GR & Mercury Precession: Explaining the Diff.

[nikko lyng]

Im curious what is it about space-time or Einsteins field equations which explains Mercurys precession correct while Newton's formula doesn't? So exactly what difference causes GR to explain it that Newton could not?

 

[haushofer]

Basically, GR provides, in the Newtonian limit, an extra term in the potential such that the elliptical trajectory does not close anymore (compared to Newton!) after one rotation. Hence precession.

 

[nikko lyng]

Thanks, where in the field equations can I find this extra term?

 

[pixel]

Im curious what is it about space-time or Einsteins field equations which explains Mercurys precession correct while Newton's formula doesn't?

Newton's law does predict a precession of Mercury's orbit when perturbations due to nearby planets are taken into consideration, but that precession didn't quite agree with observation. Einstein's general relativity provided the remaining amount of precession to match observation.

 

[Jonathan Scott]

Thanks, where in the field equations can I find this extra term?

The extra terms (compared with Newtonian theory) appear in the solution, which in this case is based on the Schwarzschild solution for a static central mass.

If you have access to Misner, Thorne and Wheeler "Gravitation", section 40.5 gives the perihelion shift to post-Newtonian accuracy in terms of a more generalised "PPN" (parameterised post-Newtonian) metric, and then when the specific parameter values are plugged in for General Relativity, the result matches that observed for Mercury.

The overall effect for a more general theory is $(2 - \beta + 2\gamma)/3$ times the observed result for Mercury. For GR, $\beta$ and $\gamma$ are both equal to 1. (For Newtonian theory, $\beta$ is 0 and $\gamma$ is effectively -1, from the assumption that coordinate $c$ is constant, so there is no perihelion precession).

The $\gamma$ parameter describes the curvature of space. The $\beta$ parameter relates to the second-order term in the time dilation, i.e. the value of $\beta$ in the following formula for the gravitational time factor in isotropic coordinates:
$$\sqrt{1 - 2\frac{GM}{rc^2} + 2 \beta \left ( \frac{GM}{rc^2} \right ) ^2}$$

[Edited to correct typos and add brackets for clarity]

 

[nikko lyng]

The extra terms (compared with Newtonian theory) appear in the solution, which in this case is based on the Schwarzschild solution for a static central mass.

If you have access to Misner, Thorne and Wheeler "Gravitation", section 40.5 gives the perihelion shift to post-Newtonian accuracy in terms of a more generalised "PPN" (parameterised post-Newtonian) metric, and then when the specific parameter values are plugged in for General Relativity, the result matches that observed for Mercury.

The overall effect for a more general theory is $(2 - \beta + 2\gamma)/3$ times the observed result for Mercury. For GR, $\beta$ and $\gamma$ are both equal to 1. (For Newtonian theory, $\beta$ is 0 and $\gamma$ is effectively -1, from the assumption that coordinate $c$ is constant, so there is no perihelion precession).

The $\gamma$ parameter describes the curvature of space. The $\beta$ parameter relates to the second-order term in the time dilation, i.e. the value of $\beta$ in the following formula for the gravitational time factor in isotropic coordinates:
$$\sqrt{1 - 2\frac{GM}{rc^2} + 2 \beta \left ( \frac{GM}{rc^2} \right ) ^2}$$

[Edited to correct typos and add brackets for clarity]

Thank you for your answer. So it seems that time dilation and curvature of space-time somehow emerges from the field equations in order to predict reality slightly differently. At least it seems logical that Newton didnt account for time dilation. I suppose all this information came from Einstein assuming that the total momentum and its direction which defines mass equals to how coordinates of space relates to coordinates of time beyond the mass in itself? So this curvature can be interpreted as what coordinates of space relates to what coordinates of time indicating that it could be possible to take new perspectives in order to model the theory with a different ontology than "space-time" or time as a dimension and in a way which is more compatible with quantum mechanics and the rest of physics?

Well I will be banned now for saying that (to make the topic more fundamental than the theories of physics in itself, aka philosophy) :) Thanks again for your kind help it made me feel better since i couldn't stop thinking about it until now :)

 

[PeterDonis]

I suppose all this information came from Einstein assuming that the total momentum and its direction which defines mass equals to how coordinates of space relates to coordinates of time beyond the mass in itself?

I'm not sure what you mean by this, but in general the properties of a given spacetime, i.e., a given solution of the Einstein Field Equation (which is what we are talking about when we talk about how to apply GR to calculating planetary orbits--we are using the Schwarzschild solution of the EFE), are not "assumed", they are calculated from the solution.

So this curvature can be interpreted as what coordinates of space relates to what coordinates of time indicating that it could be possible to take new perspectives in order to model the theory with a different ontology than "space-time" or time as a dimension and in a way which is more compatible with quantum mechanics and the rest of physics?

