Comparing Newton & Einstein's Gravity: What Makes GR Better?

[DiracPool]

Could someone explain to me what specifically distinguishes Einsteins more advanced treatment of gravity over Newtons? Here’s what I (think I) know. Newton described gravity as a “force” of attraction between two bodies or masses. That force was given by the G constant times the two masses divided by the square of the radius between the two masses. The force of gravity was then modeled as a corresponding acceleration vector between the bodies. The classical Lagrangian using the principal of least action reproduces Newtons same equations of motion, but instead of assuming a” force of attraction” between the two bodies, models the relation between the two masses as a minimized “path.”

Now we get to Einstein. Einstein eschewed the idea of attraction and instead saw gravity as a process whereby object-masses moved along a physically ill-defined but mathematically compelling “geodesic” which traced out a complex curved space-time in the vicinity of massive bodies. Mathematically the curving of this spacetime and the geodesics that arise from it are found through the continuous redefinition of the local coordinate axis due to the local mass energy density of the system in question. This value is given by the stress-energy tensor. The particular “shape,” then, of the local coordinate axis is given by the Einstein tensor. The value of each of these tensors rely on each other in real time so as to make the equations non-linear. Do I have this right?

Anyway, my question is what is it about the Reimenian geometric approach of GR that gives it it’s advantage over the classical model, which works well enough for everyday modeling that we can use it solely to send people to the moon and back? Is the answer in the nonlinearity of the solutions, that the motions of the bodies are continuously updated in real time. Is it that GR incorporates energy and pressure into the stress-energy tensor whereas Newtons equations just include mass? Does it have something to do with the geometrical approach over a force of attraction approach? Also, I’ve read that it has something to do with the Taylor series expansion, where the higher order terms give you something that the lower terms don’t, which is where you get Newton’s equations? Finally, where does one want to go through the ordeal of using GR to model a system where Newton won’t work. The places I’m familiar with are black holes, GPS, the eclipse thing, and the precession of Mercury. But why and how does this give us a better solution here. Thanks.

 

[Dale]

Now we get to Einstein. Einstein eschewed the idea of attraction and instead saw gravity as a process whereby object-masses moved along a physically ill-defined but mathematically compelling “geodesic” which traced out a complex curved space-time in the vicinity of massive bodies.

A geodesic is physically well-defined. An object physically travels along a geodesic whenever an attached accelerometer would read 0. I.e. geodesic = free fall motion.

Mathematically the curving of this spacetime and the geodesics that arise from it are found through the continuous redefinition of the local coordinate axis due to the local mass energy density of the system in question. This value is given by the stress-energy tensor. The particular “shape,” then, of the local coordinate axis is given by the Einstein tensor. The value of each of these tensors rely on each other in real time so as to make the equations non-linear. Do I have this right?

Yes, I would say that is right.

Is it that GR incorporates energy and pressure into the stress-energy tensor whereas Newtons equations just include mass?

That definitely has something to do with it.

Does it have something to do with the geometrical approach over a force of attraction approach?

I don't think so. You can formulate Newtonian gravity as a geometrical theory also. This is called Newton-Cartan gravity.

 

[andrien]

Could someone explain to me what specifically distinguishes Einsteins more advanced treatment of gravity over Newtons?

Advance of perihelion of mercury,deflection of light twice as large as predicted by Newton theory.

 

[haushofer]

But why and how does this give us a better solution here. Thanks.

GR happens to coincide better with experiments and measurements. Why? Perhaps God doesn't like scalar potentials, I don't know.

With other words: physics does not answer such "why"-questions.

 

[Mentz114]

GR happens to coincide better with experiments and measurements. Why? Perhaps God doesn't like scalar potentials, I don't know.

With other words: physics does not answer such "why"-questions.

He seems to like local translational invariance ...

 

[haushofer]

He seems to like local translational invariance ...

Since our topic about the meaning of general covariance I would restate that into "He seems to like massless self-interacting spin-2 particles", as general covariance is not a defining property of gravity :P

 

[Mentz114]

Since our topic about the meaning of general covariance I would restate that into "He seems to like massless self-interacting spin-2 particles", as general covariance is not a defining property of gravity :P

Hmm. I was referring (very obliquely) to the fact that gravity is a gauge field that arises from enforcing local translational symmetry. Which is completely off-topic. Whoops.

 

[K^2]

Hmm. I was referring (very obliquely) to the fact that gravity is a gauge field that arises from enforcing local translational symmetry. Which is completely off-topic. Whoops.

