You probably think of gravity as curved spacetime. Surprisingly Einstein didnít, not quite. And neither should you. To understand gravity you have to take the ontological view. You have how to learn to see whatís there. And to do that, you have to put time to one side, because time isnít the same kind of dimension as the Dimensions of space. Yes, an object passing a planet traces a curved path, but you donít stare up at a plane and decide that itís a silver streak in the sky. You take a mental snapshot, flash, a picture of it in a timeless instant. Itís the same with gravity. Take the time-derivative of that curved spacetime. What you get is a gradient. And itís a gradient in space, not curved spacetime.
But letís tackle it an easier way, via an old favourite. Think about a cannonball sitting on a 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. Itís a nice analogy, but itís wrong. Itís wrong because it relies on gravity to pull the cannonball down in the first place. It uses gravity to give you a picture of gravity.
To get a better handle on it, imagine youíre standing underneath the rubber sheet. Letís make that a silicone rubber sheet. Itís transparent, like my snorkel and mask. Grab hold of the rubber around the cannonball and pull it down further to give yourself some leeway. Now transfer your grip to the transparent silicone rubber itself. Gather it, pull it down some more. Now tie a knot in it underneath the cannonball, like youíd tie a knot in the neck of a balloon. Now pull it all the way down and let go. Boinggg! The cannonball is gone. Forget it.
Now, what have we got? Weíve got a flat rubber sheet with a knot in it. The knot will stand in for a region of stress, where the rubber is under pressure. Stress is the same as pressure. Itís force per unit area, and force times distance gives us the units for both work and energy. So energy is stress times volume. The knot represents energy. Or matter if you prefer. OK hereís the deal. Surrounding the small central region of stress is a much larger region of tension extending outwards in all directions. Whenever you have a stress you always have a tension to balance it. It isnít always obvious, but itís always there, like reaction balances action, and force balances force. The tension gradually reduces as you move away from the stress. If you could measure it, you would measure a radial gradient. But measuring it is trickier than you think. Because in this analogy we canít use a marble rolling across a rubber sheet. This rubber sheet represents the world, thereís no stepping outside of it. Our ďmarbleĒ has to be within the rubber sheet, and a part of it, made out of the same stuff as that knot.
We need an extra dimension. So turn your top hat upside down and tap it with your magic wand. Abracadabra! A flash of light and a puff of smoke, and that rubber sheet is now a solid block of clear silicone rubber extending in all directions. And youíre standing inside of it. Letís make you a ghost so you can glide around unimpeded, for the purposes of gedanken. Our knot is now three-dimensional, like a moebius doughnut, maybe a little silvery like a bubble underwater. Itís not really made out of anything, it hasnít got a colour, and it hasnít even got a surface. Itís a soliton, a topological defect, a travelling stress thatís basically a photon, but going nowhere fast because itís twisted round on itself. So E = hc/λ = pc = mc≤ means the momentum is now inertia, and we call it an electron.
Our electron has replaced our cannonball, and now we need a photon to stand in for that rolling marble. Letís conjure one up, and send it propagating across our rubberworld so that it passes by our electron. We could run after it and take some snapshots with our ontological camera, but letís save that for another day. For now our photon is just a shear-wave ripple, travelling at a velocity determined by the stiffness and density of the medium. Thereís an equation for it in mechanics that goes like this:
v = √(G/ρ)
The G here isnít a gravitational constant, but is the shear modulus of elasticity, to do with rigidity. Itís different to the bulk modulus of elasticity, because itís a lot easier to bend something rather than compress its volume. The equation says a shear wave travels faster if the material gets stiffer, and slower if the density increases. In electrodynamics the velocity equation is remarkably similar. Youíve probably seen it before:
c = √(1/ε0μ0)
Here ε0 is permittivity and μ0 is permeability. The two are related by impedance √(μ0/ε0). High permittivity means a material will take a larger charge for the same voltage, for example Barium Titanate has 1200 times the permittivity of air, so we donít make capacitors out of air. High permeability means a material exhibits more magnetism when you change the charge. Iron has lots of it, wood doesnít, so magnets are made of iron. There are some marvellous similarities between mechanics and electrodynamics, though confusions abound too. With the piezoelectric effect you subject a material to mechanical stress and you get an electrical stress, a voltage, but high voltage is called high tension, which is negative stress. And electric current goes from negative to positive, so things are backwards. But letís come back to that another time, and just say higher impedance means lower velocity.
Back in rubberworld, our photon-marble is passing our electron-cannonball. We notice it veers towards it a little. Thatís because where the rubberworld tension is slightly greater, the real-world impedance is slightly higher, so the velocity is slightly lower. What weíre seeing is refraction.
Hereís the crucial point: our real world is like that rubberworld with the knot in it plus an extra dimension, and weíre made out of this stuff, along with our rulers and clocks. So we donít see the tension. We donít measure the change in c. But we can infer it. Like in the Pound-Rebka experiment, where a photon is blue-shifted at the bottom of the tower because c there is lower. Or in the Shapiro experiment, where the light takes longer to skim the sun because the c there is lower too.
