The relativity theorem

There is a relativity principle, postulate, and theorem, and they get confused. I explain the difference.

The principle of relativity

is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference.

It goes back to ancient debates about the motion of the Earth. J.C. Maxwell coined the word "relativity", and pointed out the difficulties for electrodynamics. Poincare was among the first to argue that the principle applies to electrodynamics, and defined it in 1904:

The principle of relativity, according to which the laws of physical phenomena should be the same, whether for an observer fixed, or for an observer carried along in a uniform movement of translation; so that we have not and could not have any means of discerning whether or not we are carried along in such a motion.

The Lorentz aether theory includes:

A fundamental concept of Lorentz's theory in 1895[A 1] was the "theorem of corresponding states" for terms of order v/c. This theorem states that a moving observer with respect to the aether can use the same electrodynamic equations as an observer in the stationary aether system, thus they are making the same observations.

Lorentz improved this to cover all velocities in his 1899 and 1904 papers.

The idea of using the same equations is mathematically sloppy. The better statement is not just that the equations are formally the same, but that the equations have a geometric formulation that works for all observers. The variables in the equations are really tensors where a routine change of variables gives them meaning for different observers. This modern concept is called Lorentz covariance:

In standard physics, Lorentz symmetry is "the feature of nature that says experimental results are independent of the orientation or the boost velocity of the laboratory through space". Lorentz covariance is a related concept, covariance being a measure of how much two variables change together. Lorentz covariance (from Hendrik Lorentz) is a key property of spacetime that follows from the special theory of relativity.

The covariance was discovered by Poincare in 1905, and geometrically explained by Minkowski in 1907:

In particular, Lorentz's theory gives a good account of the non-existence of relative motion of the earth and the luminiferous "Æther"; it shows that this fact is connected with the covariance of the original equation, at certain simultaneous transformations of the space and time co-ordinates; these transformations have obtained from H. Poincaré the name of Lorentz-transformations. The covariance of these fundamental equations, when subjected to the Lorentz-transformation, is a purely mathematical fact; I will call this the Theorem of Relativity; this theorem rests essentially on the form of the differential equations for the propagation of waves with the velocity of light. ...

Now if hereafter, we succeed in maintaining this covariance as a definite connection between pure and simple observable phenomena in moving bodies, the definite connection may be styled the Principle of Relativity.

So the relativity theorem is the mathematical covariance of physical variables under a change of observers. The relativity principle is the physical idea that observers in different frames see the same physics.

So what was Einstein's contribution to this? His famous 1905 relativity paper says:

They suggest rather that, as has already been shown to the first order of small quantities, the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good. We will raise this conjecture (the purport of which will hereafter be called the “Principle of Relativity”) to the status of a postulate, ...

Einstein is referring to Lorentz's 1895 theorem of corresponding states, as the 1895 paper only proved it to first order in velocity. Einstein is ignoring Lorentz's 1899 and 1904 papers proving it for all velocities. This postulate is why Lorentz said in his 1906 Columbia U. lectures on relativity that "Einstein simply postulates what we have deduced". Lorentz proved his theorem of corresponding states for Maxwell's equations, and Einstein just assumed it as a postulate and confusingly called it the relativity principle.

Einstein achieved some phony simplicity with this approach, but it was useless. There was nothing new, mathematically or physically, in postulating what someone else had proved. In 1908, everyone was convinced of the superiority of the Poincare-Minkowski approach, and adopted Lorentz covariance as the cornerstone to relativity. The Einstein approach was scrapped, and Lorentz covariance continues to be one of the central physics concepts today.

Tegmark annoys the angry atheists

I just criticized Max Tegmark, and now he is stirring up more trouble:

I'd been warned. A friend cautioned me that if we went ahead and posted our MIT Survey on Science, Religion and Origins, I'd get inundated with hate-mail from religious fundamentalists who believe our universe to be less than 10,000 years old. We posted it anyway, and the vitriolic responses poured in as predicted. But to my amazement, most of them didn't come from religious people, but from angry atheists! I found this particularly remarkable since I'm not religious myself. I have three criticisms of these angry atheists:

1) They help religious fundamentalists: ...

