# Einstein gravity versus Newton's gravity

What's the basic difference between the gravity as seen by Einstein, and that by Newton?

-
youtube.com/watch?v=DdC0QN6f3G4 –  elfmotat Mar 20 '13 at 22:45
Related: physics.stackexchange.com/q/37926/2451 and links therein. –  Qmechanic Mar 20 '13 at 22:46

Often people get confused by the additional complication that Newtonian and Einsteinian gravity are often discussed in different mathematical formalisms. This can tend to obscure the physical differences. If you are game for the mathematics then Misner, Thorne and Wheeler (check it out of a library or get it second hand unless you are really serious about this business) has a wonderful chapter which puts both theories side by side in the same language (differential geometry). The key difference is that Newtonian gravity has a privileged separation of spacetime into space and time, whereas Einsteinian gravity just has spacetime.

Edit: to be absolutely clear, Newtonian gravity can be written as spacetime curvature! This is counter to the common statements about the novel thing in GR. The key difference is that Newtonian gravity has extra absolute structures that GR does not have: absolute time and space, a preferred separation of spacetime into time and spatial parts, absolute simultaneity, and a curved connection that is not the special one derived from a spacetime metric (Christoffel).

In mathematical form:

$$\begin{array}{ll} R_{00} = 4\pi\rho;\text{all others vanish},& \ \text{Newtonian} \\ R_{\mu\nu}-\frac{1}{2} g_{\mu\nu} R = 8\pi G T_{\mu\nu}, & \ \text{Einsteinian} \end{array}$$

with a few other relations I've not written (see MTW chapter 12 for details).

A consequence of the formalism is that the Newtonian equation is a constraint equation - it does not describe a propagating degree of freedom. No gravitational waves, gravitons etc. No speed of light limit for gravity. All matter has an instantaneous gravitational effect on all other matter. This is different in GR since the field equation is a wave equation which describes the propagation of gravitational disturbances from one point to another at the speed of light.

What GR has that Newton does not is a spacetime metric of Lorentzian signature. This metric has a privileged role in that all other structures (connections, curvatures, etc.) are derived from it. There is essentially nothing else to Einstein gravity. That is why it is so elegant in the geometrical formalism. This metric actually comes from special relativity. But the metric was a fixed structure in SR, almost similar to the absolute time and space of Newton (don't tell anyone I said this). The new thing in general relativity is that Einstein lets the metric "flap around" so to speak - to change from place to place and time to time in response to what matter is doing.

-
Re your edit: Yes, but isn't it the case that the OP asks specifically of the difference as seen by Einstein and Newton? Yes, Newton's theory can be written in geometric language (and I don't mean to take away from this illuminating exercise), but is it the case that Newton "saw" gravity this way? –  Alfred Centauri Mar 21 '13 at 2:05
@AlfredCentauri I'm not sure that's what the OP meant by "see." Anyway, I really doubt Newton saw gravity in this way since the concepts of curved geometries weren't around at the time. I'm not an expert on the history of Einstein's thought process, and even less so on Newton's. But I can say a thing or two about what their theories mean physically, and I think it's relevant that what is commonly stated to be the difference between them really isn't, even if maybe Einstein saw it that way (I doubt he did for long if ever he did - he obviously understood his own theory pretty well!). –  Michael Brown Mar 21 '13 at 2:23
+1 For being right about a subtle concept, though I was tempted to -1 for the sole reason that you recommended MTW to a 16-year-old ;) –  Chris White Mar 21 '13 at 3:55
@ChrisWhite Lol, I didn't realize she was 16. Well, some people are precocious. :) And if she is 16 and asking about this then maybe she can't understand MTW yet, but she's well on her way. Samama Fahim, if you're reading this comment and didn't understand my answer don't be discouraged. It takes most people years to understand this stuff. You're asking good questions. And if you did understand it... wow. Just wow! :) –  Michael Brown Mar 21 '13 at 3:59

In at least one basic respect, both general relativity (gravity according to Einstein) and Newtonian gravity are similar; both describe gravity as a gravitational field on a space. In other words, they are both classical field theories.

In the case of general relativity, that field is a pseudo-Riemannian metric $g_{\mu\nu}$ on the space, and the space is a 4-dimensional topological space called spacetime, while in the case of Newtonian gravity, the field is a vector field (if you are describing it by the gravitational acceleration $\mathbf g$) or a scalar field (if you are describing it by the gravitational potential $\Phi$) on three-dimensional Euclidean space.

In the case of general relativity, the gravitational field tells you the geometry of spacetime, and it's the curvature of this geometry that particles "ineract with" in when they move around. The gravitational field is determined by the energy-momentum content of spacetime through Einstein's Equations.

In the case of Newtonian gravity, the gravitational field tells you the acceleration that a particle would feel at any given point in space, but in contrast to general relativity, the geometry of the space itself is not altered by the sources of gravity (masses in this case).

In a highly simplified nutshell:

General relativity describes gravity as spacetime curvature while Newtonian gravity describes it as something living on top of a static space with no curvature.

-
Have you read chapter 12 of Misner, Thorne and Wheeler? You'd probably like it. Newtonian gravity can be put in the same formalism as general relativity. The difference between the two is rather interesting and non-obvious once it is a fair contest with them side by side in the same language. :) –  Michael Brown Mar 21 '13 at 0:52
Oh cool no I haven't I'll check it out! Thanks. –  joshphysics Mar 21 '13 at 7:50

What's the basic difference between the gravity as seen by Einstein, and that by Newton?

Newtonian gravity is an instantaneous force, i.e., action at a distance, coupled to gravitational mass (conceptually different from inertial mass).

General Relativity is a local theory (no action at a distance). Einsteinian gravity is the curvature of spacetime and the coupling is between mass-energy and geometry; "matter tells spacetime how to curve, spacetime tells matter how to move".

-

In Newtonian Gravity, space is like 3-dimensional graph paper, and objects are moving through space at an absolute time. The objects path curves because a force is present. Without that force (gravity) objects will continue in a straight line.

Where Newton's idea of Gravity is 3D space with Time a constant, Einstein conceived of 4D space -- called space-time. In this structure, time is not absolute but a dimension or variable in the structure such that (x,y,z,t) exist for a given event. Objects moving through space-time curve not because they are "pulled" by the force of gravity, but because they are taking the shortest distance through curved space-time.

Amazingly, for Einstein, you can have curved space-time without matter so an object might start moving in curves even if nothing is present.

-

One more aspect. Roughly speaking, in General Relativity "energy" is attracted $(E/c^2)$, while in Newtonian gravity - only mass. And there is no time dilation in Newtonian gravity.

-