# When can we approximate General Relativity to Newtonian Gravity?

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1. Sep 6, 2014

### vlemon265

For example at very low speed (v<<c), in Special Relativity, we can approximate relativistic motion to Classical Newtonian motion.
But in General Relativity, what situation can make there an approximation to Newtonian Gravity
( just like v<<c ) ?
Thanks.

2. Sep 6, 2014

### bapowell

When spacetime is close to Minkowski: $$g_{\mu \nu} \approx \eta_{\mu\nu}$$

EDIT: More specifically, the limit corresponds to the case in which spacetime metric can be written as a small perturbation about Minkowski:$$g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$$ where $$|h_{\mu\nu}| \ll1$$

Last edited: Sep 6, 2014
3. Sep 6, 2014

4. Sep 6, 2014

### Matterwave

WBN's answer is the most general one, but perhaps in a simpler language, General relativity will reduce to Newtonian gravity in the two limits where the speeds in the system are small, and the gravitational fields in the system are weak.

5. Sep 6, 2014

### pervect

Staff Emeritus
To expand on Matterwave's point a bit by example, note that in the weak field limit, light deflects twice as much in GR as it does in Newtonian theory. This extra defelction of light was in fact one one of the first experimental tests of GR. The extra deflection of light in GR is an example of why you need both low velocities and weak fields before GR will give the same answer as Newtonian theory. It's also an example whose experimental results are consistent with GR and inconsistent with Newtonian thoery.

6. Sep 6, 2014

### Matterwave

I've always been confused on how the Newtonian deflection is derived. As light is mass less how is it affected at all by the Newtonian gravitational law?

7. Sep 6, 2014

### pervect

Staff Emeritus
I haven't researched this in depth personally. Wiki credits the following:
Einsten's paper appears to be "On the Influence of Gravitation on the Propagation of Light"
http://www.relativitybook.com/resources/Einstein_gravity.html He first uses his (Einstein's) elevator thought experiment to determine the coordinate speed of light as a function of gravitational potential, then he uses Hughens principle to calculate the deflection of light.

The approach I would think of using the equivalence principle is to calculate the deflection of an object of mass m moving at the speed of light, and by making use of the equivalence principle saying that it holds even when m=0. But I haven't gone through the math, though I'm pretty confident it will agree with the above.

I don't really see how one can rule out the possibility of no deflection of light on purely theoretical grounds, for instance Nordstrom's theory is self consistent and has no deflection of light. I suspect that even the first crude experiments were accurate enough to make it unlikely that there was zero deflection, though.

8. Sep 6, 2014

### ShayanJ

Another thing is Élie Cartan's work. He formulated a geometrical theory of Newtonian gravity. In such a theory, light will surely deflect.
I don't know much a about it, but it seems interesting.

9. Sep 6, 2014

### ChrisVer

http://mathpages.com/rr/s6-03/6-03.htm
it's more like they are working with acceleration of the gravitational field [no particle's mass] rather than the gravitational force of Newton... Of course at first the $0=0$ thing wouldn't seem correct, but one can move Newton's second law with gravity one step further and start the definitions from that point on. Of course the final newtonian results are not correct but whatever...
It's like once in SR I said that $u= \frac{p}{E}$ is also valid for massless particles (gives c, even by using $E= \gamma m, ~~ p= \gamma m u$ and not only -the more correct- $E=p$)...you first do that for massive, get rid of the mass, and then extend the definition to the massless as well..
Also you can check these:

Last edited: Sep 6, 2014
10. Sep 7, 2014

### Imager

I’m still not understanding how Newton’s theories predict light bending. Is there a simple way to think of it? (I’m not strong with math.)

11. Sep 7, 2014

### A.T.

In Newtons theory the gravitational acceleration of a small test mass is independent of it's mass. So if you model light as mass-less particles they will accelerate down, just like anything else.