## tisdag 13 februari 2024

### Speed of Light vs Speed of Gravity: Maxwell vs Newton vs Aether

Propagation of light in vacuum is described by Maxwell's equations expressed in terms of an electric field $E(x,t)$ and a magnetic field $B(x,t)$ where $x=(x_1,x_2,x_3)$ is the coordinate of an Euclidean spatial coordinate system $X$ and $t$ is a time coordinate, with dot representing differentiation with respect to time:

• $\dot B + \nabla\times E =0$  and $\dot E - \nabla\times B =0$    (1)
where $\nabla =(\frac{\partial}{\partial x_1},\frac{\partial}{\partial x_2},\frac{\partial}{\partial x_3})$, and the speed of light $c$ is normalised to 1. Observation of the speed of light in the system $X$ by an observer $O$, thus gives the value 1. Since today the meter is defined in terms of light second, $c=1$ is an agreement and not a law of physics.

So far so good, but what about the speed of the $X$? Relative to what?

Suppose a different observer $O^\prime$ relies on the same Maxwell's equations (1) expressed in a different coordinate system $X^\prime$ moving with relative constant speed $v$ vs $X$, as a so called inertial system. Analysis in Many-Minds Relativity Chap 18 shows that $O$ and $O^\prime$ will agree up to a precision scaling with $v^2$. For human observers this means a precision of $10^{-9}$, which may be enough for all practical purposes. This means that (1) is Galilean invariant up to a precision of $v^2$. More precisely both observers will consider the speed of light to be exactly 1, since they agree to use the same Maxwell's equations (1).

To use Maxwell's equations (1) requires specification of the coordinate system and the natural choice is to lock the coordinate system to the observation apparatus and so allow the possibility of different apparatus moving with respect to each other, with observations agreeing up to $v^2$ with $v<<1$ for human observers. Many-Minds Relativity expands the scope to $v<1$.

Sum up: Maxwell's equations requires specification of spatial coordinate system. Different observers may use different inertial coordinate systems moving with relative speed $v$ and will then agree up to  $v^2$, and exactly agree on the speed of light. The choice of a specific coordinate system effectively represents a choice of an aether, so there are as many aethers as coordinate systems.

Let us now turn to Newtonian gravitation described by
• $\Delta\phi =\rho$            (2)
where $\phi (x,t)$ is gravitational potential and $\rho (x,t)$ mass density, and $\Delta$ is the Laplacian in the coordinates $x$ of a Euclidean coordinate system $X$. We understand that (2) is exactly Galilean invariant since (2) reads the same independent of any motion of $X$ with constant velocity, because no time derivative is involved. All inertial coordinate systems thus give the same description of gravitation.

In the sense of Einstein it means that (2) satisfies Einstein's definition of a (perfect fundamental) physical law, as a law of physics which takes exactly the same form in all inertial systems (as an expression of Galilean invariance).

The speed of gravity in (2) is formally infinite if $\rho$ is viewed to be primary from which $\phi$ is created by formally instant action at distance, which is unthinkable. Viewing instead $\phi$ as primary with $\rho$ the result of differentiation replaces instant action at distance by instant local action, which is thinkable. It is also possible to view (2) as a side condition without specifying cause-effect. In the latter perspectives the notion of speed of gravity is not needed.

Conclusion:
1. Newton's law of gravitation (2) is Galilean invariant an so is a thinkable prefect physical law for which a notion of speed of gravity is not needed. No aether enters the discussion.
2. Maxwell's equations is Galilean invariant up to $v^2$, where for human observers $v^2<10^{-9}$, with $c=1$ acting as an agreement. Each choice of coordinate system represents and aether.
3. The speed of light serves a fundamental role, while a speed of gravity is not needed.
4. Massless electromagnetics and mass gravitation are fundamentally different, which contradicts Einstein. Search of gravitons as gravitational analog of photons is fruitless.
5. There is no need to modify Newtonian mechanics, and so Einstein's relativity serves no purpose.
6. A Grand Unified Theory as Maxwell + Newton is readily available.