2) They could use more modesty:
If I've learned anything as a physicist, it's how little we know with certainty. In terms of the ultimate nature of reality, we scientists are ontologically ignorant. For example, many respected physicists believe in the so-called Copenhagen Interpretation of quantum mechanics, according to which a fundamentally random process called "wavefunction collapse" occurs whenever you observe something. This interpretation has been criticized both for being anthropocentric (quantum godfather Niels Bohr famously argued that there's no reality without observation) and for being vague (there's no equation specifying when the purported collapse is supposed to happen, and there's arguably no experimental evidence for it).

Let's compare the ontological views of Niels Bohr to those of a moderate and tolerant religious person. At least one of them is incorrect, since Bohr was an atheist. Perhaps neither is correct. But who's to say that the former is clearly superior to the latter, which should be ridiculed and taunted? Personally, I'd bet good money against the Copenhagen Interpretation, but it would be absurd if I couldn't be friends with those believing its ontology and unite with them in the quest to make our planet a better place.

3) They should practice what they preach: ...

One of those angry atheists, leftist-atheist-evolutionist Jerry Coyne, writes:

I’m curious to hear Richard’s answer to Krauss’s question, “Richard, what’s more important in some sense: if you had a choice — to explain science or destroy religion?”

That would be a tough one for me. How do you think Dawkins answered? And how would you have answered if you possessed Dawkins’s proficiency at explaining evolution as well as his enormous public profile as an atheist?

Really? I would have guessed that nearly all science professors would rather explain science than destroy religion, especially since many religions are supportive of science.

Tegmark is correct that most Christians accept evolution and most of the rest of modern science. Yes there are some ontological differences, but you can also find those within physicists discussing quantum mechanics.

That being said, Tegmark slanders the Copenhagen Interpretation. The interpretation is an explanation of how we perceive the world, and how we can update our knowledge of atomic phenomena. I would not call the collapse a fundamentally random process. But I don't want to nitpick here. His point is that physicist disagree about quantum ontology, and that is certainly correct.

An electron is not a list of numbers

MIT cosmologist Max Tegmark just gave this lecture:

In 1939, Paul Dirac observed that “the physicist, in his study of natural phenomena, has two methods of making progress”: experiment and observation, and mathematical reasoning. Although he said, “there is no logical reason why the second method should be possible,” nevertheless it works, and to great effect. The key, Dirac felt, was beauty, leading him to his principle that successive theories of nature are characterized by increasing mathematical beauty. The results of this were rich and included some predictions not confirmed until after Dirac’s death. Nevertheless, the powerful guidance Dirac found in mathematics did sometimes lead him astray, as he rejected the principle of “renormalization,” developed by Feynman, Schwinger, and Tomonaga, to remedy the nonphysical infinities that kept cropping up in Dirac’s equations for quantum electrodynamics. Even as other physicists accepted it, Dirac never did, saying it was “just not sensible mathematics.” Nevertheless, it was powerful physics.

He followed with this interview:

Why do the atoms have those properties? Because they’re made of quarks and electrons. What about the electron? What properties does it have? And the cool thing is, all the properties that electrons have are purely mathematical. It’s just a list of numbers. So in that sense, an electron is a purely mathematical object. In fact, there’s no evidence right now that there’s anything at all in our universe that is not mathematical.

Yes, there is evidence. There is no purely mathematical description of an electron because, according to the rules of quantum mechanics, measurements on an electron can depend on measurements of a possibly-distant entangled electron.

The usual explanation is that the two electrons have a joint mathematical description as a spinor wave function in a tensor product Hilbert space. This leads to paradoxes of relativity and causality.

There are different interpretations of quantum mechanics. The electron is never just a list of numbers. All attempts to reduce the electron to some lcoal set of numbers have failed. We can predict measurements of an alectron based on previous measurement of that electron as well as any entangled electrons. But we cannot reduce the electron to numbers.

Dirac got Einstein's disease, and was unproductive once he got the idea that physical theories could be predicted by mathematical beauty.

Tegmark says he is writing a book, and will promote it on his Facebook page.

