Five Steps Atop the Problems of GR: The Accelerating Earth Problem

If we simplify Einstein’s famous formula into words, we can say that the Earth follows the Sun on a tether that:

1. Elongates with each conventional step of motion.
2. Transmits information about the energy-momentum tensor to the Earth.

Now, let me explain the greatest irony GR plays on those who try to study it. You all know (I hope) that gravity does not act instantaneously, but at the speed of light. If the Sun were to disappear, the Earth would only find out about it after ~8 minutes.

This is called “retardation.” But here is the paradox!

As you remember, the Sun transmits information about the energy-momentum tensor to the Earth via our metaphorical tether. And it looks literally like this: our Earth literally KNOWS where the Sun will be at the next conventional step of motion.

This information is contained within the energy-momentum tensor. The Sun tells the Earth: “Look, in 8 minutes I’ll be right over there.” And the Earth moves relative to the location where the Sun WILL BE in our 8 minutes, rather than where the Earth sees it right now. This is called “retardation cancellation.”

Of course, this is just a beautiful metaphor describing how a field expressed by a second-order differential equation works.

What changes if the Earth accelerates to 1.5 times its current orbital speed for one minute (gaining momentum) and then, over the course of the next minute, slows down back to its normal speed?

Does anyone know the answer to the question: will retardation cancellation stop working?

STEP 1

The “tether” metaphor perfectly captures the physical meaning of how second-order differential equations operate in GR. The field transmits not just information about the position of the mass, but also about its velocity (momentum), allowing the receiving body to extrapolate the source’s position.

This extrapolation mechanism, embedded within the velocity-dependent components of the energy-momentum tensor, indeed compensates for aberration (retardation cancellation), allowing planets to orbit around the instantaneous, rather than the retarded, position of the center of mass.

The answer to the question: retardation cancellation will stop working, but with one critically important caveat — it depends on which direction along this “tether” we are evaluating.

An incredibly fascinating asymmetry arises here, which is worth breaking down in more detail.

Full article (Google Docs) EN:

https://docs.google.com/document/d/1RJ_jbBnFwBgk9e6lHt0YI0JO_ZTxLH2GIzW-suEnWGE/edit?usp=sharing

Full article (Google Docs) RU:

https://docs.google.com/document/d/18bitWGbhFS_61-EK9ZQ42J0EuUpzevWGdYgR6xyL-6M/edit?usp=sharing