tisdag 20 februari 2024

Speed of Gravity in a Static Gravitational Field?

To save General Relativity GR assuming that the speed of gravity is finite equal to the speed of light from collapse when confronted with observations apparently requiring a very much larger speed of gravity, it is commonly stated that in a static gravitational field there is no effect of time delay from finite speed of of propagation. And so common wisdom claims that there is no contradiction between GR and observations conforming to a speed of gravity much larger than the speed of light. 

It is this convincing? No problem in GR from finite speed of gravity? Let us see: A gravitational model with finite speed of propagation $c$ in a Newtonian approximation of GR takes the form

  • $\frac{1}{c^2}\ddot\phi -\Delta\phi =\rho $,       (*)
where $\phi (x,t)$ is the gravitational potential corresponding to a mass distribution $\rho (x,t)$, where $x$ is a Euclidean space coordinate, $t$ a time coordinate and the dot signifies differentiation in time. 
Now a static gravitational potential is characterised by $\ddot\phi =\dot\phi =0$ and so the value of $c$ can be anything, in particular as large as desired even larger than the speed of light without changing anything. In other words it is meaningless to speak about speed of gravity in a static gravitational field. 

To state that in a static gravitational field in GR there is no effect of finite speed of gravity does not make sense. There is no speed at all. 

Further, gravitational fields are not static, not even between the Sun and Jupiter, and so this case lacks interest. 

Yet in GR the speed of gravity is viewed to be finite = speed of light c, which requires a theory of quantum gravity to explain finite speed. But no theory of quantum gravity has been found despite intense search for 100 years. Further, gravitational waves in GR are viewed to require merge of black holes to appear... 

The idea of a finite speed of gravity = speed of light is the main road block to a Theory of Everything ToE combining Newton, Maxwell and Schrödinger. What would happen if we simply remove the block by replacing Einstein by Newton? What would be missed? Nothing? And then?


måndag 19 februari 2024

Akelius, AcadeMedia och DigiMat/BodySoul

Roger Akelius lade förra veckan ett bud på Mellby Gårds innehav på 22.4 % av aktierna i AcadeMedia med en premie på 22.6 %. Budet accepterades förstås direkt av Mellby Gård, men stoppades sedan under helgen av AcadeMedias övriga aktieägare. Orsaken var att Roger utryckt tankar om statligt finansierade friskolor vs privat finansierade, samt att hans främsta ambition är att erbjuda en god skola för många, inte att tjäna en massa pengar på skattefinansierad verksamhet, vilket alltså inte gillades.  

Roger har en uttalad mycket vällovlig ambition att utveckla pedagogik och undervisning, inte minst av skolans matematik, som verkar vara i linje med det reformprogram jag länge drivit tillsammans med medarbetare under parollen Body and Soul samt DigiMat.

Jag har sökt kontakt med Roger för att presentera vårt reformprogram och kommer att meddela resultat. 

torsdag 15 februari 2024

Emergence vs Playing Dice

Follow the emergence of structure when dropping a stone in a pond  in this code from Model Workshop Leibniz World of Math exhibiting a very rich world of emerging from simple basic laws with playing dice.

The wisdom of modern physicists since 100 years is that the Schrödinger wave function of quantum mechanics supposedly describing atom physics, represents a probability distribution as if electrons and protons forming atoms play dice or roulette to evolve in time from which the World emerges with all its amazing features. 

In other words (connecting to the previous post), modern physicists embrace an idea of:

  • Emergence of complex coordinated ordered phenomena from playing dice. 
But is it possible that order can emerge from playing dice? This question was addressed in the book The Dice Man by George Cockcroft and the answer was No. It does not work to play dice to take the decisions required to go through life. It ends in chaos without order. 

The same result is to be expected concerning the physics of atoms forming the world. The idea of atoms playing dice lacks scientific logic. An atom is an elementary simple thing and as such, cannot include a roulette wheel/ball as something complex and so unpredictable. An atom does not have the capacity to play roulette! RealQM offers an new Schrödinger equation as a continuum model in 3 space dimension which is free of roulette statistics. 