Moon Man Galileo

Adam Gopnik writes a long Galileo article in the New Yorker magazine:

But, when you’ve read through his collected evidence, the myth seems pretty much right: Galileo wrote a book about the world saying that the earth goes around the sun, and the Church threatened to have him tortured or killed if he didn’t stop saying it, so he stopped saying it. Mayer believes that had Galileo been less pugnacious things would have worked out better for science; yet his argument is basically one of those “If you put it in context, threatening people with hideous torture in order to get them to shut up about their ideas was just one of the ways they did things then” efforts, much loved by contemporary historians.

To be sure, Galileo’s trial was a bureaucratic muddle, with crossing lines of responsibility, and it left fruitfully unsettled the question of whether Copernican ideas had been declared heretical or if Galileo had simply been condemned as an individual for continuing to promote them after he had promised not to. But what is certain is that, in 1633, Galileo was threatened with torture, forced on his knees to abjure his beliefs and his book, and then kept under house arrest and close watch for the rest of his life. (Albeit of a fairly loose kind: John Milton came to see him, and the image of the imprisoned scientist appears in Milton’s defense of free speech, the “Areopagitica.”)

This gives the impression that Galileo was going to be tortured for telling a scientific truth. No such thing was ever considered.

The threat of torture was only part of the oath to tell the truth. Any witness at any trial could be said to have been under such a threat.

Furthermore, Galileo was not restricted from announcing and publishing scientific observations, facts, and theories. What he did was to publish a book arguing the entirely false notion that the daily tides prove the motion of the Earth. The Church officials explained to him why he was wrong, and warned him not to publish it.

For the last century, the favored understanding is that motion is relative to a frame of reference. It is nonsense to say that the Earth moves, unless you also supply a reference frame. And Galileo's theory of tides was wrong, contrary to what was established knowledge for millennia.

Quoting Poincare on relativity

Wikipedia has many excellent articles on relativity, including its history. Arguments about credit for the theory are in Relativity priority dispute. These articles over-credit Einstein in their opinions, but have many facts and source links so that you can decide for yourself.

I found that editing these pages was frustrating, as you can see in the discussion at Talk:History of special relativity. The articles explain who did what, but then deny credit to Lorentz and Poincare because some of their terminology was allegedly defective.

Here are examples of Poincare quotes:

"some day, no doubt, the aether will be thrown aside as useless." Science and hypothesis [1902]

"The watches adjusted in that manner do not mark, therefore, the true time; they mark what one may call the local time, so that one of them goes slow on the other." The Principles of Mathematical Physics [1904]

"since the fictitious electromagnetic mass depends upon this velocity, the total apparent mass, alone observable, must depend upon it, though the real mass does not depend upon it and may be constant." The New Mechanics [1908]

Historians use these quotes to argue that Poincare did not really believe in relativity. The argument is that "true time" must have meant time as measured by an observer who is stationary with respect to the aether, and that belief in the aether is contrary to relativity.

They also argue that terms like "apparent mass" indicate that Poincare did not believe that relativistic effects were real. Poincare does say that the apparent mass is what is observable, but they say that the reference to "real mass" suggests that there is something not real about the (observed) apparent mass.

These arguments are purely terminological, and have no substance. Poincare's "apparent mass" and "real mass" are the same as what some textbooks call "relativistic mass" and "rest mass". There is not even a consensus today about the best terms for these quantities.

The argument about "true time" is even stranger. Poincare is plainly denying true time and the aether, not endorsing them. It is as if he said, "I went to the zoo and I saw a gorilla, not bigfoot." Such a statement would not imply a belief in bigfoot.

A NY Times column answers this question:

Q. When I read that “the universe is 13.7 billion years old,” I wonder: Don’t scientists use some more universal measurement than years, something not tied to the orbit of one tiny planet?

The answer is true time, of course.