Any more fundamental theory, even if it had a different "ontology" underneath (such as string theory, or loop quantum gravity, or whatever), would still have to have GR emerge from it in the appropriate limit; otherwise such a more fundamental theory could not explain why GR is so successful within its domain of validity. So I would not expect a more fundamental theory to involve "new perspectives" on solutions to problems like planetary motion, which are well within GR's proven domain of validity. Certainly neither of the ones I mentioned (string theory and LQG) do that; they explain planetary motion by having GR emerge from them in the appropriate limit, and within that domain, GR is not changed at all.

 

[nikko lyng]

I'm not sure what you mean by this, but in general the properties of a given spacetime, i.e., a given solution of the Einstein Field Equation (which is what we are talking about when we talk about how to apply GR to calculating planetary orbits--we are using the Schwarzschild solution of the EFE), are not "assumed", they are calculated from the solution.
Any more fundamental theory, even if it had a different "ontology" underneath (such as string theory, or loop quantum gravity, or whatever), would still have to have GR emerge from it in the appropriate limit; otherwise such a more fundamental theory could not explain why GR is so successful within its domain of validity. So I would not expect a more fundamental theory to involve "new perspectives" on solutions to problems like planetary motion, which are well within GR's proven domain of validity. Certainly neither of the ones I mentioned (string theory and LQG) do that; they explain planetary motion by having GR emerge from them in the appropriate limit, and within that domain, GR is not changed at all.

Yes I agree with that the results of GR would have to emerge from a more fundamental theory. But would time as a dimension be a necessity or could it be substituted by "how coordinates of space relates to coordinates of time" where time is not a dimension?

If you are bored enough you can check my fundamental attempt of explaining physics in terms of properties of consciousness. Modeling physics is in part 3 though (anyone is welcome to email me in the mail given in the article whether they agree or disagree): http://www.scirp.org/Journal/PaperInformation.aspx?PaperID=64011

I didn't get as far as distinguish between forces but I feel like I got quite far in modeling fundamental properties in physics, even the measurement problem in quantum mechanics. I would say the way I defined time in this article explains many fundamental properties in physics in a very simple way.

 

[PeterDonis]

would time as a dimension be a necessity

Spacetime as a 4-dimensional manifold would be a necessity. Whether you call the fourth dimension "time" is a matter of words, not physics.

could it be substituted by "how coordinates of space relates to coordinates of time" where time is not a dimension?

If this is just an alternate way of choosing coordinates in 4-dimensional spacetime, then yes. (There are plenty of cases of coordinate charts with no timelike coordinate.) If it's a way of trying to get by with only 3 dimensions instead of 4, then no.

If you are bored enough you can check my fundamental attempt of explaining physics in terms of properties of consciousness.

Off topic for this thread. Also quite possibly off topic for PF in general, since PF is not intended for discussion of speculative theories or for discussion of philosophy.

 

[nikko lyng]

Spacetime as a 4-dimensional manifold would be a necessity. Whether you call the fourth dimension "time" is a matter of words, not physics.
If this is just an alternate way of choosing coordinates in 4-dimensional spacetime, then yes. (There are plenty of cases of coordinate charts with no timelike coordinate.) If it's a way of trying to get by with only 3 dimensions instead of 4, then no.
Off topic for this thread. Also quite possibly off topic for PF in general, since PF is not intended for discussion of speculative theories or for discussion of philosophy.

Well yeah, maybe one could assume that there are 4 dimensions but that the fourth isn't time? However a 4th dimension implies that it has similar properties to the other three. Are you sure it is not possible that something which is still different could appear to have the properties of a 4th dimension?

 

[PeterDonis]

maybe one could assume that there are 4 dimensions but that the fourth isn't time?

What would this mean? Remember that, as I said above, it's perfectly possible to construct coordinate charts on 4-D spacetime in which none of the coordinates are timelike. But that doesn't mean there are no timelike curves in that spacetime; there are timelike curves in every spacetime.

a 4th dimension implies that it has similar properties to the other three.

Why?

Are you sure it is not possible that something which is still different could appear to have the properties of a 4th dimension?

This looks like just empty speculation to me. There's no point in discussing empty speculation.

In any case, the statement that spacetime is a 4-dimensional manifold has a perfectly definite physical meaning. So does the statement that there are timelike curves in spacetime. These are not hypotheses; they are physical facts that are reflected in the models we use in GR. Any alternate model would also have to account for the same physical facts, so any alternate model would also have to accommodate the existence of a 4-D spacetime with timelike curves in it.

 

[nikko lyng]

What would this mean? Remember that, as I said above, it's perfectly possible to construct coordinate charts on 4-D spacetime in which none of the coordinates are timelike. But that doesn't mean there are no timelike curves in that spacetime; there are timelike curves in every spacetime.

For example there could be a fourth dimension which is not time, but which the other three dimensions curve relative to? This fourth dimension may be proportional to time. Simple as that?