The emergence of a surface wave of water comes from coordinated motion of water particles reacting to forces without the help of any dice playing. Emergence of patterns in physical systems is the result of increasing difference while playing dice represents decreasing difference. 

In real physics both play a role: Increasing difference without limit leads to break-down. Without difference no structures. This aspect is investigated in setting accessible for a general audience in The Clock and the Arrow, 

There is a connection to Darwin's theory of Evolution as emergence of complex living organisms from playing dice combined with selection of fittest. This is a very simple theory and as such cannot explain much. In particular, there is evidence that evolution of life is driven by environmental pressure as if there is presence of some form of intelligent design. 

onsdag 14 februari 2024

Macroscopics Emerging from Microscopics (or the other way around)

Fluid flow/electromagnetics shows emergence of large scale effects in the form of vortices and waves from small scale interaction of atoms and molecules, as an expression of 

  • Macroscopics =  Emergent Phenomenon of Microscopics.       (E)
To give a meaning to (E) involves upscaling of a mathematical model of microscopics/quantum mechanics into a model for macroscopics/fluid flow in a bottom-up process

The standard Schrödinger equation for quantum mechanics is not well suited for upscaling since it is a multidimensional probabilistic model, while fluid flow is described by Navier-Stokes equations and electromagnetics by Maxwell's equations, as deterministic 3d continuum models. 

RealQM is based on a different Schrödinger equation as a deterministic 3d continuum model with seamless coupling to macroscopics of fluid flow/electromagnetics. 

The trauma troubling modern physics since 100 years is the perceived incompatibility of multidimensional probabilistic quantum mechanics and Newton's theory of gravitation as a deterministic 3d continuum model, as well as electromagnetics. The trauma gets even worse replacing Newton by Einstein into the present crisis of modern physics. 

RealQM as a deterministic 3d continuum of model of microscopics is compatible with Newton's theory of gravitation as a deterministic 3d continuum model of macroscopics, which is not at all traumatic. Even better, it appears as a unified field theory for quantum mechanics + gravitation/electromagnetics.  

We may then ask if gravitation is an emergent phenomenon of RealQM? Or does it not matter? 

On microscopic scale gravitation is a very very weak force, and so it appears to be very difficult to find the origin of macroscale gravitation in some form of quantum gravity, because the required upscaling is of size $10^{40}$. 

In this situation it may be more fruitful to turn the telescope around and view the mass $\rho$ of an electron to be given as $\Delta\phi$ of a macroscopic gravitational potential $\phi$ in a top-down process as explored in recent posts with label New view on gravitation. 

In both bottom-up and top-down it is helpful if top and bottom are described by the same mathematical model (3d deterministic continuum model). This is possible with RealQM.

Particles vs Fields vs Standard Model

The Standard Model of particle physics describes atomic physics in terms of 

  • fermions as matter particles like electrons and protons 
  • bosons as force carriers like (massless) photons and gluons.
While matter particles as electrons and protons carrying charge intuitively appear to connect to some reality, the idea of photon as carrier of electromagnetic force is more difficult to grasp. In fact, in all of electromagnetics so well described by Maxwell's equations, photons as carrier of electromagnetic force do not appear. 

In particular the electric force appears as the gradient $\nabla\phi$ of an electric potential $\phi (x,t)$ as a continuous function or field depending on a Euclidean space coordinate $x$ and $t$ a time coordinate. 

Similarly, in Newtonian gravitation, the gravitational force appears as the gradient of a gravitational potential field. There is here no need to worry about a carrier of gravitational force in the form of some hypothetical particle named graviton, which has evaded all form of detection. 

In field theories there is thus no need to introduce particles as force carriers, since forces come out as gradients of fields. 

In particular, the idea of photon as carrier of electric force has no role to serve in electromagnetics described by Maxwell's equations. 

Confronted with this fact, a modern physicist would say that particle theory and field theory in fact are just the same with a particle simply a fluctuation of a field and so to say that the photon is the carrier of electric force makes perfect sense. 

