HOW GRAVITY WORKS
You probably think of gravity as “curved spacetime”, but that's the effect, not the cause. Curved spacetime is just curvilinear motion, and to see the cause you have to take a derivative of that curved spacetime. What you get is a gradient. It’s a g
μν gradient, essentially a gradient in the properties of
space. And this gradient in space is caused in turn by the central energy locked up in the matter of the planet or star that “conditions” the surrounding space. It’s all to do with stress-energy and pressure, and the best way to conceptualize it is to start with an old favourite.
Think about the cannonball in the rubber sheet. The cannonball is heavy, and it makes a depression that will deflect a rolling marble, or even cause the marble to circle like an orbit. It’s a nice analogy, but it’s wrong. It’s wrong because it relies on gravity to pull the cannonball down in the first place. It’s circular. It uses gravity to give you a picture of gravity.
Wikipedia commons public domain image by BenRG
To get a better analogy, imagine you’re standing underneath the rubber sheet. Grab hold of the rubber around the cannonball and pull it down to give yourself some leeway. Now tie a knot underneath the cannonball, get rid of the cannonball, and let go. Now we’ve got a flat rubber sheet with a knot in it. The knot is a stress configuration, and surrounding it is tension. The tension gradually reduces as you move away from the knot, so if you could measure it, you would measure a radial gradient. But measuring it is trickier than you think. Because in this analogy we can’t use a marble. This rubber sheet represents the world, there’s no stepping outside of it. Our “marble” has to be within the sheet, and a part of it. What we need is a ripple. A photon will do, because a photon is a transverse wave, also known as a shear wave. In mechanics a shear wave travels at a speed determined by the stiffness and density of the medium: v = √(G/ρ). The G here is the shear modulus of elasticity, to do with rigidity. The ρ is the density. The equation says a shear wave travels faster if the material gets stiffer, and slower if the density increases. You can’t directly apply material concepts to space, but in electrodynamics the equation is somewhat similar: c = √(1/ε
0μ
0). Here ε
0 is electric permittivity and μ
0 is magnetic permeability. People don't quite understand these terms because they don't understand the electromagnetic field. They forget about the dualism of
Jefimenko's equations, and about Minkowski's wrench, which is two pages from the end of
Space and Time:
"Then in the description of the field produced by the electron we see that the separation of the field into electric and magnetic force is a relative one with regard to the underlying time axis; the most perspicuous way of describing the two forces together is on a certain analogy with the wrench in mechanics, though the analogy is not complete".
A moving electric field doesn't
generate a magnetic field, it
is a magnetic field, because Minkowski's wrench is referring to a screw mechanism. This goes back to Maxwell’s
On Physical Lines of Force see
wikipedia page 53 and there it is. Find a drill bit or reamer, grip it in your right hand, and put your left thumb on the bottom of it, and push upwards. It turns, just like the right-hand rule. That's because the electric field is in essence a "twist field", and the magnetic field is a "turn field" view of the self-same thing. Permittivity is telling you the twistability of space, and permeability is telling you how good it is at making things turn. Hence they are similar to stiffness and density, and the photon is similar to a ripple in a rubber sheet. But for space, speed doesn’t increase as we approach our central stress. Instead it reduces. It reduces because the pressure is outward rather than inward. The energy conditions the surrounding space to create a pressure gradient rather than a tension gradient. And of course it's in three dimensions, so instead of thinking of a rubber sheet stretched over a frame, we need to think of space as being like a ghostly transparent rubbery jelly squeezed into a glass box. When you insert a hypodermic needle into the centre and inject more jelly, you create a pressure gradient in the surrounding jelly. You've essentially added energy, the result is a pressure gradient, and that results in gravity.
