# Newton vs Einstein

1. Dec 20, 2012

### 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.

2. Dec 20, 2012

### Staff: Mentor

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.

Yes, I would say that is right.

That definitely has something to do with it.

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

3. Dec 20, 2012

### andrien

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

4. Dec 20, 2012

### haushofer

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.

5. Dec 20, 2012

### Mentz114

He seems to like local translational invariance ...

6. Dec 20, 2012

### haushofer

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

7. Dec 20, 2012

### Mentz114

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.

8. Dec 20, 2012

### K^2

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.

9. Dec 21, 2012

### andrien

there is no yang-mills for gravity.

10. Dec 21, 2012

### haushofer

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.

11. Dec 21, 2012

### 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.

12. Dec 21, 2012

### 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.

13. Dec 21, 2012

### 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 :)

14. Dec 22, 2012

### Naty1

A perhaps minor point:

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

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

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.

Last edited: Dec 22, 2012
15. Dec 23, 2012

### DiracPool

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?

16. Dec 23, 2012

### Mentz114

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

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.

17. Dec 23, 2012

### pervect

Staff Emeritus
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.

Last edited: Dec 23, 2012
18. Dec 23, 2012

### Naty1

Pervect: Thanks or the PPN link posted above.

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

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

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'.....

19. Dec 23, 2012

### Naty1

In post #2, Dalespam posted:

by DiracPool
Dalespam:

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."

20. Dec 23, 2012

### DiracPool

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.