In particular, modern physics insists that both blackbody radiation and the photoelectricity must be understood in terms of a stream of photons as particles of light, and then not in terms of fields even if particle and field is the same. 

Computational BlackBody Radiation offers an alternative Maxwellian field theory, where radiative heat transfer appears as a resonance phenomenon and photoelectricity is a threshold phenomenon, both free of particles.  

In summary, there does not seem to be any real need to view a photon to be a particle carrying electromagnetic force, and so the whole idea of fundamental particles as force carriers may lack sound reason. Yet it is a fundamental part of the Standard Model and so may be a suspect behind the crisis of modern physics witnessed by leading physicists.

tisdag 13 februari 2024

Grand Unified Theory: 3 Attempts

Einstein spent the last half of his life searching for a unified field theory as an extension of his General Theory of Relativity GR from 1916 describing the force of gravitation as an effect of curved space-time, to cover  all fundamental forces including electromagnetic and weak/strong nuclear forces. But the basic idea of describing forces as effects of curved space-time did not show to be fruitful. That was the first attempt to create a Grand Unified Theory GUT.  

The next attempt initiated in the 1950s was to extend quantum mechanics/electromagnetics QM/EM to cover also gravitation in the form of quantum gravity, but has not led to success. 

Modern physicists thus confront the following failed attempts of forming a GUT:

  • A1: Extend GR to include QM/EM.
  • A2: Extend QM/EM to include gravitation.

The reason A1 fails is that curved space-time does not serve to describe QM/EM. The reason A2 fails is that standard QM as atomic physics microscopics is viewed to be fundamentally different from the macroscopic world of gravitation, thus with a glitch between microscopics and macroscopics.  

This leaves modern physics in a dead-lock of incompatible theories for microscopics and macroscopics, which is the root of the present crisis of modern physics witnessed by many including prominent physicists. 

Here RealQM may offer a way forward, because in RealQM the microscopics of atoms takes the same form as macroscopics as continuum models in 3d, as exposed in previous posts. This means that a GUT may be possible in the form of 
  • A3: RealQM + EM + gravitation  
captured in the following mathematical models:
  • RealQM: New Schrödinger equation in 3d.
  • EM: Maxwell's equations 
  • Newton's law of gravitation
which can freely be combined into a GUT without glitch. Here Einstein has no role to play and the right part of the above diagram is left out.

While A1 and A2 have already failed, A3 has not yet. Want to challenge? 

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.  

  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. 

måndag 12 februari 2024

Gravitation and Continuum Models

In the CNPS talk on Febr 3 I tried to expose the virtues of a continuum as a spatial 3d Euclidean x-coordinate system without smallest scale as the reference system of continuum mechanics in Eulerian form.  As a basic example let us consider the Euler equations for incompressible flow expressing balance of momentum (Newton's 2nd Law) combined with incompressibility in the form 

  • $\nabla\cdot u = 0$        (1)
stating that divergence of velocity field $u(x,t)$ vanishes for all $x$ and time $t$. Here (1) appears as a stipulation or side condition for which the Lagrange multiplier is the pressure $p$, which appears as a pressure force $\nabla p$ in the momentum equation with connection through Gauss Law:
  • $\int p\nabla\cdot u\, dx = -\int \nabla p\cdot u\, dx$.
The bottom line is that $\nabla p$ appears in the momentum equation as a force effectively imposing (1) while not specifying the physical nature of the force in a pressure law. The beauty is now that solving the Euler equations computationally gives full information about incompressible flow with vanishingly small viscosity, as shown in this book and this book. The divergence zero condition (1) is in computation replaced by an effective computational pressure law of the form 
  • $-\Delta p = \frac{\nabla\cdot u}{\delta}$,     (2)  
where $\delta $ is a small parameter scaling with the mesh size, for which true physics is not needed. The Euler equations as a continuum model thus in computational form constructs a pressure law imposing near incompressibility. The continuum model in computational form thus invents physics which shows to describe reality in the form of physics as computation. 

