Once you know that an optical clock goes slower when it’s lower because light goes slower when it’s lower, you soon understand why light curves. Not because it follows the curvature of spacetime. Because the speed of light is spatially variable, like Einstein said. Then once you know about the wave nature of matter and electron spin, you soon understand why matter falls down, and why the Newtonian deflection of matter is only half the deflection of light. Then once you know how gravity works, you soon understand why there is no magical mysterious action at a distance, like Newton said.
The equivalence principle is only valid in the infinitesimal
You also come to understand is that the principle of equivalence is something of a myth. In the Wikipedia equivalence principle article you can read that “being on the surface of the Earth is equivalent to being inside a spaceship (far from any sources of gravity) that is being accelerated by its engines”. However the two situations are not exactly equivalent, because standing still in inhomogeneous space is not exactly the same as accelerating through homogeneous space:
Reference frame pictures by me
Yes, when you’re in a windowless room either on Earth or in the accelerating spaceship, you might say light curves and matter falls down. But like Wikipedia says, the room has to be small enough so that “tidal forces may be neglected”. As for how small, note that in chapter 20 of Relativity: The Special and General Theory Einstein said it’s “impossible to choose a body of reference such that, as judged from it, the gravitational field of the Earth (in its entirety) vanishes”. You cannot “transform away” the Earth’s gravitational field. That isn’t true for the accelerating spaceship, That’s why in Fundamental ideas and methods of the theory of relativity Einstein said the special theory of relativity is “nowhere precisely realized in the real world”. He said it’s only valid “in the infinitesimal”. Your room has to be an infinitesimal room for the principle of equivalence to apply. So it doesn’t apply at all.
The midwife should be buried with appropriate honours
This is why on pages ix and x in his 1960 preface to relativity: the general theory, John Synge said the equivalence principle performed the essential office of midwife at the birth of general relativity, but should “be buried with appropriate honours”. It started in 1907 with Einstein’s happiest thought, wherein the falling observer doesn’t feel his own weight. This was Einstein’s train ticket to understanding, to be discarded once he reached his destination. That’s when he appreciated that exact equivalence demanded an infinitesimal room. However the equivalence principle is said to have been successfully tested on multiple occasions:
Image from the 2010 paper Quantum tests of the equivalence principle with atom interferometry from the QUANTUS team
That’s because the waters have been muddied. The Eötvös experiment is said to be a test of the equivalence principle, but it was first performed in 1885, to test whether different materials were equivalently affected by gravity. It’s the same for the Eöt-Wash experiments, which tested “the universality of free fall”. What they tested was the weak equivalence principle, which is not Einstein’s equivalence principle. See Kevin Brown‘s mathspages article on the many principles of equivalence where you can read that the equivalence principle has undergone several changes over the years. Brown talks about the strong equivalence principle, and says this: “the modern statement of the strong equivalence principle, of the assertion that the laws of physics are the same for all frames of reference (i.e. independent of velocity) is also conceptually quite distinct from the original meaning of Einstein’s equivalence principle”. OK, but see the Einstein equivalence principle in the Wikipedia equivalence principle article. It says “the outcome of any local non-gravitational experiment in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime”. That isn’t Einstein’s equivalence principle either. It doesn’t square with what Einstein said in 1907. Or with what he said in 1939 about light rays taking an infinitely long time to reach a black hole event horizon. Which doesn’t square with the idea that an object falling from infinity will be moving at the speed of light when it crosses the event horizon.
The fine structure constant is a running constant
The Wikipedia article also says the fine-structure constant must not depend on where it’s measured. That doesn’t square with the fine structure constant being a running constant. It “depends upon the energy at which it is measured”, and so must surely vary with gravitational potential. Solar probe plus was going to test this alleged violation of the equivalence principle, and so would be a “test of relativity”, but it looks like it’s now going to be purely a heliophysics mission. That’s a shame, because it might have brought the issue to a head. It’s an issue that’s been around for way too long. John D Norton talked about it in his 1985 paper what was Einstein’s principle of equivalence? He said it was a special relativity principle that dealt only with fields that could be transformed away. He talked of an old view and a new view, and said “the equivalence of all frames embodied in this new view goes well beyond the result that Einstein himself claimed in 1916”. This new view is why there’s so many papers on testing the equivalence principle when the old view doesn’t apply at all. Unfortunately the new view doesn’t apply either. We don’t have gamma ray bursters for nothing.
The ascending photon does not lose energy
Something else that doesn’t apply is gravitational redshift. Not the way they say. Take a look at page 149 of Relativity, the Special and General Theory. Einstein said “an atom absorbs or emits light at a frequency which is dependent on the potential of the gravitational field in which it is situated“. When the ascending photon ascends, its E=hf energy does not reduce, and nor does its frequency. There is no magical mysterious outflow of energy from the photon. It doesn’t get redshifted at all. That’s a myth. Instead the photon was emitted at the lower frequency at a lower elevation. Conservation of energy applies. It appears to have less energy at the higher elevation because that’s where optical clocks go faster, along with everything else. So we measure the photon frequency to be reduced, even though it didn’t change at all.
