Once you understand that there is no time flowing inside an optical clock, then you understand that an optical clock goes slower when it’s lower because light goes slower when it’s lower. Because the speed of light varies in the room you’re in. After that you understand that light waves curve downwards in a gravitational field for the same reason that sonar waves curve downwards in an ocean. Search the ES310 sonar propagation webpage and you find where the US Navy said it: “Recall how differences in the index of refraction (which are a measure of the propagation speed) affected electromagnetic waves”. Search the Einstein digital papers and you can find where Einstein said it: “the curvature of light rays occurs only in spaces where the speed of light is spatially variable”.
Light curves because the speed of light is spatially variable
Einstein said it time and time again, year after year. Yes, he came up with special relativity in 1905 with his two postulates, one of which was “the speed of light is constant”. But only two years later in 1907 he was saying light curves in a gravitational field because the speed of light isn’t constant. He said it again in 1911, 1912, 1913, 1914, 1915, and 1916. You don’t find Einstein saying light curves because spacetime is curved. Instead you find him talking about the energy-density of space varying. You can liken this to the density of air varying. Sound waves in air tend to curve downwards at night because the cooler air near the ground is denser:
Image from Rod Nave’s hyperphysics, see refraction of sound
In similar vein a light wave “veers” downwards rather like a car veers left when it encounters mud at the side of the road. We can do something similar with light with something as simple as a piece of glass. That’s why we have lenses in spectacles. But we can also do it with an energy-density gradient in space. That’s why we have gravitational lensing. See Professor Ned Wright’s Deflection and Delay of Light article for more. He doesn’t say the light is deflected because spacetime is curved. Instead he says this: “In a very real sense, the delay experienced by light passing a massive object is responsible for the deflection of the light. The figure below shows a bundle of rays passing the Sun at various distances”:
Gif from Ned Wright’s Deflection and Delay of Light
Once you know this, you can make sense of the rubber-sheet analogy. It’s sometimes said to be a tautology, in that gravity is used to try to explain gravity. However it can explain gravity rather well provided you know what it’s really depicting.
The rubber sheet analogy
The starting place for that is that the curvature you can see in the images relates to spacetime curvature. That’s associated with a “curved metric”, and a metric is associated with measurement. As for what you’re measuring, imagine you could place a 15 x 15 array of optical clocks throughout a horizontal slice of space around the Earth. Then you plot all the clock rates, such that the lower slower clock rates generate data points lower down in a 3D image, and the higher faster clock rates generate data points higher up in the 3D image. When you join the dots, your plot looks like this:
CCASA image by Johnstone, see Wikipedia
That’s an image from the Wikipedia Riemann curvature tensor page. It’s effectively the rubber-sheet depiction of curved spacetime. And because it’s derived from optical clock rates, it’s also a plot of the speed of light. Some might say that it’s just a plot of the “coordinate” speed of light, but it’s more than that. There is no time flowing inside an optical clock, so the height at some location on the plot depicts the real speed of light at that location. It also depicts the gravitational potential at that location. Meanwhile the slope at some location depicts the first derivative of gravitational potential, and therefore the force of gravity at that location. The curvature at some location depicts the second derivative of gravitational potential, and therefore the tidal force at that location. That’s where the force of gravity changes most. That’s spacetime curvature. If you don’t have any spacetime curvature, your plot can’t get off the flat and level, which is why spacetime curvature is said to be the defining feature of a gravitational field. But note that a marble rolls down where the sheet is sloping rather than curved, and that your plot is what’s curved, not space. Your plot of measurements is curved so your metric is curved, so spacetime is curved, but space is not.
Light does not follow the curvature of spacetime
This is why light does not follow the curvature of spacetime. You can appreciate this if you zoom in on a section of the plot. If we represent a light beam with a yellow line, it curves wherever there’s a gradient in gravitational potential. That’s where the plot has a slope, where the grid lines are tilted as opposed to curved:
The force of gravity and so the curvature of light is greatest where the tilt is greatest. The tilted light-cones in the 2009 Stanford singularities and black holes article by Erik Curiel and Peter Bokulich depict this. Alternatively you can emulate the tilt with a piece of stiff board. Lift one side up, and roll a marble across it. It follows a curved path because the board is tilted, not because the board is curved. It’s similar for the room you’re in. The force of gravity is 9.8 m/s² at the floor and at the ceiling, so there’s no detectable tidal force, and so no detectable spacetime curvature. But your pencil still falls down. That’s detectable. The point to note is that gravity isn’t there because spacetime is curved, it’s there because there’s a gradient in potential, and a gradient in the speed of light.