We compare the continuum model with a particle model of a fluid asking for full specification of force between particles, and understand that a computational continuum model relieves us from a very difficult if not impossible task coming with a particle model. 

We now turn to Newtonian gravitation where the analog of (1) is Newton's Law of Gravitation in the form 
  • $-\Delta \phi = \rho$       (3) 
connecting gravitational potential $\phi$ to mass density $\rho$ by the Laplacian differential operator $\Delta$. The corresponding Lagrange multiplier appears in the momentum equation as 
  • $\rho\nabla\phi$                (4)
interpreted as gravitational force analogous to the pressure force connected to (1). Computationally (3) may take the following form allowing time-stepping:
  • $\frac{\dot\phi}{C}-\Delta \phi = \rho$           
  • $\frac{\ddot\phi}{C^2}-\Delta \phi = \rho$     (5)

where $C$ is a large constant representing effective speed of propagation, and the dot signifies differentiation with respect to time. Comparing computations with observation indicates that $C$ is much larger than the speed of light. 

Recall that it is well understood by everybody, except Einstein and his followers, that (3) expresses that (i) gravitational force $F$ is conservative, thus given by a potential $\phi$ as $F=\nabla\phi$,  and that (ii) $F$ is conserved in the sense of Gauss Law with $\nabla\cdot F = 0$ where there is no mass. To question (3) lacks rationale as it would violate (i) or (ii). In fact (3) is the prime jewel of all of physics, and to seek to modify it makes no sense. 

The beauty is here that the Euler equations augmented by gravitation in the form (3) and (4) (see this book) appears to describe a very rich world on a very wide range of scales, without having to specify the exact nature of the real physics of gravitation, which is still hidden, thus following the spirit of Newton.

The beauty is enhanced by realising that also quantum mechanics can be captured as a continuum model over a 3d Euclidean coordinate system without smallest scale allowing microscopics and macroscopics to have the same seamless conceptual form as shown in Real Quantum Mechanics.  This is shocking to modern physicists educated to view microscopics beyond comprehension for humans with only macroscopic experience.

Continuum models like the Euler equations thus appear as realisations of physics as computation expressing physics in possibly new forms open to understanding. 

PS1 The total energy for incompressible flow based on (2) includes a positive contribution of the form
  • $\delta\int\vert \nabla p\vert^2dx$ 
and similarly total energy balance with gravitation in the form (4) contributes (with details here)
  • $\int\vert\nabla\phi\vert^2dx$     
 as a natural expression of gravitational energy (as a source of kinetic energy) and in the form (5):
  • $\int\vert\nabla\phi\vert^2dx+\frac{1}{C^2}\int\dot\phi^2dx$,
where the real physics of the second term with the time derivative $\dot\phi$ is less clear, and so may be interpreted rather as computational artefact allowing time-stepping. Recall that the presence of a time derivate in an energy expression represents kinetic energy from motion of matter, which is not an aspect of $\phi (x,t)$ expressing spatial presence of gravitational potential/force.   

PS2 Multiplying (3) by $\phi$ and integrating gives:
  • $\int\vert\nabla\phi\vert^2dx = \int\rho\phi dx$           
where the right hand side commonly is referred to as gravitational potential energy. We see that the left hand side includes only the gravitational potential $\phi$, which connects to viewing $\phi$ as primary, as suggested in previous posts on New Newtonian gravitation. 

PS3 We may compare (3) with a law of the form 
  • $\phi = \rho$
which expresses instant local action and  connects to gas law of (isothermal) compressible flow of the form $p=\rho$ with $p$ pressure, with $\nabla\phi$ corresponding to $-\nabla p$.   

söndag 11 februari 2024

The Wobbling Sun Gravito-Radiative Global Warming Effect

In PS4 of the previous post a comparison was made between (one-way) electromagnetic-radiative heat/energy transfer by (two-way) electromagnetics waves, which is very real, and hypothetical gravity-radiative heat transfer by gravitational waves for which experimental support is very weak. 

