First of all, I find it very important that all the discussions on the two blogs above are about physics topics that have been settled for 100 years, about the high-school understanding of relativity. I think it is desirable to emphasize this point because much of the confusion arises when complete crackpots such as Lee Smolin say or write totally wrong things about relativity and they sell these totally wrong things as a cutting-edge research.
Special relativity is a 108 years old or new theory of space and time that correctly accounts for new phenomena that are known to occur when the observers' speeds approach the speed of light. It is a principled theory that constraints what particular constructive theories of individual phenomena and their classes may say and what they mustn't say.
All other theories must be made compatible with the two postulates of special relativity:
- Relativity postulate: the laws of physics have the same form in the coordinate systems of all observers moving by constant speeds in a constant direction (inertial frames)
- Constancy of the speed of light: the speed of light is constant, \(c\), regardless of the speed of the source and the speed of the observer
So Einstein's special relativity primarily modified mechanics – mechanics was forced to change. For example, the relative speed between two bodies on a collision course on a line whose speeds are \(u,v\) is no longer \(u+v\) but it is \((u+v)/(1+uv/c^2)\). But any other kind of phenomena if there were one aside from mechanics and electrodynamics – hydrodynamics, aerodynamics, thermodynamics etc. (although they're really derived theories from mechanics and perhaps electrodynamics, not really fundamentally new) – had to be adjusted to agree with the two postulates. Particle physics only accepts theories that agree with relativity, too. For particle physics, this is so automatic – quantum field theory and string theory are the frameworks of choice and all of them are relativistic – that we don't even realize how much the possible "theories of particles" have been constrained by relativity.
The postulates imply – and Einstein was able to prove from them – that the length of objects shrinks in the direction of motion; the rate at which (any) clocks are "ticking" is slowing down if the clocks are speeding up; the total relativistic mass is increasing with the speed; its conservation law is merged with the momentum conservation law to the 4-momentum conservation law; this also implies that the mass and energy conservation laws become one identical part of the 4-momentum law and (what we used to call) energy may be converted to (what we used to call) mass and vice versa via \(E=mc^2\), the most well-known equation of relativity among the laymen.
While mechanics (I really mean kinematics, the description of motion influenced by forces that are given and whose origin isn't analyzed) was adjusted to relativity in 1905 – it was the main point of it – the physics of gravity (the description of a particular force that causes the motion – and such descriptions belong to "dynamics", not "kinematics") remained mysterious because (among related problems), Newton's gravity seems to operate instantaneously which violates the speed limit, \(c\), that relativity imposes on the speed of propagation of any usable information.
Einstein spent the decade after the discovery of special relativity, 1905-1915, by attempts to reconcile the laws of gravity with the principles of special relativity. The result of this long but successful work, the general theory of relativity, pretty much inevitably and uniquely follows from special relativity (that is required to hold whenever the gravitational fields are negligible) and the equivalence principle (the statement that all bodies accelerate in gravitational fields by the same acceleration which means that freely falling frames are indistinguishable from the life outside gravitational fields, and must therefore locally preserve special relativity).
I will discuss GR as an unavoidable extension of SR momentarily. But let me first address a more trivial question:
Is SR applicable to phenomena in which objects accelerate?The answer is, of course, Yes. Special relativity would be useless if it were requiring all objects to move without any acceleration; after all, almost everything in the real world accelerates, otherwise the world would be useless. The correct claim similar to the proposition above is that special relativity has the same, simpler form in coordinate systems associated with non-accelerating observers. But that doesn't mean that we can't translate the predictions of a special relativistic theory to an accelerating frame. Yes, we can. It's as straightforward as a coordinate transformation. Fictitious forces will appear in the description. All of them are fully calculable.
We should point out that if it were impossible to consider accelerating observers, special relativity couldn't tell us anything about the twin "paradox". At least one of the twins, the astronaut, has to intensely accelerate during his life. But the total time measured by his clock – and by the aging of his organs, which is just another type of clocks (not too accurate one) -´is clearly composed of the proper times of tiny line intervals into which his world line may be divided. The infinitesimal pieces of his world line are straight so special relativity simply has to hold. When we compute i.e. integrate the total proper time along the world line, of course that we will find out that the twin-astronaut will be younger than his brother who spent decades on Earth.
