According to Einstein, mass curves spacetime and objects in the nearby field tends to travel in the shortest possible path to reach their heavier counterparts. My question is was not Newton's interpretation better; i.e. considering gravity as a force that acts on $n$ masses and tends to attract? I know that photons don't hold up with this definition, but how can spacetime be visualized (saw those trampoline models) in real life? How to define motions of celestial objects by considering an invisible 'fabric'? Any help will be appreciated.

# [Physics] Conceptual problem with general relativity

general-relativitygravitypopular-science

#### Related Solutions

Eric's answer is not really correct (or at least not complete). For instance, it doesn't tell you anything about the motion of two comparably heavy bodies (and indeed this problem is very hard in GR, in stark contrast to the Newtonian case). So let me make his statements a bit more precise.

The correct approach is to treat the Newtonian gravity as a perturbation of the flat Minkowski space-time. One writes $g = \eta + h$ for the metric of this space-time ($\eta$ being Minkowski metric and $h$ being the perturbation that encodes curvature of the space-time) and linearize the theory in $h$. By doing this one actually obtains a lot more than just Newtonian gravity, namely gravitomagnetism, in which one can also investigate dynamical properties of the space-time not included in the Newtonian picture. In particular the propagation of gravitational waves.

Now, to recover Newtonian gravity we have to make one more approximation. Just realize that Newtonian gravity is not relativistic, i.e. it violates finite speed of light. But if we assume that $h$ changes only slowly and make calculations we will find out that the perturbation metric $h$ encodes the Newtonian field potential $\Phi$ and that the space-time is curved in precisely the way to reproduce the Newtonian gravity. Or rather (from the modern perspective): Newtonian picture is indeed a correct low-speed, almost-flat description of GR.

**Update**: I made a number of mistakes in the original version of this post, although I think all the big ideas are right. I tried to fix everything, but I wouldn't be at all surprised if I've made additional errors.

I'm pretty sure that this is a very hard problem! I think I know how to get started, but I doubt I can finish.

I'll work in Schwarzschild coordinates, with $c=1$, Schwarzschild radius $R$, and $(+---)$ signature, so the metric is $$ ds^2=(1-R/r)dt^2-(1-R/r)^{-1}dr^2-r^2(d\theta^2+\sin^2\theta\,d\phi^2). $$ All the action can be taken to lie in the equatorial plane $\theta=\pi/2$.

For a particle traveling on a geodesic in this geometry, the energy-like conserved quantity (i.e., the one arising from time-translation invariance of the metric) is $u_0$ where $u$ is the 4-velocity. I'll call this quantity $E$ (it's really energy per unit mass): $$ E=\left(1-{R\over r}\right)\left(dt\over d\tau\right). $$

I'm going to *assume* that this quantity is conserved even for our particle whizzing along in the tube. I'm pretty sure this is the correct generalization of the assumption that the constraint forces due to the tube do no work. I think you could prove this by looking at the physics in a local inertial reference frame in which the tube is at rest as the particle whizzes by. In that frame, the above energy conservation law is equivalent to the statement that the particle's speed is constant, which follows from a special-relativistic analysis in that frame, because the tube pushes in a direction perpendicular to the velocity.

Next we use the fact that the four-velocity has unit norm: $$ 1=u_\mu u^\mu=(1-R/r)\dot t^2-(1-R/r)^{-1}\dot r^2-r^2\dot\phi^2, $$ where dots mean $d/d\tau$.

Divide through by $\dot t^2$: $$ \dot t^{-2}=1-{R\over r}-\left(1-{R\over r}\right)^{-1}\left({dr\over dt}\right)^2-r^2\left({d\phi\over dt}\right)^2. $$ Express $\dot t$ in terms of the energy, and rearrange: $$ \left(1-{R\over r}\right)^{-1}\left(dr\over dt\right)^2+r^2\left(d\phi\over dt\right)^2 =1-{R \over r}-\left(1-R/r\over E\right)^2. $$ Say the particle starts from rest at position $r_0$. Then $E=(1-R/r_0)^{1/2}$. So $$ dt=\sqrt{(1-R/r)^{-1}(dr/d\phi)^2+r^2\over 1-R/r-(1-R/r)^2/(1-R/r_0)}\,d\phi. $$ If our initial points are $(r_0,0)$ and $(r_0,\alpha$), then the quantity we want to minimize is $$ t=\int_0^\alpha \sqrt{(1-R/r)^{-1}(dr/d\phi)^2+r^2\over 1-R/r-(1-R/r)^2/(1-R/r_0)}\,d\phi. $$ You can in principle use standard calculus of variations techniques from here to get $r(\phi)$.

That's enough for me! You said in the comments that you'd be happy with just the functional. Are you happy?

## Best Answer

The trampoline models do not show spacetime; they show space at one instant of time. To be precise, they offer an attempt to visualize space by taking a 2-dimensional cross-section (e.g. the equatorial plane) and then showing how spatial distances are affected by gravity by plotting a surface such that the distances along the surface match the distances in the cross-section through the gravity-affected space.

This visualization of space does offer some good intuition, but unfortunately it is not much use at understanding the idea of a

geodesicor 'straightest possible line' in the temporal direction. For that you need a diagram showing time as well, and such a diagram is not so easy to draw. What I think people working in this area do is use the spatial diagram to get a feel for the notion of a spatial geodesic (the shortest spatial line between two points at some given time) and then mostly trust the algebra when they calculate timelike geodesics. These are the lines in the temporal direction that show how things move when they are moving solely under gravity.To get an intuition about these timelike geodesics, picture the spatial diagram but flatten it out, without forgetting that the distances are distorted really, and then allow the vertical direction to represent time. A timelike geodesic extends upwards and turns towards the central axis. For a circular orbit it would be a helix. Imagine lots of little tick marks on this line, representing the ticking of a clock moving along it. If you fix the two ends of this line and then pull the middle of the line outward a little, there are fewer clock ticks along it because the clock has to move faster along the line and it gets a time dilation associated with this motion. If you push the middle of the line inwards a little, so that the clock takes a shortcut to its destination, then it can move more slowly, but now there is a gravitational time dilation that makes it tick slower on average. The line actually followed by the falling clock is the one which makes a compromise between these two effects and thus has the highest number of clock ticks between the given start and end events.

So there is one attempt at visualization. I am aware that it all seems rather abstract but in the end of course we have to go with the theory that matches experimental observation. But in this case there is also an added feature: it is the feeling that the theory has an extraordinary beauty in and of itself. The very fact that we do not need to mention the concept of force is itself to do with the fact that we can consider the whole description in geometric terms. If one day you get to study this more fully then you will be able to appreciate the beauty more fully.

postscriptIf you ask "why? why does the falling clock follow the line with maximum proper time?" then one way to answer is to focus on each tiny segment of the line. The answer is that each tiny segment just goes straight ahead! But how can lots of straight segments add up to a curved line? For that your best answer is to think about beetles walking around on the surface of a sphere. A beetle walks in a "straight" line when the legs on the two sides of its body move through the same distance. But two beetles setting off from the south pole of a sphere in two different direction, and walking "in a straight line" like this, will find that their lines meet up again at the north pole. This illustrates the notion that a sequence of segments that do not themselves turn to the right nor to the left nevertheless make up a non-trivial overall line if the space (or the spacetime) is itself warped or curved.