But it does speed up
Another myth is a gravitational field that’s so strong it can drag light back down. It’s a myth that’s been around for a long time now. See for example Stephen Hawking’s paper singularities and the geometry of spacetime dating from 1966. On page 76 Hawking talked of “such a strong gravitational field that even the ‘outgoing’ light rays from it are dragged back”. This is incorrect, because a gravitational field is a place where the speed of light is spatially variable. It varies in the room you’re in. That’s why your pencil falls down. The speed of light at the ceiling is greater than the speed of light at the floor. Hence the ascending photon doesn’t slow down as it ascends. Au contraire, it speeds up.
Black and white falling gif by James Zanoni, inverted and cropped by me
If I could somehow make the gravitational field in your room much stronger, there would be a much bigger difference between the speed of light at the ceiling and the speed of light at the floor. So the ascending photon would speed up all the more. We don’t measure it as speeding up because we use the local motion of light to define our second and our metre, and then we use them to measure the local motion of light. So we always say it’s 299,792,458 m/s. But nevertheless that ascending photon does speed up. It doesn’t slow down. It isn’t like an ascending brick.
The mass deficit
Talking of which, when you throw a 1kg brick up in the air you do work on it. You give it kinetic energy. Whilst conservation of momentum p=mv means there is an effect on the Earth, it’s very slight. The kinetic energy KE=½mv² is not shared equally because the Earth doesn’t move in any detectable fashion, so the brick gets virtually all of the kinetic energy. It’s akin to what happens when you fire a projectile from a cannon, only more so:
Image drawn by me
As the brick ascends it slows down, because gravity converts the brick’s kinetic energy into gravitational potential energy. Note that this gravitational potential energy is in the brick, not in the gravitational field, and not in the Earth-brick system. You did work on the brick. There is no magical mysterious mechanism outflow of energy from the brick. The kinetic energy you gave to the brick merely gets converted into potential energy, which is in the brick. This potential energy is mass-energy. Hence when the brick is at the top of its arc, its mass is greater. Then when the brick falls back to Earth the situation is reversed. Gravity converts potential energy, which is mass-energy, into kinetic energy. Once the brick hits the ground this kinetic energy gets dissipated, and you end up with a mass deficit. See the mass-energy relation section of the Wikipedia binding energy article: “a bound system is typically at a lower energy level than its unbound constituents because its mass must be less than the total mass of its unbound constituents”. Binding energy is not some actual thing that consists of negative energy. It’s the reduction in the mass-energy of the real things that consist of positive energy. Hence the mass of the brick at rest on the ground is less than the mass of the brick at the top of its arc.
Escape velocity means the kinetic energy leaves the system
You can check this by throwing the brick upwards at 11.7 km/s, so giving it escape velocity. It will end up leaving the system, taking the original 1kg worth of mass-energy away, along with 11.7 km/s worth of kinetic energy. This gets converted into gravitational potential energy, which is in the brick, increasing its mass. You did work on the brick. Once the brick escapes the Earth’s gravitational field, it leaves the system, and so does the kinetic energy that is now mass-energy. Hence the mass of the brick at rest on the ground is less than the mass of the brick in free space.
The mass of an electron varies
The same applies to a 511keV electron. When you throw the electron up in the air, you do work on it. You give it kinetic energy, and gravity turns this into gravitational potential energy, which is mass energy. Hence the mass of the electron at rest on the ground is less than the mass of the electron at the top of its arc. In similar vein the mass of an electron in a hydrogen atom is less than that of a free electron. The proton is only 1836 times the mass of the electron, so the situation isn’t quite so unequal as the Earth and the brick. But if it was, when an electron “falls” towards a proton to form a hydrogen atom, some of its mass-energy gets converted into kinetic energy, which gets dissipated as a 13.6eV photon:
Image from Rod Nave’s hyperphysics
The electron mass is then circa 13.6eV less than 511keV, and this 13.6eV is the binding energy. Unfortunately there’s a myth that says the electron has an invariant mass of 510.9989461(13)keV. The irony of course is that invariant mass varies. It varies even more in a Helium-4 nucleus which consists of two protons and two neutrons. The binding energy there is circa two million times greater than the electron-proton binding energy.