If gravitational wave transfer of energy is real, a wobbling Sun would have a heating effect on the Earth transmitted by gravitational waves, which would add to the familiar electromagnetic-radiative heating, and so add to global warming. But no such a scenario is mentioned in the reports by IPCC.  We can thus conclude that it is not real, since IPCC supposedly takes in all real effects on global warming. 

Of course, a lack of gravity radiation effect from the Sun, could be that in fact the Sun is not wobbling enough to give a measurable signal. In any case, its effect on global warming appears to be small. This should come as a consolation in difficult times.


onsdag 7 februari 2024

Speed of Gravity? Newton or Einstein?

In Newtonian mechanics gravitational potential $\phi (x,t)$ is connected to mass density $\rho (x,t)$ by the Laplacian differential operator $\Delta$ through the equation 

  • $\Delta\phi (x,t) = \rho (x,t)$                       (1)
where $x$ is a Euclidean coordinate and $t$ a time coordinate. 

The standard way is to view the gravitational potential $\phi (x,t)$ and gravitational force $\nabla\phi (x,t)$ at some space-time coordinate $(x,t)$ as somehow being generated from the distribution of mass density $\rho (y,t)$ for all $y$ different from $x$ in an apparent instant action at distance at time $t$ as if gravitational force is propagated with infinite speed.  Solving the differential equation in terms of $\phi$ is a global operation of integration/summation. 

But instant action at distance is inexplicable and so in modern physics Newton's mechanics has been replaced by Einstein's General Theory of Relativity GR, where gravitational force is propagated with the finite speed of light c. 

On the other hand, if we in (1) view the gravitational potential $\phi (x,t)$ as primary from which mass density $\rho (x,t)$ is generated by the action of the differential operator $\Delta$ as 
  • $\rho (x,t) =\Delta\phi (x,t)$                      (2)
which is a local operation at $x$ of differentiation, which is possible to view to be instant. Mass density is here generated by instant local action from gravitational potential, and then the problem of instant action at distance evaporates and it is no longer necessary to replace Newton by Einstein. 

Newtonian mechanics describes celestial dynamics in the form an initial value problem 
  • $\dot x(t) = f(x(t))$ for $t>0$ with $x(0)$ given  
where $x(t)$ represents positions of celestial bodies at time $t$, and $f(x)$ is a given function of $x$ including Newton's law of gravitation. The  differential equation can be solved by explicit time stepping of the form
  • $x(t) = x(t-dt) + dt*f(x(t-dt))$                      
where $dt>0$ is a time step. The position $x(t)$ at time $t$ is thus computed from previous position $x(t-dt)$ with a correction $dt*f(x(t-dt))$ determined at the previous time $t-dt$, thus with a time delay of $dt$. We can view the time delay $dt$ as an expression of finite speed of propagation $C$, and we now ask if observations can give information about the size of $C$. 

We do this in the simplest case of one small body (Earth) orbiting a big body (Sun) as expressed in this code, where we can test the effect of different time steps $dt$. By normalisation we can connect $dt$ to $\frac{1}{C}$ as the time required for a gravitational signal from the Sun to reach the Earth. The effect on position $x(T)$ at time $T>>0$ of explicit time stepping with time step $dt$ at best scales with $T*dt$ and if we ask for a precision of $\epsilon$, we have 
  • $dt < \frac{\epsilon}{T}$, that is $C>\frac{T}{\epsilon}.$                              
Relevant values may be $T>10^4$ and $\epsilon <10^{-4}$, that is 
  • $C>10^8$.
If we put this number in the perspective of the Sun-Earth system with the speed of the Earth about $0.0001$ times the speed of light $c$, we get $C>10^{12}$ to be compared with $c=3\times 10^8$ meter/second with thus a factor at least $10^3$. We can compare with the estimate $10^7$ made by Laplace and even $10^{10}$ from the PS below.  

In any case, observations indicate that the required speed of propagation of gravitational effects in (1) is several orders of magnitude bigger than the speed of light.

A planetary system based on (1) with a time delay from finite speed of gravity equal to the speed of light would not persist over time. 