Gravitational time dilation does not occur because spacetime is curved
Another point to note is that gravitational time dilation does not occur because spacetime is curved. That’s confusing cause and effect. Curved spacetime corresponds to your curved plot of optical clock rates, and optical clocks don’t go slower when they’re lower because your plot of clock rates is curved. They go slower when they’re lower because light goes slower when it’s lower, along with all other electromagnetic phenomena. And light goes slower when it’s lower because space near the Earth is different to the space further away. Because a concentration of energy in the guise of a massive planet “conditions” the surrounding space, this effect diminishing with distance in a non-linear fashion.
Space is a polarizable medium
Einstein talked about it in his 1920 Leyden Address. He said this: “According to this theory the metrical qualities of the continuum of space-time differ in the environment of different points of space-time, and are partly conditioned by the matter existing outside of the territory under consideration. This space-time variability of the reciprocal relations of the standards of space and time, or, perhaps, the recognition of the fact that ’empty space’ in its physical relation is neither homogeneous nor isotropic, compelling us to describe its state by ten functions (the gravitation potentials gμν), has, I think, finally disposed of the view that space is physically empty”. Einstein didn’t talk about spacetime curvature. Instead he talked about space that was neither homogeneous nor isotropic. He finished up saying this: “recapitulating, we may say that according to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an ether”. He also spoke of “the refraction of light rays by the gravitational field”. So did Newton, see Opticks query 20: “Doth not this aethereal medium in passing out of water, glass, crystal, and other compact and dense bodies in empty spaces, grow denser and denser by degrees, and by that means refract the rays of light not in a point, but by bending them gradually in curve lines?” All this talk of aether might sound archaic, but it isn’t. Julian Schwinger wrote a paper quantum electrodynamics II : vacuum polarization and self-energy. Space is a polarizable medium. You can also find modern authors saying much the same thing. See for example the 2008 paper Inhomogeneous vacuum: an alternative interpretation of curved spacetime.
Space is not some ideal Newtonian emptiness
Also see the Wikipedia aether theories article and note the quote by Robert B Laughlin: “it is ironic that Einstein’s most creative work, the general theory of relativity, should boil down to conceptualizing space as a medium when his original premise [in special relativity] was that no such medium existed”. Laughlin also said space is more like a piece of window glass than ideal Newtonian emptiness. He finished up saying this: “the modern concept of the vacuum of space, confirmed every day by experiment, is a relativistic ether. But we do not call it this because it is taboo”. Einstein is supposed to have done away with the aether in 1905 but in the end, he didn’t. He thought of space as a something rather than a nothing. In his 1929 essay on the history of field theory, Einstein described a field as a state of space. He was talking about gravitational fields and electromagnetic fields, and he said this: “it can, however, scarcely be imagined that empty space has conditions or states of two essentially different kinds”. The point to appreciate is that according to Einstein a field isn’t something that exists in space, it’s a state of space. As to what sort of state, I’d say a gravitational field is a place where space has an energy-density gradient. Or perhaps it’s better to say it’s a pressure-gradient, and that space has an elastic quality.
Space is modelled as an elastic solid
This is why if you google on Einstein elastic space, there’s plenty of hits. This is also why general relativity is related to continuum mechanics. This is why we have the stress-energy-momentum tensor, which “describes the density and flux of energy and momentum in spacetime”:
Public domain image by Maschen, based on an image by Bamse see Wikipedia
The shear stress term on the right tells you we’re dealing with something that could be modelled like some kind of elastic solid. The energy-pressure diagonal tells you it’s an elastic solid subject to pressure. For an analogy, imagine you have a block of gin-clear ghostly elastic jelly, with grid lines in it so you can see what’s going on. You slide a hypodermic needle into the centre of the block, and inject more jelly. This represents a concentration of energy bound up as the matter of a massive star. It creates a pressure gradient in the surrounding jelly. Stress is directional pressure, the pressure is outwards, and Einstein’s equation Gμν = 8πTμν is modelling the way gin-clear ghostly elastic space is conditioned by the energy you added. But don’t forget that you added jelly to represent energy, and that the jelly also represents space. Space doesn’t just have some kind of innate intrinsic vacuum energy. At some deep fundamental level, space and energy are the same thing.