We don't need general relativity because the presence of acceleration doesn't mean that there's a gravitational field. The curvature of the spacetime is still zero. Acceleration is locally equivalent to gravity by the equivalence principle but the clever way to use it isn't to envision unnecessary gravitational fields but, on the contrary, to undo the gravity whenever we can by replacing it with acceleration combined with no gravity – and for this combination, special relativity is sufficient.
Not being able to produce this right answer to the twin "paradox" means not to understand special relativity at the high-school level (at least we did learn basics of special relativity at the high school, a pretty ordinary high school). It's not wise, deep, clever, or sophisticated to be doubtful about the usual resolution to the twin "paradox". It is nothing more than a sign of brutal ignorance. (Christine Dantas is among those who believe that special relativity doesn't imply that the astronaut-twin will be younger because acceleration makes it impossible to use the theory. Holy cow. This lady has had a full big mouth about quantum gravity while high school physics is apparently way too hard for her.)
Now, let me switch to general relativity again. Sabine promotes a particular definition of special relativity:
Ask some theoretical physicist what special relativity is and they’ll say something like “It’s the dynamics in Minkowski space” or “It’s the special case of general relativity in flat space”. (Representative survey taken among our household members, p=0.0003). But open a pop science book and they’ll try to tell you special relativity applies only to inertial frames, only to observers moving with constant velocities.I don't think that it is downright incorrect to describe special relativity in Sabine's way. But I don't think it's the deepest or most natural way, either. More importantly, I do agree with the criticized books that at least something in special relativity does apply to observers moving with constant velocities only – the Lorentz symmetry only mixes the viewpoints of these observers and, consequently, the laws of physics only have the usual simple form in the coordinate systems connected with these observers. The difference between inertial and non-inertial systems is essential in special relativity and if that's the claim that Sabine criticizes, she is completely wrong.
Moreover, her "definition" of special relativity is useless. A definition is meant to be helpful to someone whose knowledge is at a lower level than the level at which the defined object is "obvious". If someone doesn't know special relativity, you won't help him much if your explanation will assume the knowledge of general relativity because, you know, general relativity is harder than special relativity.
But there's another, more conceptual reason why I consider Sabine's definition to be a sign of her (and her spouse's, as we were told) shallow knowledge of the subject. What is the reason? Her definition implicitly says that general relativity is the fundamental set of insights, rules, and principles and special relativity is just a minor corollary of it. While it's true that special relativity is a limit of general relativity obtained for gravitational fields going to zero, the actual "hierarchy of power" is the opposite: general relativity is just one application of special relativity – the incorporation of the gravitational field in a special-relativity-invariant way. While general relativity is arguably the prettiest (and geometrically most non-trivial) classical application of the rules of special relativity, in principle it is on par with Yang-Mills theory or any other (special) relativistic field theory.
This claim of mine may be interpreted as a modern interpretation of the philosophy underlying relativity – and widely appreciated by most of the competent modern theoretical/phenomenological particle physicists (people who were clearly not included in Sabine's low-brow survey). But there's a sense in which it's ancient, too. What's the sense? Well, the insight is ancient because Einstein simply didn't have a choice when he was searching for a relativistic theory of gravity between 1905 and 1915. General relativity is the unique theory obeying the postulates of special relativity that describes the gravitational force – by which I mean a force (and we can prove that it's the force because such a force must be unique for a physical system) that respects the equivalence principle.
The gravitational field must be given by some components of the mass/energy/momentum-encoding stress-energy tensor. Because the strength of the field around a physical system as measured at infinity cannot change (in analogy with the field around a charge in electrostatics), it must be conserved quantities that source the gravitational field/influence. Because our goal is a gravitational force that depends on the mass, it's clearly the whole stress-energy tensor \(T_{\mu\nu}\) that must be involved in sourcing the gravitational field (\(T_{00}\) which must surely influence the gravitational field isn't a Lorentz-invariant quantity and the Lorentz transformations of this quantity involve all other components of the tensor). The corresponding "potentials" of the gravitational field must be organized as a symmetric tensor with two indices, too. It's \(h_{\mu\nu}\).