As to how it works in the real world, it's to do with vacuum impedance, which is Z
0 = √(μ
0/ε
0). Impedance is like resistance, but for alternating current rather than direct current:
Wikipedia commons GNU FDL image by Jacobsk, see http://commons.wikimedia.org/wiki/File:Wisselstroom.png
You might wonder why alternating current is important here, but it's very simple. That's what a light wave is:
Wikipedia commons GNU FDL image by Heron, see http://commons.wikimedia.org/wiki/File:Light-wave.svg
There's an electromagnetic field variation, first one way, then the other, and you just can't have this field variation without some form of current. There’s no charge in there, but it really is current, like
displacement current. But because a photon conveys energy in a three-dimensional space, and the dimensionality of energy is pressure x volume, it's better to think of a photon in terms of a pressure pulse, shaped something like a lemon. Think of it as a "pulse of spacewarp", where the sinusoidal waveform is telling you the twist and tracing a slope. It grows to a maximum a quarter of the way along the lemon, goes to zero half way along, increases to a negative maximum three-quarters of the way along, and then goes back to zero. But it's still a wave, and its rate of propagation is determined by impedance, because impedance is describing "the strength of space". If space is stronger, and you’re a photon, it's harder to twist and it turns you more easily. Impedance increases as we approach the central matter/energy stress, the photon is a quantum of alternating current, a higher vacuum impedance equates to more resistance, and that results in a lower velocity, hence a lower c because c = √(1/ε
0μ
0).
Hence when a photon passes a massive body, it's travelling through
inhomogeneous vacuum where there's a gradient in c across the photon wave-front. Hence it veers towards the body a little. What we’re seeing is refraction. It’s not quite the same as refraction through a glass block, but it’s similar. It’s so very similar that when we see it through our telescopes, we call it gravitational lensing:
Wikipedia commons public domain image, NASA/ESA, uploaded by Ladsgroup
Here’s the crucial point: our real world is like ghostly transparent block of rubber containing pulses of stress, some of which are trapped in circular paths and are called matter. And we are made of this ghostly, intangible, insubstantial "fabric". We are so totally immersed in it and so much a part of it that we cannot directly measure any change in vacuum impedance. And nor can we directly measure any change in the velocity of light, because we calibrate our rods and clocks
using the motion of light:
Under the International System of Units, the second is currently defined as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom...
The metre is the length of the path travelled by light in vacuum during a time interval of 1/299792458 of a second...
So when we measure the motion of light, we're taking that measurement using units derived from... the motion of light. And as pair production is telling us, we are made of this stuff. Anything that affects the speed of light affects electrons, and we can say the same for protons and neutrons. It affects all processes. That's why we can’t measure the change in c locally. It's like trying to measure the length of your shadow using the shadow of your ruler. It always measures the same, be it morning, noon, or dusk. But we can infer the change in c. We can measure it from afar, by comparison. It’s there in the gravitational time dilation, programmed into our GPS, in the Shapiro delay. The evidence is hiding in plain view, we can see it but we don’t know what we’re seeing. For many of us c is set in stone, and to challenge the constancy of the speed of light would be to challenge relativity. But Einstein
told us about the variable speed of light. The space is not homogeneous, there's a gradient in vacuum impedance and hence a gradient in c, and that's why things fall down.
The bald truth is that a gradient in c is what a gravitational field is, and when you can appreciate this, you can allow yourself the epiphany of understanding gravitational potential energy. We know that E=mc², so a cannonball sitting quietly in space represents maybe 10^11 Joules of energy. If a planet now comes onto the scene, the cannonball will fall towards it, and just before impact will also have kinetic energy of say 10^8 Joules. Now, we ask, where did this kinetic energy actually come from? Has it been sucked out of the planet via some magical mysterious action at a distance? Let's ask an expert, somebody who was right on the money:
“That gravity should be innate, inherent, and essential to matter, so that one body may act upon another at a distance through a vacuum, without the mediation of anything else, by and through which their action and force may be conveyed from one to another, is to me so great an absurdity that I believe no man who has in philosophical matters a competent faculty of thinking can ever fall into it”.
That's a no from Sir Isaac Newton, and he should know. So, has it been magically extracted from the surrounding space? No, the energy density in the surrounding space is not reduced when a cannonball falls through it. The force of gravity doesn’t work like that in relativity. A free-falling body is not accelerating, the body on the ground is the one accelerating. No acceleration means no force and no energy is being delivered. So the only source of that kinetic energy is the cannonball itself. It hasn’t come from its mass because mass is “invariant”. So E=mc² and we’ve got a cannonball kinetic energy that hasn’t come out of the m. There’s only one place left it can have come from: the c.