Thereís an equivalence going on here between General Relativity and Special Relativity, but itís tricky to spot. Imagine that I stay here on earth while you travel to Alpha Centauri in a very fast rocket travelling at .99c. We can use 1/√(1-v≤/c≤) to work out that you experience a sevenfold time dilation. (Multiply .99 by itself to get .98 and subtract this from one to get a fiftieth, which is roughly a seventh multiplied by a seventh). We normally think of time dilation as being matched by length contraction, but thatís only in the direction of travel. Hold up a metre ruler transverse to the direction of travel and itís the same old metre. Your metre is the same as my metre, and your time is dilated by a factor of seven, which means it takes a beam of your light seven times longer to traverse your transverse metre. Looking at it another way c = s/t and your t changed, your s didnít, so your c did. Your c is a seventh of mine. Donít get confused about this. Donít tell yourself that your lightbeam is following a diagonal path and has to cover a greater distance. Thatís introducing an absolute reference frame, mine. Stay in your own frame. Then when you come back after your year-long round trip, I aged seven years, but you only aged one. You aged less because your c was slower than mine, but you never noticed it at the time. The equivalence comes in because I could have slid you into a black box and subjected you to high gravity instead of sending you to Alpha Centauri. We know that ďclocks run slowĒ in a high gravity situation, just as they do when youíre travelling fast. And itís for the same simple reason. The c is reduced. But you wonít measure it as reduced, because itís just a distance/time conversion factor. Just like you when you go to the moon you donít get three ounces to the pound.
I know itís difficult to stop thinking c is a constant. Yes itís always measured to be the same in all frames. But when you step back to see the big picture that is the whole gallery, when you look at all the frames side by side, you see what distinguishes them is the way c changes. Itís a constant, but it isnít constant. Once you realise that c changes in a ďgravitational fieldĒ you can allow yourself the epiphany of understanding gravitational potential energy. We know that E=mc≤, so a cannonball sitting quietly in space represents maybe 1011 Joules of energy. If the earth now trundles on to the scene, the cannonball will fall towards it, and just before impact will also have kinetic energy of say 109 Joules. Now hold it right there. Freeze frame. Where did that kinetic energy actually come from? Has it been sucked out of the earth? Has it been magically extracted from some zero-point bottomless bucket? Has it come from the ďgravitational fieldĒ? No. Thereís no free lunch from Mister Gravity. The energy came from the cannonball. And it hasnít come from its mass because mass is ďinvariantĒ. Only it isnít invariant because the mass has actually increased, check the Pound-Rebka experiment. So E=mc≤ and weíve got a pile of kinetic energy that hasnít come out of the m. Thereís only one place left it can have come from. The c. The c up there is greater than the c down here, and thereís a gradient in between.
Thereís always a gradient in c when thereís gravity. Even across the width of an electron. Yes, the gradient might be very small. But it isnít negligible. If you think it is, as per the General Relativity Equivalence Principle, youíve just thrown the baby out with the bathwater. An accelerating frame with no tidal gradient isnít the same as a proper gravity situation. Thereís always a tidal force. The gradient has to be there. There can be no Uniform Gravitational Field. Because without that gradient, things donít fall down.
Letís go back to rubberworld. But itís time we did a Reverse Image and made the rubber the ghost. Now youíre back to normal again take a look at that electron once more. Itís a travelling stress localised because itís going round in a circle. Stick this ring of light in a real gravity gradient, caused by a zillion other electrons some distance downaways. Whatís going to happen? Flash, take a picture. At a given instant we have a quantum of light travelling down like this ↓. Thereís a gradient top to bottom, but all it does is gives the photon a fractional blueshift. A little later take another picture. Flash. Now the photon is moving this ← way, and the upper portion of the photon wavefront is subject to a slightly higher c than the lower portion. So it bends, refracts, curves down a little. Later itís going this ↑ way and gets fractionally redshifted, and later still itís going this → way and curves down again. These bends translate into a different position for our electron. The bent photon path becomes electron motion. Only half the cycle got bent, so only half the reduced c goes into kinetic energy. The other half goes into mass, but itís only a scale-change falling out of the clear blue sky:
So hereís your free lunch:
Now you can understand why gravity is not some magical, mysterious, action-at-a-distance force. There is no curvature of spacetime, no hidden dimensions, no gravitons sleeting between masses. Thereís no energy being delivered, so gravity isnít even a force. Itís just the tension gradient that balances the stress that is mass/energy. And weíre just rubberworld Fatlanders getting to grips with our wrinkles and bumps.
No energy delivered, extra mass to use as collateral... that means thereís no energy cost. So if we could somehow contrive a gradient that goes the other way... whoo, itíll be The Stars My Destination. But first of all we must also understand the thing we call Space. We must learn how light is a ripple of nothing, and how all the somethings are made from it. Itís a tale of something and nothing, and since nothing comes for free, there will be a Charge...
Acknowledgements: thanks to J.G. Williamson and M.B. van der Mark for Is the electron a photon with toroidal topology? http://members.chello.nl/~n.benschop/electron.pdf, to Peter M Brown for his many papers on his excellent website http://www.geocities.com/physics_world/, to Robert A Close for for Is the Universe a Solid? http://home.att.net/~SolidUniverse/]home , to Reg Norgan for http://www.aethertheory.co.uk/pdfRFN/Aether_Why.pdf, to G S Sandhu for The Elastic Continuum http://www.geocities.com/gssandhu_1943/index.html to all the forum guys with their relevant posts and links, Wikipedia contributors, and to anybody who Iíve forgotten or whose pictures Iíve used. Thanks guys.