Not exactly. If it was, we'd have quantum gravity already. The conserved charge is right. The covariant derivative you get is right. But you can't write down a QFT Lagrangian that includes gravitational field as a gauge field, and that's a problem.

But you're right. That is getting a bit off topic.

 

[andrien]

Hmm. I was referring (very obliquely) to the fact that gravity is a gauge field that arises from enforcing local translational symmetry. Which is completely off-topic. Whoops.

there is no yang-mills for gravity.

 

[haushofer]

there is no yang-mills for gravity.

No Yang-Mills, but one can obtain GR by gauging the Poincaré group. The difference with YM-theorie lies in the dynamics: one gauge curvature (those of the translations) is put to zero in order to make the spin connection dependent and remove the local translations, and the dynamics is given by the action, which is linear in the remaining (Lorentz) gauge curvature.

 

[haushofer]

I think at the fundamental level the big difference between GR and Newton is that GR is basically a theory of massless self-interacting spin-2 (wrt the Lorentz group) particles, while the Newtonian theory is a theory of massless nonself-interacting spin-0 (wrt the homogeneous part of the Galilei group: rotations and boosts) particles. All the other properties should be consequences of this, I would say.

 

[dextercioby]

You mean massive particles in Newtonian gravity. And the way you formulated your sentence makes people think of quantum field theory, which has nothing to do with Newtonian gravity.

 

[haushofer]

No, i mean massless particles; i regard the Newton potential as a massless galilei-scalar field. What one basically does is to take the non-rel. and weak field limit of the Einstein-Hilbert action. That such a theory would be non-renormalizible is clear, as GR is non-ren. Maybe the word particle is a bit deceiving; i m not sure to which extent one can regard such a theory as an effective (quantum) field theory and interpret the excitations of the fields as particles. So if that's what you mean, i fully agree :)

 

[Naty1]

A perhaps minor point:

Dircpool :

Mathematically the curving of this spacetime and the geodesics that arise from it are found through the continuous redefinition of the local coordinate axis due to the local mass energy density of the system in question. This value is given by the stress-energy tensor.

Here is a quote I kept from a discussion in these forums...from [Misner, Thorne, Wheeler]:

...nowhere has a precise definition of the term “gravitational field” been given --- nor will one be given. Many different mathematical entities are associated with gravitation; the metric, the Riemann curvature tensor, the curvature scalar … Each of these plays an important role in gravitation
theory, and none is so much more central than the others that it deserves the name “gravitational field.”

I'll leave it to you experts to decide the relationship between 'field' and curvature...

Could someone explain to me what specifically distinguishes Einsteins more advanced treatment of gravity over Newtons?

How about "GPS"...really!...[LOL]

When I think of Einstein's gravitational theory, I now think 'geometry of spacetime has physical consequences' and especially 'cosmology'.

For example,

...Geometric circumstances create real particles e.g. Hawking radiation at BH horizon and Unruh radiation caused by acceleration or felt by an accelerated observer. So it seems that expansion of geometry itself, especially inflation, can produce matter!

...How about horizons in general, as Hubble, Event, and Black Hole for examples...the geometric solutions are amazing..Schwarszschild, Rindler, etc,

...And how about the fact that in an isotropic and homogeneous universe, geometry, leads to an unstable cosmos...expanding or contracting but not static. Where would we be with Hubble's observational finding of not only expansion but accelerated expansion without the underling theory of Einstein...

...and gravitational time dilation and gravitational redshift...

I've never read about Newton-Cartan gravity mentioned by Dalespam...is it non-relativistic...must be, right??...anyway , I suspect EFE go way beyond what would be suggested by a non relativistic theory.

 

[DiracPool]

Where would we be with Hubble's observational finding of not only expansion but accelerated expansion without the underling theory of Einstein...

Thank all of you for the replies and the discussion. I guess my central question that many comments have positioned around but haven't really addressed has to do with the neccessity of a Geometrical approach over a Mechanics approach to address the motion of bodies in a gravitational field. We live in 3 dimensional space or 4-d spacetime, and special relativity can handle 4-d spacetime just fine. From what we can experience of the motion of bodiles in our 4d world, a single cartesian coordinate system + time can specify any point we can image and the trajectories between those points. Why is it only for gravity that we have to continuously redefine the basis coordinate system? What is it about undergoing this labor intensive process of tensor calculus that allows is to calibrate the GPS and precession of Mercury that good old classical physics-special relativity can't handle?

