Undergrad General Relativity vs Newtonian Weak Field: G Force?

[e2m2a]

When does GR apply in gravitational fields as opposed to Newtonian gravitation? Another words, what would be the strength of the graviatational field before we abandon Newtonian gravity and start implementing GR? 10 g's, 20 g's?

 

[Ibix]

It depends on how precise you want your results to be. For example, Gravity Probe B measured the pure GR frame dragging effect in Earth orbit, and the GPS has to take GR time dilation into account to keep providing accurate location information. On the other hand, you can lay artillery pieces and sling space probes around the solar system with only Newtonian dynamics (plus radiation pressure for the Pioneer anomaly, which I suppose is technically a relativistic effect). One of the earliest tests of GR was that it correctly predicted the precession of Mercury's perihelion - but for aiming a telescope or a space probe, the 43 seconds of arc per century that the angle of perihelion was incorrect by is utterly irrelevant.

 

[PeterDonis]

what would be the strength of the graviatational field before we abandon Newtonian gravity and start implementing GR? 10 g's, 20 g's?

g's are not the "strength of the field" that is relevant here. There is no single number that describes the "strength of gravity" in general. But for purposes of telling when Newtonian gravity can work, the best quick heuristic is the ratio $R_s / R$, where $R_s$ is the Schwarzschild radius $GM / c^2$ associated with the mass $M$ of the relevant source of gravity (e.g., the Earth, the Sun, etc.) and $R$ is the distance from the center of the source. For situations where Newtonian gravity is a good approximation, this ratio will be much less than 1; the more accurate our measurements are, the smaller it has to be for deviations from Newtonian gravity to be non-observable.

Typical numbers in the solar system for the ratio are:

At the surface of the Earth, using the Earth's mass: $7 \times 10^{-10}$.

At the Earth's distance from the Sun, using the Sun's mass: $1 \times 10^{-8}$.

At the surface of the Sun, using the Sun's mass: $2 \times 10^{-6}$.

 

[haushofer]

The Newtonian limit in GR involves three different (!) limits (which is not stressed properly in most GR-books imo):

* slowly moving objects wrt c
* static grav.fields
* weak fields

I'd say this gives you three parameters which describe the devation from GR.

 

[PeterDonis]

I'd say this gives you three parameters which describe the devation from GR.

The first and third are not independent: they are both governed by the parameter $M / R$ that I described. It most directly describes how weak the fields are, but how weak the fields are determines how fast the orbital motions of objects in the fields are (and those orbital motions are what "slowly moving objects" refers to). Note that the orbital velocity of an object in a circular orbit at radius $R$, in coordinates in which the source is at rest, is $\sqrt{M / R}$.

The second parameter I would describe as "almost static fields"; they don't have to be exactly static, just close enough to it. For example, the field near the Earth is not exactly static, because of the Earth's rotation (and also because of the effects of the Sun and other planets). But Newtonian gravity is a very good approximation near the Earth. One possible parameter for "how static the field is" might be $J / R^2$, where $J$ is the source's angular momentum (in geometric units--this works out to $G/c^3$ times the angular momentum in conventional units, and has units of length squared) and $R$ is the distance from the center of the source. Typical numbers for that ratio are:

At Earth's surface, using Earth's angular momentum: $4 \times 10^{-16}$.

At Earth's distance from Sun, using Sun's angular momentum: $1 \times 10^{-16}$.

At Sun's surface, using Sun's angular momentum per unit mass: $6 \times 10^{-12}$.

 

[pervect]

The first and third are not independent: they are both governed by the parameter $M / R$ that I described.

I don't think I agree. Consider the deflection of light by the sun, for instance. The amount of deflection itself is (I think) proportional to m/r, but the fact that ithe deflection is twice the Newtonian value doesn't depend on the ratio m/r. That's a specific example of the more general principle, why I believe it's necessary to assume low velocities, not just small m/r.

 

[PeterDonis]

The amount of deflection itself is (I think) proportional to m/r, but the fact that ithe deflection is twice the Newtonian value doesn't depend on the ratio m/r.

Hmm. I would say that the deflection of light is an example where the first and third parameters that haushofer gave are in fact independent--weak field but an object not moving slowly. But light cannot be captured in a bound orbit in a field that is weak by this definition, so I think my claim needs to be weakened to the claim that the first and third parameters are not independent for objects that are bound in the field, but they can be for unbound objects.

 

[PAllen]

Hmm. I would say that the deflection of light is an example where the first and third parameters that haushofer gave are in fact independent--weak field but an object not moving slowly. But light cannot be captured in a bound orbit in a field that is weak by this definition, so I think my claim needs to be weakened to the claim that the first and third parameters are not independent for objects that are bound in the field, but they can be for unbound objects.

A crucial example of how these parameters are decoupled for unbound objects is to consider the head on collision of two baseballs at relative velocity c - epsilon. Per Newton, you get a big explosion, with escaping gas. Per GR, you get a black hole (if epsilon is small enough).

 

[e2m2a]

g's are not the "strength of the field" that is relevant here. There is no single number that describes the "strength of gravity" in general. But for purposes of telling when Newtonian gravity can work, the best quick heuristic is the ratio $R_s / R$, where $R_s$ is the Schwarzschild radius $GM / c^2$ associated with the mass $M$ of the relevant source of gravity (e.g., the Earth, the Sun, etc.) and $R$ is the distance from the center of the source. For situations where Newtonian gravity is a good approximation, this ratio will be much less than 1; the more accurate our measurements are, the smaller it has to be for deviations from Newtonian gravity to be non-observable.

Typical numbers in the solar system for the ratio are:

At the surface of the Earth, using the Earth's mass: $7 \times 10^{-10}$.

At the Earth's distance from the Sun, using the Sun's mass: $1 \times 10^{-8}$.

At the surface of the Sun, using the Sun's mass: $2 \times 10^{-6}$.

Thanks. This gives me a ballpark estimate.

 

[haushofer]

The first and third are not independent: they are both governed by the parameter $M / R$ that I described. It most directly describes how weak the fields are, but how weak the fields are determines how fast the orbital motions of objects in the fields are (and those orbital motions are what "slowly moving objects" refers to). Note that the orbital velocity of an object in a circular orbit at radius $R$, in coordinates in which the source

Yes. For bound systems $\{Mm\}$ one has in the Newtonian regime

$mv^2 \sim \frac{GMm}{r}$ giving

$\frac{v^2}{c^2} \sim \frac{r_s}{r}$ where $r_s$ is the Schwarzschild radius. But what I meant was that one can perfectly consider other external forces, such that you have a slowly moving mass in a strong gravitational field, or a fast moving object in a weak grav.field (as Einstein's Original calculation of the deflection of light). These limits can be disentangled.

Btw, in a paper by Dautcourt,

https://inis.iaea.org/search/search.aspx?orig_q=RN:23072570

it is shown how one can consider the speed of light in GR as contraction parameter to give the Newtonian limit. The idea is that the coupling becomes weak, the speed wrt light becomes small, and all time derivatives of the metric are multiplied by a factor $c^{-1}$, which gives ultimately Newton-Cartan theory. To be honest, I consider this approach a bit messy.