Using relativity to suppress classical inertial acceleration

Question

This is a question from a mathematics student trying to visualize the fact that general relativity is based on a concept of, “identifying gravity with the curvature of spacetime” (sincere apologies for probable physics inaccuracies and useless details).

Suppose an absolute spacetime, a time-dependent mass distribution $p(t)$ and a time-dependent mass distribution $M(t)$, both with smooth trajectories starting at $t=0$ ($t<0$ is not considered here) and momenta low enough so that Newtonian gravity approximates Einsteinian gravity.

Suppose also that total mass $m(p)$ doesn't change over time and that, at all times, $m(p)\ll M(t)$ and $p$ and $M$ are close enough so that the influence of $p$ over the gravitational field is neglected here.

Now denote $\vec{A}(t)$ the acceleration of $p$ at time $t$ and $\vec{A}_{G}(t)$ its component resulting from the gravitational influence of $M$, the goal being to have a pretty arbitrary smooth vector field acting as inertial acceleration on $p$ trajectory.

Under those hypotheses, or similar or better-formulated ones, can we state, with an error controlled by scales of approximations made, that there exists an Einstein metric on spacetime such that the spatial acceleration $\vec{A^{*}}(t)$ of $p$ at time $t$ in the new manifold is the parallel transport of $\vec{A}(t)-\vec{A}_{G}(t)$ from $p(0)$ to $p(t)$ along the trajectory ?

In this case we could write (in a broad sense)$$\vec{A^{*}}=\vec{A}-\vec{A}_{G}$$so the gravity component of acceleration woud have been “replaced” adequately.

Any explanation of any of my misconceptions would be greatly appreciated.

Answer 1

This is the simplest way I know to see the basis of General Relativity. It sacrifices math for simplicity.

See Why can't I do this to get infinite energy? for the basis of General Relativity. This tells you the physical reason why time runs slower near Earth than high above Earth.

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This shows that spacetime is curved.

Space is curved if you don't come back to your starting point when you walk around a square. Or equivalently, you wind up at different points if you walk east-then-north vs north-then-east.

The surface of the Earth is curved in this sense. It doesn't show for a small square. But try a really large square. Start on the equator.

• Walk 1/4 of the way around the world to the east. Turn left and walk 1/4 of the way around the world to the north. You are at the north pole.

• Walk 1/4 of the way around the world to the north. Turn right, and walk 1/4 of the way around the world. (OK, it isn't east because coordinates are weird at the north pole.) But you are on the equator.

Space-time is 4 dimensional, so you get an extra direction you can walk around the block. You can also wait a while.

So trace out this "square" where one side is distance, and the other time. Start at a point above the Earth.

• Have a person at the top wait a bit, then find the point/time a distance X below him at that time.

• Find the point/time a distance X below the top person right now. Have someone at that bottom point wait a bit.

Time is slower at the bottom. In his travel through time, the bottom person passes through the the point/time the top person picks out. But when he does, he isn't done waiting yet.

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Finally, the curvature of spacetime causes gravity.

The easiest example is the deflection of starlight, the effect Einstein proposed as evidence for his theory.

Light from a distant star is a plane wave by the time it arrives at the solar system. If the geometry is just right, part of the wave will skim the surface of the Sun and continue onward to arrive at Earth. Normally light from the Sun would make the starlight invisible. But an eclipse is happening at the same time. The moon is positioned just right to block the sunlight, but not the starlight.

Part of the plane wave skims the Sun, where an observer sitting some distance above the Sun sees time running slowly. Part skims the distant observer, where he sees time running normally. He sees starlight near himself traveling at the speed of light. In time $t$, light travels distance $x$. Light near the surface also travels at the speed of light. But clocks run slower. In the same time $t$, a shorter time will have passed at the surface. Light will travel a shorter distance.

This means that the wavefront will be tilted. The part near the Sun will fall behind. Since light travels perpendicular to the wavefront, light will be deflected.

On Earth, this means we will see the starlight traveling in a slightly different direction than it would have. We see this as an apparent shift in the position of the star.

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The experiment was done. The star did indeed shift, but the shift was twice as big as expected.

Einstein went back to the drawing board and this time showed that there would be distortions in distance as well as time. With this, one can derive a correct theory of gravity that predicts all the effects we measure.