However, the derivatives of the field \(h_{\mu\nu}\) contribute to the energy as well, like the derivatives of any matter field. We are led to the question how the field sources itself. We're brutally constrained by the equivalence principle because physics in the \(h_{\mu\nu}\) field that linearly depends on the coordinates must be indistinguishable from physics outside any nonzero fields: a freely falling observer (in the linear \(h\)-field) mustn't be able to figure out that he's in a gravitational field at all.
This is only possible if there is a rather large symmetry that is able to identify configurations with different profiles of \(h_{\mu\nu}\) – identify some configurations where this field is nonzero (and even non-constant) with the configuration where it's zero. So this symmetry must be mixing the gravitational field \(h_{\mu\nu}\) with something that was nonzero to start with. It must have the same tensor structure and we conclude that it must be the pre-existing metric tensor \(\eta_{\mu\nu}\). The only symmetry that is able to produce the right number of symmetries acting on these metric tensors is the diffeomorphism symmetry under which the "total metric"\[
g_{\mu\nu}=\eta_{\mu\nu} + h_{\mu\nu}
\] transforms as the tensor field. So we're led to general relativity as the only possible (special) relativistic description of gravity that uses fields.
This was a sequence of arguments that tried to be as classical as possible. Modern particle physicists would present a similar but quantum-field-theory-based version of the ideas. Because it's locally sourced by the stress-energy tensor, gravity must involve spin-two fields. In the covariant, manifestly Lorentz-invariant description, spin-two fields have some positively definite components \(h_{ij}\) and perhaps \(h_{00}\) (which will also be mostly killed, despite its good sign) and some negative-normed components \(h_{0i}\), ghosts. The latter is unacceptable because it leads to the prediction of negative probabilities for some processes. So there must exist a symmetry that decouples all the ghosts. The symmetry has to be local and have a whole "vector" of parameters at each spacetime point. Coordinate redefinitions \(\delta x^\mu\) are the only solution. For gravity in terms of quantum fields, you need spin-two fields and the diffeomorphism invariance is necessary to get rid of their pathological, negative-normed components. The rest of the GR follows; the Ricci scalar is the lowest-order (in the number of derivatives) coupling compatible with the required symmetry but there may also be higher-order corrections (whose effect becomes negligible at long distances).
Some people would declare all the derivations above to be heresies because they think it is a blasphemy to ever write the metric tensor as a sum of two or several pieces because such a blasphemy contradicts the holy beauty of general relativity as written in an unwritten commandment somewhere. ;-) The price they pay for this medieval, unjustifiable, irrational, stupid taboo (the commandment really says "you shall never make your hands dirty by any science that actually applies to a situation in the real world or answers some questions beyond the questions whose answers you have been given to start with, by science that requires you to write anything else than the most beautiful form of the basic equations") is very high: They can't understand some key facts about modern physics, e.g. that and why the general theory of relativity is unavoidable given the validity of special relativity and the existence of gravity sourced by the energy-and-momentum density and their fluxes/currents.
Many people in the Backreaction discussion are confused about many other things.
For example, is a charged object sitting somewhere on the Earth's surface emitting electromagnetic and/or Unruh and/or gravitational radiation?
The answer is, of course, No. If it were radiating in any of the three ways (to be precise, by radiation I mean sending physical photons, gravitons, or other particles to infinity), it would have to lose energy to avoid the violation of the energy conservation law. But the charged object is already sitting at a place where the energy is minimized so there's no way to extract more energy out of the particle.
Relatively to a freely falling frame, the charged object sitting on the Earth's surface is accelerating so it should emit all three kinds of radiation, some people could argue. If it emits no radiation, doesn't it violate the equivalence principle?
