General Relativity and light deflection

[Mickey Farley]

I am a student of physics at a local Junior College in Mendham NJ and am planning on transferring to a 4 year program at the University of Alabama in a year. Iam having a bit of a difficult time understanding general relativity. Why does a photon bend twice as much under a gravitational field compared to a test particle in Newtonian mechanics? I understand time dilation is a factor, but I would have thought the deviation would be proportional to the inverse of the square of the distance, not 2 times that. Doesn't this imply general relativity violates the inverse square law? Forgive my ignorance, for I am only trying to understand where terms like "inverse square law" and such fits in with modern physical models. Thanks

 

[jbriggs444]

You do understand that "proportional to x" and "proportional to 2x" are equivalent statements, right?

 

[Nugatory]

Doesn't this imply general relativity violates the inverse square law?

GR does "violate" the inverse square law. However it would be better to say that the inverse square law is not quite exact; GR produces better predictions for the behavior of objects in strong gravitational fields than the Newtonian calculations based on the inverse square law.

The difference between the planetary orbits we calculate from the inverse square law and from GR is just barely noticeable in the precession of Mercury's orbit, but it's there. This anomaly (google for "Mercury precession") was first observed in the 18th century and remained a mystery until the discovery of GR. The other planets are farther from the sun so the gravitational field is weaker and the discrepancy between the two theories is small enough that it went unnoticed.

 

[pervect]

I am a student of physics at a local Junior College in Mendham NJ and am planning on transferring to a 4 year program at the University of Alabama in a year. Iam having a bit of a difficult time understanding general relativity. Why does a photon bend twice as much under a gravitational field compared to a test particle in Newtonian mechanics? I understand time dilation is a factor, but I would have thought the deviation would be proportional to the inverse of the square of the distance, not 2 times that. Doesn't this imply general relativity violates the inverse square law? Forgive my ignorance, for I am only trying to understand where terms like "inverse square law" and such fits in with modern physical models. Thanks

The "extra deflection of light can be understood a consequence of the curvature of space, if one makes some modest assumptions about how space is split form space-time. The "curvature of space" may seem like word soup without the proper background. Basically, it means the geometry of space (suitabley defined by a particular split of space-time into space and time) is not Euclidean. A simple and hopefully familiar example of a "curved space" is the surface of a sphere.

I would guess that you haven't studied any textbooks on GR yet - there are some treatments at the undergraduate level, but even those are advanced undergraduate level, and I don't think you're there yet. Better treatments of GR come about at the graduate level, which you're definitely not at.

If you are familiar with special relativity and space-time diagrams, you can better think of General relativity as the curvature of space-time. At the simplest level, this basically involves drawing the same space-time diagrams one used to draw on flat sheets of paper for special relativity on curved surfaces , such as the surface of a sphere, instead. If you're not familiar with special relativity, you need to learn it before you learn GR - General relativity is built on top of special relativity. Special relativity has relatively modest mathematical requirements to gain a basic understanding, only high school algebra is needed for the most basic treatment of SR. GR is much more demanding mathematically. Some things in SR, like "the relativity of simultaneity" are not necessarily mathematically difficult, but may be conceptually difficult, basically due to previously established beliefs about the nature of time that have to be unlearned to understand the theory.

I haven't really given a proper treatment of curvature in this short post - that is one of the mathematically challenging parts of General relativity. At this point, I only hold up the example of a spherical surface as a familiar example of something that is curved, and note that the geometry on this curved surface is not Euclidean. Generalizing this simple examle of one curved surface in 2 dimensions to a full treatment of curvature in arbitrary dimensions (at least 4 needed for space-time) requires tensors and differential geometry, both rather advanced topics.

 

[A.T.]

Why does a photon bend twice as much under a gravitational field compared to a test particle in Newtonian mechanics? I understand time dilation is a factor, ...

The gradient of gravitational time dilation accounts for the locally observed effect, that the Newtonian model explains. Additionally there is a global effect due to spatial geometry.

In pictures:
http://demoweb.physics.ucla.edu/content/10-curved-spacetime

With math:
https://www.mathpages.com/rr/s8-09/8-09.htm