 

[Mentz114]

Why is it only for gravity that we have to continuously redefine the basis coordinate system?

I don't know what you mean by this. We get good results with static GR spacetimes.

What is it about undergoing this labor intensive process of tensor calculus that allows is to calibrate the GPS and precession of Mercury that good old classical physics-special relativity can't handle?

The Good Old Newtonian Equations do not predict precessing ellipsoidal orbits, and that's what we got. So something else is needed to account for observations in our back yard. With the good algebraic software and powerful PCs available today, it is not so labour intensive any more.

 

[pervect]

Thank all of you for the replies and the discussion. I guess my central question that many comments have positioned around but haven't really addressed has to do with the neccessity of a Geometrical approach over a Mechanics approach to address the motion of bodies in a gravitational field. We live in 3 dimensional space or 4-d spacetime, and special relativity can handle 4-d spacetime just fine. From what we can experience of the motion of bodiles in our 4d world, a single cartesian coordinate system + time can specify any point we can image and the trajectories between those points. Why is it only for gravity that we have to continuously redefine the basis coordinate system? What is it about undergoing this labor intensive process of tensor calculus that allows is to calibrate the GPS and precession of Mercury that good old classical physics-special relativity can't handle?

I'm not sure what you mean by "continuously redefining the basis coordinate system". I wouldn't describe GR in those terms.

What SR can't handle is the fact that space-time is curved. With the usual time-slicing, space, as well as space-time, is curved.

Light bending (and I think Mercury's perihelion advance as well) can in particular be attributed to the purely spatial part of space-time curvature. In particular the PPN parameter gamma has to be nonzero to explain observed light deflection results. ((PPN : http://en.wikipedia.org/w/index.php?title=Parameterized_post-Newtonian_formalism&oldid=523184874). Gamma is a measure of space-curvature - and light bending isn't senstive to the other PPN parameters, just gamma.

According to wiki, current measurments place gamma equal to it's predicted-by-GR value of 1 within an error of about +/- 25 parts per million.

 

[Naty1]

Pervect: Thanks or the PPN link posted above.

I never saw all that nor discussions in a link within that article:

A table comparing PPN parameters for 23 theories of gravity can be found in Alternatives to general relativity#PPN parameters for a range of theories.

http://en.wikipedia.org/wiki/Altern...Testing_of_alternatives_to_general_relativity

What is it about undergoing this labor intensive process of tensor calculus that allows is to calibrate the GPS and precession of Mercury that good old classical physics-special relativity can't handle?

So am I correct in assuming Diracpool's question now becomes 'why one approach works when 23 others aren't quite so good?
If so, we are back to the questions that have been discussed, but not answered, in these forums previously: Why does ANY of our man made math seem to describe the world around us? Why does some math work but not others? The only possible answer I have seen so far that registered with me is the possibility that
if there is a multiverse and maybe our 'other' math would apply there.

What I still find incredible is that 'Einstein intuition' seems to have allowed him to pick an 'off the shelf' math that is SO close overall and perhaps virtually perfect on large scales. And Diracpool may be asking in the future about a more accurate theory of 'quantum gravity'...

 

[Naty1]

In post #2, Dalespam posted:

by DiracPool

Mathematically the curving of this spacetime and the geodesics that arise from it are found through the continuous redefinition of the local coordinate axis due to the local mass energy density of the system in question. ... Do I have this right?

Dalespam:

Yes, I would say that is right.

Ok, so now I am sorry I did not ask what that 'redfinition' meant; I attributed it to my lack of understanding of mathematical details and interpretations...

Has this do do with the characteristics of a smooth Riemannian manifold or metric space??...a metric space with geometric interpretations?edit: Diracpool: "..the neccessity of a Geometrical approach over a Mechanics approach to address the motion of bodies in a gravitational field."

If I understand what you are positing, I'd reply "I don't think GR is the final answer, it's the best one we have. It does not take us back to the big bang, nor to the 'singularity' within a black hole...so we need a
better theory...like quantum gravity."

 

[DiracPool]

What I still find incredible is that 'Einstein intuition' seems to have allowed him to pick an 'off the shelf' math that is SO close overall and perhaps virtually perfect on large scales.

Yes, it is amazing, and so iconoclastic for the time. Really, who would think not only to adopt a putatively unrelated "fringe" geometry maths, but to be so confident in its utility as to have the faith to stick with its development for upwards of a decade. And apparently get it so right that we are still "agog" over it, as evidenced right here. I still think that there's a more parsimonious model for gravity that will usurp Einsteins one day, and that will probably be the one that unifies the maths with QM. However, GR will always be notable if for no other reason than how the complexity of the model gave such accurate results. One could argue that there's beauty in complexity as well as parsimony.