Gravitational field energy is positive
Another myth is that gravitational field energy is negative. It’s arisen because people historically set Newtonian gravitational potential energy to be zero at infinity. That means it’s deemed to be negative at sea level. Only it isn’t really negative, it’s still positive. That’s why you can drop a brick into a hole and convert some more of that gravitational potential energy into kinetic energy. That’s not the end of the world, but unfortunately gravitational potential energy then gets confused with gravitational field energy. On page 82 of his 2002 book The Theory of Everything, Stephen Hawking said this: “in a sense the gravitational field has negative energy. In the case of the whole universe, one can show that this negative gravitational energy exactly cancels the positive energy of the matter, so the total energy of the universe is zero”. It isn’t true. In his 1916 Foundation of General Relativity Einstein said “the energy of the gravitational field shall act gravitatively in the same way as any other kind of energy”. Gravitational field energy is positive, just like electromagnetic field energy. And when you drop a 1kg brick into a black hole from some “infinite” distance, the black hole mass increases by 1kg. It doesn’t decrease by 1kg. It would be similar if you dropped stars and planets and everything else into a single supermassive black hole. Hence the zero-energy universe is just another myth. As are some other things in cosmology. I’ll talk more about them at a later date.
Misinterpretations of the theory
For now, see Peter Brown’s 2002 paper Einstein’s gravitational field. He quotes John Synge and John Ray on page 20. He also says Einstein’s general relativity and modern general relativity differ, and the latter suffers from contradiction and confusion. Chung Lo says essentially the same in 2015 in his rectification of general relativity. He blames Wheeler and others: “Wheeler led his school at Princeton University while his colleagues, Sciama and Zel’dovich (another H-bomb maker) developed the subject at Cambridge University and the University of Moscow”. Lo goes on to say “the misinterpretations of the theory and errors such as the singularity theorems have been accepted as part of the faith”. The bottom line is that the general relativity that is taught today, is not the same as Einstein’s general relativity. Instead it’s like some kind of doppelganger from Invasion of the Body Snatchers. What’s more like Einstein’s general relativity is the polarizable vacuum proposed by Robert Dicke and then Harold Puthoff: “in essence, Dicke and Puthoff proposed that the presence of mass alters the electric permittivity and the magnetic permeability of flat spacetime”. Dicke’s 1957 paper was Gravitation without a Principle of Equivalence. Puthoff’s 1999 paper was Polarizable-Vacuum (PV) representation of general relativity. Both Dicke and Puthoff refer to Harold Wilson’s 1921 paper an electromagnetic theory of gravitation. I am reminded of Julian Schwinger’s 1949 paper quantum electrodynamics II : vacuum polarization and self-energy.
Perhaps the past, if looked upon with care and hindsight, may teach us where we possibly took a wrong turn
There’s more, much more, but we’ll come on to that later. For now, there’s a more pressing lead to follow, and it’s this: a gravitational field is not a place where space is curved. Instead it’s a place where space is neither homogeneous nor isotropic. It’s inhomogeneous in a non-linear way, in line with the inverse square rule. So if space isn’t curved where a gravitational field is, where is it curved? To answer that we need to look elsewhere. We also need to take careful note of Bert Schroer’s 2003 essay on Pascual Jordan, his contributions to quantum mechanics and his legacy in contemporary local quantum physics. Because on page 9 he says this: “in times of stagnation and crisis as the one we presently face in the post standard model era of particle physics, it is helpful to look back at how the protagonists of quantum field theory viewed the future and what became of their ideas and expectations. Perhaps the past, if looked upon with care and hindsight, may teach us where we possibly took a wrong turn and what alternative path was available”. You bet Bert.
I didn’t say anything about the wavelength of the ascending photon above. But I had an interesting conversation about that recently:
We start with the photon in special relativity. A series of identical photons are released at point A, and you’re at point B. You measure the photon energy along with the frequency and wavelength, wherein E=hf and E=hc/λ. Then you accelerate away from point A until you reach some speed v. Then you repeat your measurements. The photon energy appears to have descreased, as does the frequency, whilst the wavelength appears to have increased. But those identical photons haven’t changed one jot. Instead you’ve changed.
We then move on to general relativity. A series of identical photons are released upwards at point A, and you’re above them at point B. You measure the photon energy along with the frequency and wavelength, wherein E=hf and E=hc/λ. Then you climb upwards away from point A until you reach some elevation C. Then you repeat your measurements. The photon energy appears to have decreased, as does the frequency, whilst the wavelength appears to have increased. In my conversation I said those identical photons haven’t changed one jot. Instead you’ve changed.
But I think I was wrong. Because the second is 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”. In essence you count light waves going by, and when you get to 9,192,631,770 you say a second has elapsed. So if the light goes slower the second is bigger. Then you use the second along with the light to define the metre as “the length of the path travelled by light in vacuum during a time interval of 1⁄299,792,458th of a second”. Note here that if the light goes slower the second is bigger, and these two opposites cancel each other out. So the metre is unchanged. The second changes, but the metre doesn’t.
So, when you’re at the higher elevation your seconds are shorter. So it looks like the photon frequency has decreased. Even though it didn’t. It also looks as if the photon wavelength has increased. But your metre hasn’t changed. So that should mean that the wavelength really has increased. Interesting stuff.