Newtonian mechanics describes celestial/planetary motion very accurately over long time with a from observations apparent speed of gravity much bigger than the speed of light. If (1) is viewed to express Newton's law of gravitation this essentially requires instant action at distance, which is unthinkable.

But changing view to (2) replaces instant action at distance by instant local action, which is thinkable.  

Since in GR the speed of gravity is finite, one would expect to see effects in GR of time delay, but that would contradict observations where no time delay can be detected. The situation is handled by claiming that  there is a very subtle strange effect of cancellation in GR, which means that in the end the effective speed of gravity is infinite. So GR says the the speed of gravity is finite equal to the speed of light, but the effect of finite speed is cancelled and so the net effect is zero as if the speed of gravity in fact is infinite as in Newtonian gravity. Do you buy this argument? 

Recall that Einstein when claiming that Newtonian gravitation must be replaced by GR, could not refrain from expressing "Newton, forgive me." as if he had committed a scientific crime.

It is also possible consider a potential-mass connection of the form  
  • $\Delta\phi - \rho = 0$                          (3)
where the cause-effect is not indicated. Here (3) appears rather as a side condition expressing a balance of potential and mass without worrying about casual connection, see this book and this computation exploring the Euler equations for fluid flow with gravitation.

Sum up: There is no reason to replace Newton by Einstein, and anyway doing so leads to a quagmire of mysteries. It is not necessary to view mass as primary from which gravitational potential/force is formed by apparent instant action at distance as in (1), which is unthinkable. We may as well turn (1) around into (2) viewing gravitational potential to be primary from which mass is formed by instant local action, which is not unthinkable. 

  • Laboratory, solar system, and astrophysical experiments for the “speed of gravity” yield a lower limit of $2\times 10^{10}c$.
  • But mediation requires propagation, and finite bodies should be incapable of propagating at infinite speeds since that would require infinite energy. So instantaneous gravity seemed to have an element of magic to it.
  • We will examine the explanations offered by GR for these phenomena, and conclude that in the most widely taught curved space-time interpretation of GR the acceleration of bodies through space lacks a causal connection to the source of gravity. And we will confront the dilemma that remains when we are through: whether to modify our existing interpretation of GR, or give up the principle of causality.
It thus appears that GR assuming that the speed of gravity is finite equal to the speed of light, is incompatible with experiments showing an infinite speed of gravity, which asks for modification of GR.

This modification may simply be a return to Newtonian gravitation with a law of gravitation of the form (2) with instant local action, which is not in contradiction to causality.  

The reason that mass traditionally is viewed to be primary and potential a derived quantity as in (1), is (probably) that mass may be directly visible and gravitational potential/force is not. On the other hand, all bodies directly "feel" gravitational force, and so gravitational potential/force is very present although not directly visible.   

PS2 Further evidence of speed of gravity being much larger than the speed of light from observation of satellite motion, is given here.

PS3 Maxwell's wave equations for electromagnetics describe propagation of light of all frequencies at the same speed = speed of light = c. Augmented with an Abraham-Lorentz radiation force Maxwell's wave equations also describe radiative heat transfer as Computational Blackbody Radiation as a one-way transfer of energy from warm to cold mediated by two-way electromagnetic waves, more precisely as a resonance phenomenon. The speed of radiation thus in principle can be viewed to be equal to c, even if in reality effective transfer of energy from resonance may change at a slower speed. 

Finite speed of gravity = C requires augmentation of (1) into a wave equation $-\frac{\ddot\phi}{C^2}+\Delta\phi =\rho$ with the dot signifying differentiation in time, and transfer of energy by gravitational waves requires some form of Abraham-Lorentz force. Observations show that C is much bigger than c, and so both theory and observation supporting C=c, is lacking. 

While radiative wave energy transfer is a reality, there seems to be little evidence that gravitational wave energy transfer is real. The proclaimed experimental detection of very faint gravitational wave energy transfer from distant merging black holes suffers from the difficulty of finding a needle in a haystack.