 

[Dale]

I still think that there's a more parsimonious model for gravity that will usurp Einsteins one day

I doubt that very much. GR has only two free parameters, the cosmological constant and the Newtonian gravitational constant. It is hard to see how any other model will get more accurate results with fewer. At least, all of the current competitive theories that have not already been falsified have more.

However, GR will always be notable if for no other reason than how the complexity of the model gave such accurate results. One could argue that there's beauty in complexity as well as parsimony.

I think you misunderstand what makes a model simple scientifically. It isn't the mathematical notation, but the number of free parameters in the theory. GR has only two, so it is a very simple and parsimonious model. That is precisely why it dominates over other current competitive theories.

 

[pervect]

I still think that there's a more parsimonious model for gravity that will usurp Einsteins one day, and that will probably be the one that unifies the maths with QM. However, GR will always be notable if for no other reason than how the complexity of the model gave such accurate results. One could argue that there's beauty in complexity as well as parsimony.

Pretty much any quantum replacement theory that replaces GR is going to have to wind up looking very similar in the classical limit.

With the possible exception of redefining our notion of distance (and time) there isn't any way to escape the fact that the geometry of space-time is curved. Consider gravitational time dilation, for instance. There isn't any sort of "force" that's going to make a clock at a higher altitude run faster than one at a lower altitude. Something more fundamental is at work here, something that affects many different sort of clocks all in the same way.

 

[AlephZero]

The Good Old Newtonian Equations do not predict precessing ellipsoidal orbits, and that's what we got. So something else is needed to account for observations in our back yard. With the good algebraic software and powerful PCs available today, it is not so labour intensive any more.

"People today need "good algebraic software and powerful PCs" to do that? I remember doing the "precession of Mercury" example in a first-year maths degree course on dynamics and SR, with nothing more than pencil and paper.

 

[pervect]

"People today need "good algebraic software and powerful PCs" to do that? I remember doing the "precession of Mercury" example in a first-year maths degree course on dynamics and SR, with nothing more than pencil and paper.

If you make the appropriate approximations, it's not too bad. Calculating the Chrsitoffel symbols would be more work than I cared to do by hand (you could always look them up I suppose). The real need for symbolic algebra shows up when you try to calculating the Riemann / Ricci tensors et al from the metric.

 

[Naty1]

Here is another distinction between Newton and Einstein I came across reading my notes on a related subject:

Analogously to Newtonian mechanics, the four-momentum of a system is the sum of the four-momenta of its constituent particles, and the four-momentum of the system is conserved across any interaction, including particle annihilation and creation interactions. This means that a system's energy (timelike component of four-momentum), momentum (spacelike component of four-momentum), and mass ("length" of four-momentum) are also conserved and you get one conservation law which unifies three separate conservation laws from classical mechanics. To me it is one of the most elegant and compelling facets of relativity.

from pervect... [Be careful what you say, I may quote you!] [LOL]

 

[Naty1]

And another I just stumbled upon:

...the Austrian physicists Josef Lense and Hans Thirring,... predicted that the rotation of a massive object would distort the spacetime metric, making the orbit of a nearby test particle precess. This does not happen in Newtonian mechanics for which the gravitational field of a body depends only on its mass, not on its rotation. The Lense–Thirring effect is very small—about one part in a few trillion.

http://en.wikipedia.org/wiki/Frame-dragging

Also the rotating massive object in free fall does not following the same geodesic as when non spinning.

 

[DiracPool]

This does not happen in Newtonian mechanics for which the gravitational field of a body depends only on its mass, not on its rotation.

Ah, thar you go Naty, 25 posts and finally what I was looking for. A good quote is worth a thousand pictures. And you got quite a few quotes in your quiver I see.

I wonder if Einstein knew this when he set about the problem of calculating the precession of Mercury with his new field equations. Or if he just sort of blindly tried his new model with the unexplained anamolies of the day to see if it worked. Anyone know?

 

[George Jones]

Yes, it is amazing, and so iconoclastic for the time. Really, who would think not only to adopt a putatively unrelated "fringe" geometry maths, but to be so confident in its utility as to have the faith to stick with its development for upwards of a decade. And apparently get it so right that we are still "agog" over it, as evidenced right here. I still think that there's a more parsimonious model for gravity that will usurp Einsteins one day, and that will probably be the one that unifies the maths with QM. However, GR will always be notable if for no other reason than how the complexity of the model gave such accurate results. One could argue that there's beauty in complexity as well as parsimony.