While some people claim that Lorentz's understanding of local time was deficient, Poincare clearly and correctly understood Lorentz's local time as the time observed on clocks. As Michel Janssen explains in chap. 3 of his thesis:

The passage I have in mind occurs in the section “The principle of relativity” in Poincaré’s famous 1904 lecture in St. Louis. Poincaré vividly describes the situation in ether theory around the turn of the century. While the dominant theories posit a stationary ether, the experiments aimed at detecting the earth’s presumed motion through this medium consistently give negative results. The task of explaining these experimental findings theoretically, Poincaré writes “was not easy, and if Lorentz has got through it, it is only by accumulating hypotheses” (Poincaré 1904, p. 99). Starting a new paragraph, he continues: “The most ingenious idea was that of local time” (ibid.). Poincaré proceeds to explain that if an observer in uniform motion through the ether synchronizes his clocks using what a modern reader immediately recognizes as the light signaling method from Einstein 1905a, these clocks will not read the true Newtonian time, but Lorentz’s local time. Poincaré does not indicate in any way that this interpretation is entirely his own and is not to be found in any of Lorentz’s writings up to this point. For Poincaré, the notion of local time clearly involves a physical assumption. Poincaré assumes that the local time is, in effect, the time registered by moving observers, which helps to account for the fact that such observers do not detect ether drift. After making this point (ibid, pp. 99–100), Poincaré starts his next paragraph saying: “Unhappily, that does not suffice, and complementary hypotheses are necessary; it is necessary to admit that bodies in motion undergo a uniform contraction in the sense of the motion” (ibid., p. 100). So, for Poincaré the notion of local time and the contraction hypothesis (to be discussed in detail in sections 3.2 and 3.3) have essentially the same status. They are both physical assumptions. This is a far cry from Lorentz’s understanding of the situation. He obviously looked upon the contraction hypothesis as a physical assumption, but local time for him is no more than a convenient purely mathematical auxiliary quantity.

Einstein wrote in 1907 that local time was the crucial idea for relativity:

Surprisingly, however, it turned out that it was only necessary to grasp the concept of time sharply enough in order to get around the above difficulty. It required only the recognition that the auxiliary quantity introduced by H.A. Lorentz, and called by him "local time," can be defined sas simply "time." If one adheres to the indicated definition of time, then the basic equations of Lorentz's theory accord with the principle of relativity, provided only the above transformation equations are replaced by transformation equations that agree with the new time concept.

Whether or not Lorentz had this idea, Poincare published it in 1900 and 1904. Einstein acts as if it is his own idea, but does not explicitly say so. My guess is that he did not dare claim that the idea was original to him, because too many people knew that it was Poincare's idea. But Einstein does not reference Poincare's papers either.

I don't even see why it makes any difference whether Lorentz and Poincare believed in the aether. The aether played no part in their theories. Lorentz refused to express assumptions about the nature of the aether. What used to be called Lorentz electron theory is now called Lorentz aether theory, but it hardly has anything to do with the aether. It is only in Kuhnian paradigm shift philosophy that terminology is so important, and not in science. None of these terminology differences had any difference to how observations were explained.

Wikipedia guidelines are to prefer secondary sources over primary sources. Most of the historians do indeed say that Poincare believed in the aether and in true time, based on the above quotes. So that is what Wikipedia says. Wikipedia is great but sometimes you have to read the quotes for yourself, and parse the article statements carefully.

For another view of Poincare contributions to relativity, read my my book.

Local time measured by clocks

To a lot of people, the essence of relativity is the Lorentz transformations, or the spacetime geometry. Since Einstein did not have anything to do with discoverying those, the historians who credit Einstein usually point to his superior understanding of time. Einstein himself tried to claim credit that way, and wrote in 1907:

Surprisingly, however, it turned out that a sufficiently sharpened conception of time was all that was needed to overcome the difficulty discussed. One had only to realize that an auxiliary quantity introduced by H. A. Lorentz, and named by him 'local time', could be defined as 'time' in general. If one adheres to this definition of time, the basic equations of Lorentz's, theory correspond to the principle of relativity ...

Poincare had that realization before Einstein, even if Lorentz did not.

Mario Bacelar Valente writes in a new relativity paper:

In a way this change in the metrology of time was antecipated in developments in theoretical physics. In the late 19th and early 20th century several thinkers were involved with issues related to the so-called electrodynamics of moving bodies. In his criticism of Lorentz's electron theory and its extension to the case of matter in (inertial) motion, Poincaré noticed that what for Lorentz was a mathematical artifice – that of rewriting his equations for the case of moving bodies in terms of auxiliary variables one of which Lorentz had called local time –, could have a completely different physical interpretation. According to Poincaré the local time can be the time being measured by observers in motion with the material bodies in question:

