I Domain of Applicability of Post-Newtonian Formalism

[ohwilleke]

The parameterized post-Newtonian formalism is an approximation of general relativity that is easier to calculate with that applies in "weak" gravitational fields where objects are "slow" moving.

My question is: "What is the approximate domain of applicability of the post-Newtonian formalism?"

In other words, how weak is "weak" and how slow is "slow" and whether there are other circumstances where the formalism breaks down.

I could imagine that another way of answering the question would be to identify fact patterns where the post-Newtonian formalism makes sense (e.g. perhaps the precession of Mars), and fact patterns where the post-Newtonian formalism doesn't make sense (e.g. perhaps a binary set of black holes spiraling into each other starting at a fairly "short distance").

I see that there are several previous discussions on the post-Newtonian formalism at PF but none seem to address this particular question.

To be clear: I am not asking about the range of post-Newtonian formalism parameters that are consistent with empirical measurement. I'm simply asking when a simplification of GR often used as a calculation short-cut is valid.

 

[PeterDonis]

The parameterized post-Newtonian formalism is an approximation of general relativity

Actually, it's broader than that. The introduction to the Wiki article explicitly distinguishes the "post-Newtonian formalism", which is what you describe, from the "parameterized post-Newtonian formalism", which extends the post-Newtonian formalism to allow it to be used to test GR against other possible theories of gravity, by adding parameters to the equations. The post-Newtonian limit of GR specifies particular values for all of those parameters; but the parameters themselves can be tested experimentally (at the end of the article the results of our best current experimental tests are summarized), so we don't have to assume that those parameters take their GR-specified values; we can test that empirically.

how weak is "weak" and how slow is "slow"

"Weak gravity" means all differences between the actual metric coefficients and the Minkowski metric coefficients are small.

"Slow motion" means all relative motions are at speeds much less than the speed of light.

 

[ohwilleke]

Actually, it's broader than that. The introduction to the Wiki article explicitly distinguishes the "post-Newtonian formalism", which is what you describe, from the "parameterized post-Newtonian formalism", which extends the post-Newtonian formalism to allow it to be used to test GR against other possible theories of gravity, by adding parameters to the equations. The post-Newtonian limit of GR specifies particular values for all of those parameters; but the parameters themselves can be tested experimentally (at the end of the article the results of our best current experimental tests are summarized), so we don't have to assume that those parameters take their GR-specified values; we can test that empirically.

Thanks. You are right and I was really just thinking about the post-Newtonian formalism and not the parameterized generalization of it. My bad.

"Weak gravity" means all differences between the actual metric coefficients and the Minkowski metric coefficients are small.

"Slow motion" means all relative motions are at speeds much less than the speed of light.

With regard to "weak gravity" this is all good and well, but what does this mean, either in terms of numerically or in terms of situations where one or the other would be appropriate? What kind of observables in what kind of units would you even use to assess the situation? Obviously, it would defeat the purpose to calculate the actual metric coefficients to determine if you could save time by approximating them with the post-Newtonian formalism.

With regard to "slow motion", yes I understood that it was relative to "c", but how slow is slow? For example, is 0.1c (i.e. 18,600 miles per second) "fast" or "slow"? What about "0.01c" (i.e. 1,860 miles per second)? I understand that there isn't any exact cutoff, but what would be safe harbors on either side and what kind of fact patterns would present gray areas or would be clearly appropriate or inappropriate?

 

[PeterDonis]

Obviously, it would defeat the purpose to calculate the actual metric coefficients to determine if you could save time by approximating them with the post-Newtonian formalism

You're thinking of it backwards. We know what the Minkowski metric coefficients are. And we know what approximate metric coefficients we are using in our post-Newtonian approximation. If the difference between the two is small, that means the approximation is working ok. (More precisely, if the difference between the two is small, and the post-Newtonian model is still making good predictions for the things we are measuring, then the approximation is working ok.)

how slow is slow?

It depends on how accurate your measurements are and what order you want to carry the post-Newtonian approximation to. There is no single answer.

 

[ohwilleke]

You're thinking of it backwards. We know what the Minkowski metric coefficients are. And we know what approximate metric coefficients we are using in our post-Newtonian approximation. If the difference between the two is small, that means the approximation is working ok. (More precisely, if the difference between the two is small, and the post-Newtonian model is still making good predictions for the things we are measuring, then the approximation is working ok.)

Ah. That makes sense.

 

[PAllen]

It is also true that PPN has been used for regimes well outside its expected validity, with such results verified by numerical relativity. The following famous paper discusses this, characterizing this success as a mystery:
https://arxiv.org/abs/1102.5192

 

[ohwilleke]

It is also true that PPN has been used for regimes well outside its expected validity, with such results verified by numerical relativity. The following famous paper discusses this, characterizing this success as a mystery:
https://arxiv.org/abs/1102.5192

The conclusion of that paper is worth quoting:

Wigner remarked that the effectiveness of mathematics in the natural sciences was mysterious. The unreasonable effectiveness of the post-Newtonian approximation in gravitational physics is no less mysterious. There is no obvious reason to expect PN theory to account so well for the late stage of inspiral and merger of two black holes. The Strong Equivalence Principle of general relativity undoubtedly plays a role, by making the internal structure of the bodies irrelevant until they begin to distort one another tidally. But it does not explain why PN waveforms should agree so well with numerical waveforms when the orbital velocities are almost half the speed of light, or why recoil velocities calculated using PN methods should agree so well with those from numerical methods. Our colleague Robert V. Wagoner once speculated during the 1970s that, because the gravitational redshift effect makes processes near black holes appear slower and “weaker” from the point of view of external observers, the PN approximation should somehow work better than expected, even under such extreme conditions. Because of the redshift effect, “strong” gravity is not as strong as one might think. But nobody has been able to translate Wagoner’s musing into anything quantitative or predictive. And yet the unreasonable effectiveness of postNewtonian theory will likely be an important factor in the anticipated first detection of gravitational waves.