The c up there is greater than the c down here, and there’s a gradient in between. The Pound-Rebka experiment backs this up. There’s a gravitational blue-shift because the c is reduced so it looks like there’s more energy in the photon at the bottom of the tower. But the photon hasn't changed, the environment has changed, and the measuring devices, because of the gradient in c. There’s
always a gradient in c wherever there’s gravity. Yes, the gradient might be very small. But don't neglect it like the tidal gradient, which is a gradient
in the gradient in c. Because without that local gradient in c,
things don't fall down.
Let’s take a look at an electron to see why things fall down. It has angular momentum aka spin and a magnetic dipole moment, and this evidence along with electron-positron annihilation tells us its something like a self-trapped 511keV photon going round and round in circles. Stick a circle of light that looks like this: O, into a gravitational gradient, caused by a very large number of other electrons and protons some distance off. What’s going to happen? Let’s divide the circle into four flat quadrants and make it very simple:
..←
↓
....↑
..→
Starting from the left and going anticlockwise, at a given instant we have a photon travelling down like this ↓. There’s a gradient in c from top to bottom, but all it does is make the photon look blueshifted. A little while later the photon is moving like this → and the lower portion of the photon wave-front is subject to a slightly lower c than the upper portion. So it bends, refracts, curves down a little. Later it’s going this way ↑ and looks redshifted, and later still it’s going this way ← and bends down again. These bends translate into a different position for our electron. The bent photon path becomes electron motion. The electron falls down:
○
↓
The reducing speed of light effectively bleeds motion out of the component photon and into the electron. But only half the cycle got bent, so only half the reduced c goes into kinetic energy aka relativistic mass. That’s why light is deflected twice as much as matter. That’s why gravity is not some magical mysterious action-at-a-distance force. There are no hidden dimensions, there's no blizzard of gravitons sleeting between the masses. There’s no energy being delivered, so gravity isn’t a force in the usual sense, and it isn't negative energy. There is no location in a gravitational field where the matter/energy of a planet, or the vacuum energy of space, is negative. It’s just the gradient in the properties of space caused by the central stress-energy. And it makes things fall down like this:
Imagine a swimming pool. Every morning you swim from one end to the other in a straight line. In the dead of night I truck in a load of gelatine powder and tip it all down the left hand side. This starts diffusing across the breadth of the pool, imparting a viscosity gradient from left to right. The next morning when you go for your swim, something's not right, and you find that you're veering to the left. If you could see your wake, you'd notice it was curved. That's your curved spacetime, because the pool is the space round a planet, the viscosity gradient is Einstein's non-constant gμν, and you're a photon. As to how the gradient attracts matter, consider a single electron. We can make an electron along with a positron from light, via pair production. Since the electron also has spin, think of it as light trapped in a circular path. So if you're swimming round and round in circles, whenever you're swimming up or down the pool you're veering left. Hence you find yourself working over to the left. That's why things fall down.
NB: note that it’s
energy that causes gravity, not matter per se. Matter only causes gravity because of the energy content. See
The Foundation of the General Theory of Relativity and look at page 185 where Einstein says
"the energy of the gravitational field shall act gravitatively in the same way as any other kind of energy". A gravitational field is a region of space that contains extra energy and in itself causes gravity, hence an integration approach is required, as per page 201. But we don't consider a gravitational field to be dark matter. We don’t go looking for WIMPs. Yes, space is "dark", and the mass of a system is a measure of its energy content, so if you defined the space around a planet as a system, it has a mass of sorts. But it isn’t matter. It’s just space. What did Einstein say about space?
Neither homogeneous nor isotropic. What does the
FLRW metric say?
”The FLRW metric starts with the assumption of homogeneity and isotropy of space.” Spot the difference? Gravitational anomalies aren’t evidence for dark matter. Dark matter is just a hypothesis that attempts to explain them. And those who promote it sweep the raisins-in-the-cake analogy under the carpet. The universe expands, but the space within the galaxies doesn’t, because galaxies are gravitationally bound. So each and every galaxy is surrounded by a halo of inhomogeneous space. That’s a g
μν gradient. It’s a gravitational field
without any matter on the end of it. So when you hear people talking about the hunt for dark matter, bear this in mind.
Thanks for the paper – reading it now... spin(11,3) is news to me. 
Their ansatz for Φ (a Hodge dual) differs from yours since yours is, as yet, undefined.
