General Relativity and Newtonian Mechanics

[alchemistf9]

Hello,

I wanted to know how Einstein's General Relativity and its equations simplify to Newton's Universal Law. Einstein's equation is obviously a much more generalized version of Newton's law and gives much more accurate predictions that Newton's laws even if we don't think of Newton's gravity as spacetime warping.

On a side note, I've seen simulations of space time bending and demonstrations describing it as a trampoline with a ball in the middle causing a curvature of the space around it... if this analogy is accurate then how "stretchy" is the fabric of spacetime? The Sun is 99% the mass of our entire solar system so it must bend spacetime strongly and as far as Pluto to keep all the planets in orbit.

 

[Nugatory]

I wanted to know how Einstein's General Relativity and its equations simplify to Newton's Universal Law. Einstein's equation is obviously a much more generalized version of Newton's law and gives much more accurate predictions that Newton's laws even if we don't think of Newton's gravity as spacetime warping.

Einstein's equations don't ever simplify to Newton's law - they'll always yield a slightly different result. However, when the tidal effects (Newtonian language) and curvature (GR language) are not very great, the difference between the two predicted results is too small to notice. This isn't a coincidence; we have centuries of observational data that agrees with Newton's law to the limits of experimental accuracy, so if GR didn't make predictions that also agreed to within the limits of experimental accuracy it would be wrong and we wouldn't bother with it.

You can google for "General relativity weak field approximation" to see how the calculations work out.

On a side note, I've seen simulations of space time bending and demonstrations describing it as a trampoline with a ball in the middle causing a curvature of the space around it... if this analogy is accurate then how "stretchy" is the fabric of spacetime? The Sun is 99% the mass of our entire solar system so it must bend spacetime strongly and as far as Pluto to keep all the planets in orbit.

That trampoline analogy is hopelessly misleading. If you search this forum for "rubber sheet" you'll find many posts about why it cannot be trusted, and you will also find some videos by member @A.T. that provide a much better explanation.

As for whether the mass of the sun is curving spacetime "strongly"? That's somewhat in the eye of the beholder; the sun is about 99% of the mass of the solar system, but it's also only about .000000001% of the mass of our galaxy.

 

[alchemistf9]

Einstein's equations don't ever simplify to Newton's law - they'll always yield a slightly different result. However, when the tidal effects (Newtonian language) and curvature (GR language) are not very great, the difference between the two predicted results is too small to notice. This isn't a coincidence; we have centuries of observational data that agrees with Newton's law to the limits of experimental accuracy, so if GR didn't make predictions that also agreed to within the limits of experimental accuracy it would be wrong and we wouldn't bother with it.

You can google for "General relativity weak field approximation" to see how the calculations work out.That trampoline analogy is hopelessly misleading. If you search this forum for "rubber sheet" you'll find many posts about why it cannot be trusted, and you will also find some videos by member @A.T. that provide a much better explanation.

As for whether the mass of the sun is curving spacetime "strongly"? That's somewhat in the eye of the beholder; the sun is about 99% of the mass of the solar system, but it's also only about .000000001% of the mass of our galaxy.

Can you give me the links to the videos? I'm just tryin to understand visually as best as possible how the Sun curves spacetime so much that al the planets orbit it. Also, is there a specific reason as to why the orbit of Mercury processes as much as it does compared to the other planets?

 

[A.T.]

Can you give me the links to the videos? I'm just tryin to understand visually as best as possible how the Sun curves spacetime so much that al the planets orbit it. Also, is there a specific reason as to why the orbit of Mercury processes as much as it does compared to the other planets?

https://www.physicsforums.com/threads/3d-gravity.885685/#post-5569815

 

[Nugatory]

Also, is there a specific reason as to why the orbit of Mercury processes as much as it does compared to the other planets?

Mercury is closest to the sun so the gravitational forces and curvature are greater. Thus, the deviation between the predictions of GR and of Newtonian theory is greater, and it's large enough to measure. Even then, it's not much in absolute terms: roughly 40 seconds of arc in a century.

 

[Ibix]

Can you give me the links to the videos? I'm just tryin to understand visually as best as possible how the Sun curves spacetime so much that al the planets orbit it. Also, is there a specific reason as to why the orbit of Mercury processes as much as it does compared to the other planets?

The problem with Mercury isn't that its orbit precesses, it's that the precession doesn't match Newtonian predictions. It turns out that Newtonian gravity is a weak field approximation to general relativity. The Sun's gravity is least weak closer to it, so it is here that the shortcomings of Newtonian gravity are most obvious.

 

[alchemistf9]

Why do physicists speak of spacetime as a "fabric" that bends or warps when a mass is introduced if it's a misleading analogy? Is it that it's too difficult to visualize for the average Joe that it's the best analogy they can come up with?