I suppose that observers placed in different points set their watches by means of optical signals; that they try to correct these signals by the transmission time, but that, ignoring their translatory motion and thus believing that the signal travel at the same speed in both directions, they content themselves with crossing the observations, by sending on signal from A to B, then another from B to A. The local time t' is the time indicated by watches set in this manner. (Poincaré 1900; cited in Darrigol 2003, 359)

This procedure for synchronizing clocks had been presented by Poincaré, for the particular case of clocks taken to be at rest, in an earlier work published in 1898. In this work Poincaré mentions that even the best clocks, by that time still mechanical clocks, had to be calibrated to the sidereal time:

In fact, the best clocks must be adjusted from timeto time, and these adjustments are made with the help of astronomical observations; arrangements are made so that the sidereal clock marks the same hour when the same star passes over the meridian. In other words, it is the sidereal day, that is the duration of the Earth's rotation, which is the constant unit of time. (Poincaré 1898, 3; my translation)

We can still read Poincaré’s 1900 remarks in the light of his 1898 memoir, i.e. implicitly, the clocks are taken to have been calibrated to sidereal time. However, due to the issue of the setting of the initial phase of distant clocks (the synchronization of the clocks), Poincaré discusses the relation between the time read by the clocks in an apparently autonomous way.

That is correct. Einstein first wrote about relativistic time in 1905, long after Poincare's ideas were well-known throughout Europe.

No more Einsteins

A SciAm blog asks:

There’s a short rumination in this week’s Nature in which Dean Keith Simonton, a psychologist from the University of California, Davis asks a question that often surfaces: Is the age of scientific genius over? Will we see another Einstein, Darwin or Newton or is the idea of the lone genius assiduously scribbling at his desk and making a breakthrough a relic of the past?

No, Einstein was not a lone genius working in isolation. He is widely credited with special relativity being his solitary accomplishment, but it was really a restatement of much more original work by Lorentz and Poincare.

His work on general relativity was not solitary at all, as he collaborated with Grossmann, Levi-Civita, Hilbert, and others. The details are in my book.

Newton and Darwin also relied heavily on the work of others.

When people wonder why there are no more Einsteins, the simple answer is that there never was an Einstein that matched the myth about him.

Stuck at three qubits

Scott Aaronson attacks a new paper by Ross Anderson and Robert Brady on Why quantum computing is hard and quantum cryptography is not provably secure. The paper tries to give an "possible explanation of why we’re stuck at three qubits".

The UK Register reports:

Two killjoy researchers from the University of Cambridge have cast doubt on whether quantum cryptography can be regarded as ‘provably secure’ – and are asking whether today’s quantum computing experimentation is demonstrating classical rather than quantum effects. ...

Anderson and Brady are asserting that experiments conducted to date observe coherence between distant particles, but fail to eliminate the possibility that the two particles are responding to an identical third stimulus – like the bouncing droplet in the video. In other words: two boats on a lake, bouncing on the waves, aren’t demonstrating quantum physics, they’re responding to the same ripples.

Hence their doubts about cryptography: “As the experiments done to test the Bell inequalities have failed to rule out a classical hidden-variable theory of quantum mechanics such as the soliton model, the security case for quantum cryptography based on EPR [The Register - Einstein – Podolsky Rosen] pairs has not been made.”

The reasoning is a little fuzzy here. I say that the quantum crypto security is dubious for other reasons.

Update: Scott drew some interesting comments, with Lumo saying:

I understand that you consider a quantum computer running Shor’s or different algorithm to be something in between the Earth and Heaven – after all, we don’t have it yet. My suspicion is that you also want to say that its possible existence is disputable because this makes your field look like unsettled, ongoing research near the frontiers of physics knowledge – which it’s actually not because in reality, quantum computation is just an advanced engineering application of rudimentary physics insights of the 1920s and 1930s.

And Scott going off the rails:

Even if they predict a huge “temporary” setback (e.g., losing a war), they typically also predict that far enough in the future, the world will come to see the martyrdom and heroism of their cause.

For this reason, I submit that, while it’s not necessary as a matter of principle, in fact most political ideologies are pretty tightly coupled to empirical predictions about the future of humankind: for example, “the world will come to see the rightness of superior races enslaving or exterminating inferior ones.” And many of those predictions have been pretty dramatically falsified.