Curved spacetime is not curved space and time
Yes, some people talk about elastic spacetime as opposed to space, wherein gravitational waves are elastic ripples that propagate through spacetime. I think that’s the wrong approach myself, because spacetime models space at all times, so there is no motion in spacetime. So it’s space that’s elastic, not spacetime. People also talk about curved spacetime as if it’s curved space and curved time, but I think that’s the wrong approach too. See John Baez and Emory Bunn’s preliminaries article dating from 2006: “Similarly, in general relativity gravity is not really a ‘force’, but just a manifestation of the curvature of spacetime. Note: not the curvature of space, but of spacetime. The distinction is crucial”. Space isn’t curved where a gravitational field is. Instead as Einstein said, space is “neither homogeneous nor isotropic” where a gravitational field is. Because a concentration of energy conditions the surrounding space. Because of this the speed of light is spatially variable, and because of that light curves. When you plot the spatial inhomogeneity using optical clocks at different elevations, your plot is curved because the inhomogeneity decreases with distance. That’s why the force of gravity diminishes with distance in a non-linear fashion, in line with the inverse square rule. Eventually when you’re a long long way from Earth your optical clock readings no longer exhibit any variation with elevation. At such a location space is homogeneous, spacetime is flat, light goes straight, and your pencil doesn’t fall down..
A better analogy
It’s important to appreciate that the curvature in the rubber sheet picture is not the curvature of space in some higher dimension. We have no scientific evidence for any such higher dimension. However we do have evidence that space is three dimensional. That’s why the three-dimensional gin-clear ghostly elastic jelly is a better analogy than the rubber sheet. You can draw it by imagining you’re looking at the Earth in the rubber-sheet lattice from underneath. Think in 3D, and you can get a feel for the way the surrounding jelly is pressed outwards rather than pulled down or inwards. Like this:
Image credit: NASA (I added the lattice lines)
However this depiction can still cause issues because the Earth is spherical. As a result the grid lines are curved, sending the wrong message about curved spacetime. To clear the air on that we need to zoom in to get rid of the curvature of the Earth. Like this:
Image credit: NASA (I removed the moon and added the lattice lines and the light beam)
The height of each rectangle in the lattice relates to the optical clock rate at that location. Because they’re optical clocks, the height of each rectangle also relates to the speed of light at that location. And because the speed of light is spatially variable, a beam of light going across the picture will curve downwards like the sonar wave. Hopefully you can see how this relates to Ricci curvature. Whilst the image above shows rectangles instead of geodesic balls, you can easily imagine that the more flattened the rectangle, the lower the gravitational potential. And usually when gravitational potential is lower, the gradient in gravitational potential is steeper, so light curves downwards all the more.
Why matter falls down
Anyway, once you understand why light curves, it’s easy to understand why matter falls down. All you need to know about is the wave nature of matter, as demonstrated by the Davisson-Germer experiment. And about the Einstein-de Haas effect which “demonstrates that spin angular momentum is indeed of the same nature as the angular momentum of rotating bodies as conceived in classical mechanics”. And about pair production, wherein we can make an electron and a positron out of light. See Hans Ohanian’s 1984 paper what is spin? He said this: “the means for filling the gap have been at hand since 1939, when Belinfante established that the spin could be regarded as due to a circulating flow of energy”. It’s like what Feynman said, if you look at the energy flow, you find that it “just circulates around and around”. So just think of an electron as light going around and around. Then simplify it to light going around a square path. Like this:
Now imagine it’s in a gravitational field. The vertical parts of the path stay vertical, but the horizontal parts bend down a little. So the electron falls down:
In essence the reducing speed of light is transformed into the downward motion of the electron. Internal kinetic energy is converted into external kinetic energy. But only the horizontal component bends down, so the Newtonian deflection of matter is only half the deflection of light. Since you can diffract protons and neutrons and other things too, the same principle applies to matter in general. To things like your pencil. Like Newton said in his 1692 letter to Richard Bentley, there is no magical mysterious action at a distance. There are no gravitons flying back and forth either. That might come as an unpleasant surprise to some. It isn’t the only one.