 

[vanhees71]

Well, usually physicists are much less poetic than popular-science-book writers, and the question how to "visualize" abstract mathematical concepts is very individual. E.g., for me even to have a visual/intuitive idea about four-dimensional spaces is very limited (often I struggle even with 3D). So for me I rather think about gravity more as about any other interactions, i.e., it's a relativistic field theory described by a mathematical object called a 2nd-rank tensor field. The physical feature that distinguishes gravity from the other fundamental forces is the validity of the equivalence principle, which says that for all local phenomena, if observed in a freely falling cabin (like the international space station) the laws of special relativity are valid as long as these phenomena are extended only over a very small space-time interval. In other words, in each space-time point I can define a local inertial frame of reference, where free particles are not accelerated, i.e., locally I can always neglect (approximately) gravity when I fall freely in the gravitational field. It was Einstein's ingenious insight that this equivalence principle is "most intuitively" described by a curved space-time, mathematically a socalled pseudo-Riemannian manifold, and that the fundamental laws should be independent of the choice any (local) reference frame.

Now all this bases of an older mathematical theory, namely non-Euclidean geometry which reaches back to Gauss, who realized that you can have a consistent geometry without having necessarily a Euclidean one. He showed that you can describe a curved surface (like a cylinder or a sphere) in 3D Euclidean space completely without ever making references to the embedding three-dimensional space. You can determine the lengths, the utmost straight lines (geodesics), curvature, etc. without ever referring to the embedding space. So some being (like an intelligent ant) not aware of the embedding 3D Euclidean space would develop a geometry on the surface, which does not fulfill all the axioms of Euclidean geometry. This idea was further developed and generalized to more than 2 dimensions by Riemann, Felix Klein, and others leading to a generalize analytical geometry on abstract manifolds, and general-relativistic spacetime again is a slight extension of these Riemannian spaces to pseudo-Riemannian spaces.

You can always locally (i.e., for a not too large region of spacetime) introduce reference frames, according to which an observer "splits" spacetime into time and 3D "space". That's, however an observer dependent split. Nevertheless for any such observer it leads to a valid description of what's going on. In this sense you can still think about gravity as about any other interaction, and Einsteins Ansatz to describe it as a pseudo-Riemannian manifold, where free falling test particles move along geodesics as a mathematical powerful tool to write down the correct equations of motion for all kinds of quantities (like the trajectories of particles subject to gravity and/or other forces, the equations of the electromagnetic field, etc.). Particularly there is a nearly unique field equation for the gravitational field itself. In the geometric picture the gravitational field as a potential that is described as a 2nd rank symmetric tensor field which at the same time is the pseudo-metric of the pseudo-Riemannian spacetime manifold, which means it determines what the proper distance in time and space for any observer, independent of his/her choice of reference frames are.

The most important consequence of this quite restricted possibilities of the dynamics is that for the theory to be consistent, the sources of gravity is not (only) mass but all kinds of distributions energy, momentum and stress of matter and radiation. Mathematically speaking the source is the energy-momentum tensor of matter and radiation coupling with a universal strength (given by Newton's gravitational constant $G$) to the gravitational field (aka pseudometric). Applied to massive freely falling test particles (described effectively by the energy-momentum tensor of an ideal fluid in the spirit of continuum mechanics) this implies that no matter of which material they consist and which mass they have (note that mass is only part of the energy of matter) they all are following identical trajectories due to gravity, i.e., the universality of the coupling between the energy-momentum-stress tensor of matter to the gravitational field leads back to the equivalence principle, proving that Einstein's theory indeed fulfills this fundamental postulate on the nature of gravity. It tells you also more: The electromagnetic field is massless, i.e., its proper (or invariant) mass is strictly zero. Nevertheless it has energy and momentum, which means it is influence by gravity too. That's why Einstein immediately concluded when thinking about the consequences of his theory that also light rays (i.e., the propagation of electromagnetic waves in the sense of ray optics, which is an approximation to full wave optics) are no longer straight in a 3D space according to an observer far away from the source of gravity. This is the famous deflection of the star light which has been first observed by the solar-eclipse mission lead by Eddington in 1919. Within the (by the way not that small) uncertainties it confirmed Einstein's prediction, which made Einstein to the first pop star of theoretical physics :-).

 

[Nugatory]

Why do physicists speak of spacetime as a "fabric" that bends or warps when a mass is introduced if it's a misleading analogy? Is it that it's too difficult to visualize for the average Joe that it's the best analogy they can come up with?

Actually not many physicists use that fabric analogy, even when speaking to laypeople (unfortunately the ones who do are among the most prolific and visible populizers). It's one of those things that the popular press picked up years ago, and science writers have been repeating it ever since. If you're writing a non-technical article on gravitational waves (choosing this example because the LIGO success has made it big news right now) you need a sentence or so about the thing that is waving... and it's hard to do that without either oversimplifying by implying the existence of some physical fabric or totally failing to explain by saying things like "second-rank tensor field".

 

[Ibix]

...or as xkcd puts it: https://xkcd.com/895/

 

[alchemistf9]

...or as xkcd puts it: https://xkcd.com/895/

LMAO that was great! Hahaha