Origin of the term relativity

The first founder of relativity was the Scottish physicist James Clerk Maxwell. I say this because the created the first relativistic theory, coined the word "relativity", and inspired the crucial experiment. In the long run, Maxwell will be recognized as a much more important physicist than Einstein.

By a relativistic theory, I mean one where motion is relative, and there is no action-at-a-distance. A more precise definition is a theory with Lorentz covariance, but that concept was not understood in Maxwell's lifetime. Motion is relative in Newtonian, Lagrangian, and Hamiltonian mechanics, but action-at-a-distance was necessary to explain gravity.

Here is what Maxwell said in an elementary textbook, Matter and Motion (1876):

102. RELATIVITY OF DYNAMICAL KNOWLEDGE

Our whole progress up to this point may be described as a gradual development of the doctrine of relativity of all physical phenomena. Position we must evidently acknowledge to be relative, for we cannot describe the position of a body in any terms which do not express relation. The ordinary language about motion and rest does not so completely exclude the notion of their being measured absolutely, but the reason of this is, that in our ordinary language we tacitly assume that the earth is at rest.

As our ideas of space and motion become clearer, we come to see how the whole body of dynamical doctrine hangs together in one consistent system.

Our primitive notion may have been that to know absolutely where we are, and in what direction we are going, are essential elements of our knowledge as conscious beings.

But this notion, though undoubtedly held by many wise men in ancient times, has been gradually dispelled from the minds of students of physics.

There are no landmarks in space; one portion of space is exactly like every other portion, so that we cannot tell where we are. We are, as it were, on an unruffled sea, without stars, compass, soundings, wind, or tide, and we cannot tell in what direction we are going. We have no log which we can cast out to take a dead reckoning by; we may compute our rate of motion with respect to the neighbouring bodies, but we do not know how these bodies may be moving in space.

We cannot even tell what force may be acting on us; we can only tell the difference between the force acting on one thing and that acting on another.

Since then, the cosmic microwave background (CMB) radiation has been discovered to be a landmark for motion in space. Our galaxy is moving at 627±22 km/s relative to the CMB rest frame.

Relativity books sometimes make fun of Maxwell because he wrote the 1878 9th Ed. Encyclopædia Britannica article on the aether. He died in 1879, long before the discovery of the Lorentz transformation. But I think the essay holds up well as a summary of scientific knowledge of the aether at the time. That essay concluded:

No theory of the constitution of the aether has yet been invented which will account for such a system ... Whatever difficulties we may have in forming a consistent idea of the constitution of the aether, there can be no doubt that the interplanetary and interstellar spaces are not empty, but are occupied by a material substance or body, which is certainly the largest, and probably the most uniform body of which we have any knowledge.

The uniformity of the aether is the essence of relativity. That uniformity makes light and electromagnetism the same for all observers.

Einstein did not like the term "relativity" initially, but Poincare and others used the term before Einstein wrote about the subject, and the term caught on from them.

Finding invariant subspaces

NewScientist reports:

Would a basketball spinning on a fingertip behave the same way in an infinite number of dimensions? The question has flummoxed mathematicians for 80 years, but now it looks as if the answer is yes – a find that could have implications for quantum theory.

The invariant subspace problem was studied in the 1930s by John von Neumann, a pioneer of operator theory, the mathematics behind quantum mechanics. The problem asks whether carrying out certain changes, or operations, will always leave part of an object unaltered, or invariant. In the case of the basketball, the operation is rotation. In three dimensions, the sphere's rotational axis remains unchanged, but you can't take that for granted in infinite dimensions.

On 25 January, a solution was unveiled at a meeting of the Royal Spanish Mathematical Society in La Coruña. Eva Gallardo of the Complutense University of Madrid in Spain and Carl Cowen of Purdue University in West Lafayette, Indiana, said they have proved that part of the object will always be unaltered.

No great prize rests on the proof, but, if it is correct, the method of solving it should enable innovations in operator theory, says Miguel Lacruz of the University of Seville. "Operator theory is the language of quantum mechanics," he adds.

No, this Invariant subspace problem has almost nothing to do with basketball or quantum mechanics.

This is a hard problem, so the proof might be wrong.