Explaining Perihelion Shift of Mercury in GR Theory

[Herbascious J]

TL;DR Summary

Is there some fundamental/simple explanation as to why the perihelion shift of the orbit of Mercury can be attributed to General Relativity?

Recently, when reading an entry about Mercury's perihelion shift, someone mentioned a "hand-wavy" explanation as to why GR predicts the orbit so precisely. I was wondering if there was some elementary way to expound on what he was saying. Fundamentally, the comment said something to the effect that the curvature of space-time itself was contributing to the gravitational potential and so the equation was not exactly Newton's familiar equation. Does this mean that the gravitational field surrounding the orbiting bodies somehow itself emits a gravitatitonal field of it's own and does this continue in some kind of mild, cascading, propagation of gravity fields?

 

[A.T.]

Fundamentally, the comment said something to the effect that the curvature of space-time itself was contributing to the gravitational potential and so the equation was not exactly Newton's familiar equation.

This effect is related to the curvature of space itself. This is visualized here (book pages 171-181, document pages 184-194):
https://archive.org/details/L.EpsteinRelativityVisualizedelemTxt1994Insight/page/n183/mode/2up

 

[Herbascious J]

This effect is related to the curvature of space itself. This is visualized here (book pages 171-181, document pages 184-194):
https://archive.org/details/L.EpsteinRelativityVisualizedelemTxt1994Insight/page/n183/mode/2up

So, if I'm to understand the exercise in the above example, the reason the perihelion is shifting is due to geometry? By "creating a cone like bump" somehow the elliptical-like orbital path is shifting slightly as it goes around, similar to the small lost segment of the circle that is being buckled into a cone. Let's say a very small arc-second of the circle is missing allowing it to become conical in geometry and that would represent the amount the elliptical orbit is sliding around each pass? Is it then purely a geometry thing, and there is nothing about the gravitational field somehow having a contribution to the gravitational field because it is a source of energy or something?

 

[PeterDonis]

an entry about Mercury's perihelion shift

Can you give a link?

 

[A.T.]

...the reason the perihelion is shifting is due to geometry?

Everything in GR is "due to geometry". But these pages discus just one aspect of the geometry (the spatial part), which creates just minor effects.

The main gravitational effects require to include the time dimension. See the previous chapter 10:
https://archive.org/details/L.EpsteinRelativityVisualizedelemTxt1994Insight/page/n157/mode/2up

Is it then purely a geometry thing, and there is nothing about the gravitational field somehow having a contribution to the gravitational field because it is a source of energy or something?

The energy/mass distribution determines the space-time geometry.

 

[Herbascious J]

Can you give a link?

https://physics.stackexchange.com/a/26410

The second answer down the page. He mentions an explanation at the end.

 

[PeterDonis]

https://physics.stackexchange.com/a/26410

The second answer down the page. He mentions an explanation at the end.

Ok, thanks.

Does this mean that the gravitational field surrounding the orbiting bodies somehow itself emits a gravitatitonal field of it's own and does this continue in some kind of mild, cascading, propagation of gravity fields?

Not really. What it means is that the "gravitational field" of the Sun is not entirely describable in terms of a Newtonian "potential"--i.e., the effect of the field on the motion of an object is not entirely describable in terms of a Newtonian "force" that depends only on distance from the Sun. The effect also depends on the speed of the object relative to the Sun. One way of looking at this is that the Sun's gravitational field has a "magnetic" component as well as the "electric" component that is the familiar Newtonian field.

 

[Herbascious J]

Ok, thanks.
Not really. What it means is that the "gravitational field" of the Sun is not entirely describable in terms of a Newtonian "potential"--i.e., the effect of the field on the motion of an object is not entirely describable in terms of a Newtonian "force" that depends only on distance from the Sun. The effect also depends on the speed of the object relative to the Sun. One way of looking at this is that the Sun's gravitational field has a "magnetic" component as well as the "electric" component that is the familiar Newtonian field.

I hope I am not taking your explanation out of context here. I cannot help but think of Special Relativity when I read this. Is the reason that the speed of the object changes the gravitational field is because the object's mass increases due to it's approaching the speed of light? The speed would directly increase the energy/mass content of the object and therefore it's gravitational effect on space-time?

 

[A.T.]

I cannot help but think of Special Relativity when I read this. Is the reason that the speed of the object changes the gravitational field is because the object's mass increases due to it's approaching the speed of light?

No, this not a good way to think about it. The "mass increase" itself is a misleading idea within SR, and relating it to GR makes no sense, because it is a frame dependent effect.

In the GR the relationship between mass/energy and the geometry of space time is rather complex:
https://en.wikipedia.org/wiki/Einstein_field_equations

 

[Herbascious J]

No, this not a good way to think about it. The "mass increase" itself is a misleading idea within SR, and relating it to GR makes no sense, because it is a frame dependent effect.

In the GR the relationship between mass/energy and the geometry of space time is rather complex:
https://en.wikipedia.org/wiki/Einstein_field_equations

Ok, I see. That actually seems cleaner, albeit more complicated. In a final attempt to wrap my head around this, is it safe to say that the velocity of an object has an impact on the geometry of spacetime relative to itself and nearby objects? An example would be a comet traveling at two different speeds would have slightly different gravitational fields in proximity to a passing star?

 

[PeterDonis]

Is the reason that the speed of the object changes the gravitational field is because the object's mass increases due to it's approaching the speed of light?

No. The object that is orbiting has no effect on the spacetime geometry at all. (Strictly speaking, the object will have some effect on the spacetime geometry in its immediate vicinity--for example, satellites can orbit Mercury--but this has no effect on the orbit of the object, such as Mercury, around the Sun.)

 

[PeterDonis]

is it safe to say that the velocity of an object has an impact on the geometry of spacetime relative to itself and nearby objects?

No. See my previous post just now.

An example would be a comet traveling at two different speeds would have slightly different gravitational fields in proximity to a passing star?

No.

 

[A.T.]

In a final attempt to wrap my head around this, is it safe to say that the velocity of an object has an impact on the geometry of spacetime relative to itself and nearby objects?

These would be frame dependent effects, because they cannot exist in the rest frame of the object itself. The gravitational field might look different in different frames.

 

[phyzguy]

Summary:: Is there some fundamental/simple explanation as to why the perihelion shift of the orbit of Mercury can be attributed to General Relativity?

One thing to be aware of is that the perihelion of Mercury's orbit shifts even without General Relativity. In fact, more than 90% of the perihelion shift can be understood through Newtonian gravitation alone. It is only the small, unexplained residual perihelion shift that was explained by General Relativity