Why?

Because if it does not have any similar properties it would not be logical to name it similar as well i suppose.

This looks like just empty speculation to me. There's no point in discussing empty speculation.

I would call it a fruitful question though. Because considering how the ontology which GR is based on (infinitesimally small space-time) doesn't fit well with anything else in physics than the predictions it makes correctly.

In any case, the statement that spacetime is a 4-dimensional manifold has a perfectly definite physical meaning. So does the statement that there are timelike curves in spacetime. These are not hypotheses; they are physical facts that are reflected in the models we use in GR. Any alternate model would also have to account for the same physical facts, so any alternate model would also have to accommodate the existence of a 4-D spacetime with timelike curves in it.

I agree about that another model would have to predict the same. This is why I am curious of how necessary is it that time is a dimension instead of for example that time is a by product of causation itself. It even seems redundant that time must be a byproduct of causation and a dimension at the same time.

 

[PeterDonis]

For example there could be a fourth dimension which is not time, but which the other three dimensions curve relative to?

You're still missing the point. What does it mean to say the fourth dimension "is not time"? If you try to actually give this a precise meaning, I think you will find you can't. "Time" is not a property of one specific dimension. It's just a property of spacetime as a whole that it has four dimensions and that it contains timelike curves.

Because if it does not have any similar properties it would not be logical to name it similar as well i suppose.

"I suppose" is not an argument.

I would call it a fruitful question though.

There are plenty of people working on possible more fundamental models that would have GR emerge from them in the appropriate limit. I've already mentioned two, string theory and loop quantum gravity. So in that sense the idea is certainly fruitful, yes.

What is not fruitful, IMO, is your particular approach to this idea, which is to throw out vague ordinary language statements instead of trying to construct precise models with precise consequences.

I am curious of how necessary is it that time is a dimension instead of for example that time is a by product of causation itself.

See my comments above about dimensions; "time" is not a dimension, nor is it a property that can be assigned to one particular dimension.

As for time being a "by product of causation", I think the statement "spacetime contains timelike curves" basically captures that. So I don't think the redundancy you are describing is actually there in GR.

 

[nikko lyng]

You're still missing the point. What does it mean to say the fourth dimension "is not time"? If you try to actually give this a precise meaning, I think you will find you can't. "Time" is not a property of one specific dimension. It's just a property of spacetime as a whole that it has four dimensions and that it contains timelike curves.

That's like saying that velocity is not a property of motion. It is a property of s=vt as a whole. I cannot really make more sense of what you're saying here than that.

"I suppose" is not an argument.

Then you are free to remove "i suppose" and then you got one.

There are plenty of people working on possible more fundamental models that would have GR emerge from them in the appropriate limit. I've already mentioned two, string theory and loop quantum gravity. So in that sense the idea is certainly fruitful, yes.

As far as I've heard they failed to, but if they succeeded that's interesting. If you have any links to any sources please share.

What is not fruitful, IMO, is your particular approach to this idea, which is to throw out vague ordinary language statements instead of trying to construct precise models with precise consequences.

I disagree. Besides I have modeled all the laws of physics in the link to my article. It is in fact the simplest model of physics existing today, so I find it strange that you accuse me of not constructing precise models when they precisely predict the maths as well.

See my comments above about dimensions; "time" is not a dimension, nor is it a property that can be assigned to one particular dimension.

All the experts who explain general relativity seem to call time the fourth dimension. So they are wrong? I don't follow what you are trying to say.

As for time being a "by product of causation", I think the statement "spacetime contains timelike curves" basically captures that. So I don't think the redundancy you are describing is actually there in GR.

What do you mean by that space-time has timelike curves?

 

[PeterDonis]

I cannot really make more sense of what you're saying here than that.

Then I suggest spending more time learning the details of GR and how it models things. What I am saying is a fairly basic observation.

As far as I've heard they failed to

I said they were "possible" models that might do this, not they they actually do. Nobody currently has a more fundamental theory that does all of the things we would like it to do. However, the theories I mentioned are at least precise models with precise consequences.

I have modeled all the laws of physics in the link to my article.

I am extremely skeptical of this claim. For some insight as to why, see my last comment in this post.

All the experts who explain general relativity seem to call time the fourth dimension.

Please give quotes from actual textbooks or peer-reviewed papers. Lots of pop science articles on GR call time the fourth dimension, yes, but pop science is not science.

What do you mean by that space-time has timelike curves?

Again, I suggest spending more time--lots more time in view of this comment--learning the details of GR and how it models things. This statement is an extremely basic one in GR. Someone who claims to have modeled "all of the laws of physics" in one single article should not be asking this question; it's the equivalent of someone who claims to have modeled "all possible mathematical systems" in one article asking what the natural numbers are.

 

[PeterDonis]

The original question in the OP has been answered and the thread is degenerating into speculation. Thread closed.