Comparing Gravity in Newtonian and Relativistic Frameworks

[John15]

I understand there is a slight difference in the value of gravity when worked out using Newton or relativity. The obvious example being mercury where Newtonian is slightly out and relativity accurate.
Does relativity show a higher or lower figure or does the difference in working out mean you cannot compare the 2.

 

[A.T.]

I understand there is a slight difference in the value of gravity when worked out using Newton or relativity. The obvious example being mercury where Newtonian is slightly out and relativity accurate.
Does relativity show a higher or lower figure or does the difference in working out mean you cannot compare the 2.

The only effect you can directly compare it the gravitational acceleration. Here is a thread about it:
https://www.physicsforums.com/showthread.php?t=310397

But GR predicts other effects that also affect trajectories like time dilation and space distortion.

 

[John15]

Why do Newtons equations not correctly describe the orbit of mercury. How precise are they with the other orbits.

 

[A.T.]

Why do Newtons equations not correctly describe the orbit of mercury.

We recently discussed what a good explanation is here:

https://www.physicsforums.com/showthread.php?t=545174

But the thread died.

 

[Passionflower]

Why do Newtons equations not correctly describe the orbit of mercury. How precise are they with the other orbits.

Newton's theory is pretty accurate already, the error is only about 1 part in 10⁷.

The perihelion shift applies to all non circular orbits but the effect is largest for Mercury because Mercury is closest to the Sun.

We can actually express equatorial orbits in common form both for GR and Newton's theory:
$$\Large {\frac {{d}^{2}u}{{d\varphi }^{2}}}=1/2\,\mbox {D} \left( f \right) <br /> \left( u \right)$$

Where f(u) is:
$$\Large f \left( u \right) =2\,{\beta}^{2}u+2\,k-{u}^{2}$$
for Newton and

f(u) is:
$$\Large f \left( u \right) =2\,{\beta}^{2}u+2\,k-{u}^{2}+2\,{u}^{3}$$
for GR.

The only difference is an extra term (u is defined here as m/r).

For details see for instance "General Relativity" - Woodhouse, chapter 8.2

With a lot of hand waving we get an approximate advance of:
$$\Large 6\,{\frac {Gm\pi}{r_{{0}}{c}^{2}}}$$
where r₀ is the approximate radius.

 

[John15]

Have looked at the other thread, I personally don't like the cone idea, it is similar to the rubber sheet example unfortunately for me it does not work in 3d space all the cones cancel out leaving a series of expanding spheres and, for example, the Earth ofbits the suns equator not around its top.
Out of interest if the distance in Newton was taken from the shwartzchild radii rather than exact centers would it bring Newton and einstein any closer, it would obviously slightly increase g in Newton especially close to a large body like the sun.
Not sure if the +2u3 above causes an increase or decrease in GR.