it is deflected, "accelerating into the sun", by only a small amount.

I have no doubt the additional bend is due to light traveling by and crossing different altitudes of time. And the correct number for that part of it is:
$$\alpha = \frac{2GM}{Rc^2}$$ radians
But we have yet to see how arriving at that part of the solution can be described or was derived.

As I said above

To do it properly you have to solve the equation of motion of a null-geodesic grazing the Sun. There is no easy alternative, see
Weinberg "Gravitation & Cosmology" page 188 ff. or
MTW "Gravitation" page 184-5 (a simpler derivation) or
Wald "General Relativity" pages 144-8.

Garth

 

[RandallB]

You are confusing the orbit curving in space (Newtonian) and the curvature of space. GR explains the curvature of an orbit, such as the circular orbit of the Earth around the Sun as a 'straight line' - that is a geodesic - along a hyper-surface of space-time that itself is "curved". For such curvature you do not have to embed the hypersurface in a higher, fifth dimension, although it might help to visualise it, curvature is described by its intrinsic geometrical properties.

Now you’re just grabbing at clichés – no one needs the 5th dimension hyper-surface here stick w/ GR.

You’ve only assumed that GR curve doesn’t fit with those STiger explanations.
And give no rational explanation to claim they should be compared with time dilation other than to say “It must be the non-space part of space-time curvature” based on an assumption.

The correlation between STiger explanations and the GR curve in space I gave in post #42 is simple and direct.

If all you have is 5-D and hand waving to try to hold onto comparing these Newtonian explanations to GR time dilation, let someone else weigh in.
Or at least take some time to think it through.

 

[Garth]

For such curvature you do not have to embed the hypersurface in a higher, fifth dimension, although it might help to visualise it, curvature is described by its intrinsic geometrical properties

Now you’re just grabbing at clichés – no one needs the 5th dimension hyper-surface here stick w/ GR.

You obviously do not read my posts so there is no point in trying.

Garth

 

[RandallB]

You obviously do not read my posts so there is no point in trying.
Garth

No your obviousy not even trying to see the point, but at least maybe other can learn.

 

[exponent137]

However there is a further effect due to the dilation of time.
This extra effect is caused by space-time being curved rather than just space.

Instead of there being a 'curvature of time' as well as a 'curvature of space' the time component of space-time curvature reveals itself as a time dilation between clocks at different levels in the gravitational field as determined by the equivalence principle, or equivalently by the SR treatment of Newtonian gravity. [Edit I originally put this down the other way round; why do I do that?]

The time dilation component adds an extra
$$\alpha = \frac{2GM}{Rc^2}$$ radians

making a total of

$$\alpha = \frac{4GM}{Rc^2}$$ radians = 1.75" arc for the Sun, as is observed.

I hope this helps.

Garth

I search, but I do not find derivation and explanation for this time dilatition component.
Space time curvature component can be easily calculated. I can show and Tiger Space showed (not to zero precision).
But how to understand this time dilatation part?

 

[exponent137]

I tried another model: Space is not curved, but light moves faster in one height than in the other (because of time dilatation). Besides, it moves faster in one direction as in another (dilatation of lenght). Light is curved on lens. So it can be curved also on such dilatated space (because of sun).

I tried to calculate something and calculation with dilatation seem right. But I cannot calculate with length dilatation. I tried to calculate with small graviational fields, this means bending of light because of sun.

Is something wrong with my model?

Best regards

 

[RandallB]

I tried another model: Space is not curved, but light moves faster in one height than in the other (because of time dilatation). . ... .
Is something wrong with my model?

IMO yes there is,
I know of now method of applying a time dilation to only one direction of space to allow light to change speed based on the direction you pick. It will appear to slow down in all directions

Also I don't understand your comment:

Space time curvature component can be easily calculated. I can show and Tiger Space showed (not to zero precision).

What Space Tiger (not Tiger Space) showed was how Newton’s Gravity calls for light to bend simply based on his gravity formulas. You can take a “no-mass” photon, as acting the same as “any small mass” as those formulas results do not depend on the mass of the moving entity. That component cancels out and the Impulse analyses Space Tiger used as far as I can see give a completely exact and precise result; giving one half the answer that GR gives. [In fact this is the same result as Einstein’s 1914 GR prediction, but corrected before publishing in 1916, had they been able to test the 1915 eclipse with his wrong 1914 prediction who knows what would have happened to GR, but the war delayed the test to 1919 after GR was completed in 1916]
What do you mean by “not to zero precision”??

One half of the GR result is defined by a curve caused by classical gravity is understandable. (A curve in space if you like)
What is missing is an explanation for the other half in terms of Time Dilation alone, a curve caused by time dilation alone, not “space-time”.
GR gives the complete “space-time” solution.
This thread attempts to break that into two pieces:
Space Tiger derived the gravity part (IMO with precision).
What is missing is the derivation of a “Special Relativity treatment of Newtonian gravity” to give the other half due to time dilation.

 

[Pgotac]

I thought the classical way of finding the bending of light was by finding the mass of light with whatever debroglies thing is (I forget but I know frequency is in there), and then using that mass to find the gravitational attraction with F = GMm/r^2. and is it rly off by exactly a factor of 2? that seems like a huge coincidence...

Who first calculated bending of light using Newton's Laws? Did he know Debroglie mass?(h/(wavelenght x c). Could someone help?

 

[jtbell]

The calculation of the (Newtonian) gravitational deflection of light was first published by Johann Georg von Soldner in 1801.

http://www.einstein-online.info/en/spotlights/light_deflection/

Note that the Newtonian deflection doesn't depend on the mass of the particle, for the same reason that objects of different masses fall with the same acceleration towards the Earth.

 

[Hal King]

The light should be 'uneffected' -- its the observer's state that determines the answer. How the observer (or the light) got to that state is also not relevant.