No, it doesn't. First of all, the equivalence principle is only guaranteed locally. But in the previous paragraphs, we were asking whether particles are emitted to infinity. This requires us to connect the vicinity of the Earth with infinity, to compare them. But such a global connection turns the existence of the Earth's gravitational field into an objective fact. There exists no flat-space-based equivalent description of a region that would include both Earth's vicinity as well as the asymptotic region at infinity. So the equivalence principle isn't really applicable. There's no justifiable way to argue that the charged sitting object should emit radiation.
There are other ways to argue and reach the same conclusion.
For example, the equivalence principle identifies the experience of a freely falling observer with those of an inertial observer in the flat spacetime. But the identification only holds if "all other factors are equal". The freely falling observer who is going to hit the Earth's surface soon doesn't have "all other factors equal". In particular, there may be some extra radiation coming from the rest of the Universe. It just happens that the radiation is such that it perfectly cancels the would-be electromagnetic/Unruh/gravitational radiation of the charged object sitting on the Earth.
To make this discussion really complete, I would have to describe a formalism that has something to cancel at all and distinguish the different amounts of radiation as seen by a nearby static, nearby accelerating, or infinitely distant detector. The discussion could get unnecessarily messy and repetitive. But my point that shouldn't get lost in this technical material is that only the black holes emit the Hawking radiation. One actually needs the horizon for that. If there's no horizon, there's no energy loss by the Hawking or another acceleration-based radiation. (And this 1999 paper is just wrong. It's not the only one.)
Why does the horizon matter? If there's the horizon, one simple fact holds: the black hole interior can't possibly send any radiation (positive-energy one or a "compensating one") in the outward direction; nothing gets out of the black hole. That's why the frame of an observer who is freely falling into a black hole (with a horizon) is as equivalent to an inertial observer in an empty space as you can get. He could have been freely falling throughout his life which explains that no radiation was going in his direction.
On the other hand, there's no radiation going from the black hole interior, against him, either. It's forbidden by the blackness of the black hole. It's this latter property that doesn't hold for the Earth. The Earth imposes different boundary conditions on the surface than the black hole enforces on the event horizon. If the Earth were a conductor, the electrostatic potential would vanish on the surface. The relevant modes of waves would be standing waves above the Earth's surface. While the condition "killing" one-half of the modes in the black hole case says that "nothing is coming in the outward direction", the conditions are different for the Earth: "no waves are inside the conducting Earth". The latter condition is past-future-symmetric, unlike the condition for the black hole.
The vacuum is Unruh and electromagnetic radiation-free in the "most natural frames". For black holes, it's the freely falling frame because you can just freely fall and you will never notice that something is unnatural about that frame (the singularity kills you before you realize that). That's why there's no radiation in this frame while the frame of an observer keeping himself above the horizon by jets experiences Unruh radiation that penetrates through the black hole's gravitational field and becomes real, physical Hawking radiation at infinity.
For the Earth, the most "vacuum-like" frame is one associated with the surface because the freely falling observer will hit the Earth's surface and the headache will convince him it's not the frame most similar to the empty space. ;-) So the Earth stabilizes all the surrounding fields relatively to its static surface and relatively to this frame, there's no radiation – and frames accelerating relatively to the surface's frame will see some radiation. Of course, a semiclassical analysis of GR coupled to electromagnetism offers you a more reliable but less funny derivation of the same conclusion.
One should emphasize that the Unruh/Hawking radiation for the Earth, even if there were one, would be ludicrously weak. The typical wavelength of the emitted photons would be comparable to \(c^2/g\) which is about \(10^{16}\,{\rm meters}\), not far from a light year. It's clearly just an academic debate for the Earth as the very weak radiation would be totally unobservable – dozens of orders of magnitude weaker than the observable one. But it would still be an inconsistency if stable objects and particles like that would radiate because of some incorrectly applied equivalence principle.
Off-topic: Mr Ilja Hurník (*1922) died. He was a serious Czech composer of highly non-classical music for classical instruments and a piano virtuoso but people like your humble correspondent know him as the author of small pieces such as the "Merry Postman" (yes, he's ringing the bell and knocking the door) and "Little Soldier" above which I liked to play when I was 8 or so. ;-)
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