This is possible, but I think that it is likely to be wrong for at least two reasons.

First, since they move us further from everyday experience, new theories tend have greater technical complexity and/or a greater level of abstractness than the theories that they "replace".

Secondly, current non-relativistic quantum mechanics is already more complex than general relativity.

In my opinion, students could find physics courses in general relativity easier than courses in quantum mechanics. I think that students become more familiar with quantum mechanics because they spend more time studying it.

For example, when I was a student, I:

saw bits of special relativity stuck here and there into a few courses;

did not have the opportunity to take any lecture courses in general relativity;

was required to take three semesters of quantum mechanics as an undergrad and two semesters of advanced quantum mechanics as a grad student;

was required to take two semesters of linear algebra, which gives the flavour of much of the mathematics of quantum mechanics;

was not required to take any maths courses that give the flavour of the mathematics used in general relativity.

Because of the importance and widespread applicability of quantum mechanics, my programme offered much more opportunity to learn quantum mechanics than to learn relativity.

If physics students spent as much time studying general relativity and its mathematical background (say 4 or 5 semesters) as they spend studying quantum mechanics and its mathematical background, then general relativity would be understood by possibly millions of people. I understand why students spend much less time studying relativity than they spend studying quantum theory, and I am not necessarily saying that students should spend more time studying relativity (see the post above by Haelfix), but I do think that this time difference is a big part of the reason that general relativity still has a bit of a reputation.

Fortunately, there are many more good technical books on general relativity (pedagogical, advanced, physical, mathematical, etc.) available now than were available 25 years ago.

Time to put a myth to bed.

*GR is very complicated if you do it "properly" and it is very unlikely that you will get the mathematical background as part of you undergrad math courses.

*General relativity. As others have noted, the math is a bit on the advanced side even for the typical senior physics major.

The mathematics of non-relativistic quantum mechanics is, in my opinion, more difficult than the mathematics of general relativity. Students acquire more facility with the mathematics of quantum mechanics because they spend more time studying it.

At the level of mathematics taught by physicists, the mathematics of non-relativisitic quantum mechanics is somewhat more difficult than the mathematics of general relativity. Typically, an undergrad physics major is introduced to quantum mechanics in a Modern Physics course, and then takes two more semesters of quantum mechanics. In a one=semester general relativity course, the techniques and mathematics of general relativity are presented at light speed, and this perpetuates the myth that the mathematics is difficult. If the techniques and mathematics of general relativity were spread out over 2+ semesters, I don't think that things would seem nearly so difficult.

At the level of honest mathematics, functional analysis, the mathematics of non-relativisitic quantum mechanics is substantially more difficult than the differential geometry used in general relativity. For example, if operators $A$ and $B$ satisfy the canonical commutation relation $\left[ A , B \right] = i \hbar$, then at least one of $A$ and $B$ must be unbounded. Say it is $A$. Then, by the Hellinger-Toeplitz theorem, if $A$ is self-adjoint, the domain of physical observable $A$ cannot be all of Hilbert space! Also, the spectral decomposition for $A$ will be given by the the spectral theorem for unbounded self-adjoint operators. It would be crazy, if not impossible, to teach quantum mechanics this way!

 

[Naty1]

Ah, thar you go Naty, 25 posts and finally what I was looking for. A good quote is worth a thousand pictures. And you got quite a few quotes in your quiver I see.

At my age I can't remember all the stuff I'd like! When I find an explanation that creates an 'aha moment', into my notes she goes!

 

[DiracPool]

Fortunately, there are many more good technical books on general relativity (pedagogical, advanced, physical, mathematical, etc.) available now than were available 25 years ago.

There may be a few good books out there, but try to find some video tutor-age on it. I've done quite a bit of searching on the web, and have found only two sources that go into any detail, and one of those two sources isn't even a focused GR treatment, its a treatment on tensor analysis only. For those who are interested, the GR treatment is Lenny Susskinds course on the Stanford channel, and the other is the guy from digital-university.org. If anybody knows of any other, please let me know.

So we have two sources to choose from, and even these aren't very accessible to someone with a fairly decent math background watching them cold.

Now take QM, as George points out rightly. Here we have literally hundreds of video tutor treatments by many dozens of amateur and not so amateur presenters. So I think the proof is in the pudding here, and it's not plum pudding, I mind you.