Magnetic Field as the Rotational Component of Electromagnetic Field

Let (M, gab) be the background relativistic spacetime. We are assuming it is temporally orientable and endowed with a particular temporal orientation. Let ξa be a smooth, future-directed unit timelike vector field on M (or some open subset of M). We understand it to represent the four-velocity field of a fluid. Further, let hab be the spatial projection field determined by ξa. The rotation and expansion fields associated with ξa are defined as follows:

ωab = h[amhb]nmξn —– (1)

θab = h(amhb)nmξn —– (2)

They are smooth fields, orthogonal to ξa in both indices, and satisfy

aξb = ωab + θab + ξammξb) —– (3)

Let γ be an integral curve of ξa, and let p be a point on the image of γ. Further, let ηa be a vector field on the image of γ that is carried along by the flow of ξa and orthogonal to ξa at p. (It need not be orthogonal to ξa elsewhere.) We think of the image of γ as the worldline of a fluid element O, and think of ηa at p as a “connecting vector” that spans the distance between O and a neighboring fluid element N that is “infinitesimally close.” The instantaneous velocity of N relative to O at p is given by ξaaηb. But ξaaηb = ηaaξb. So, by equation (3) and the orthogonality of ξa with ηa at p, we have

ξaaηb = (ωab + θaba —– (4)

at the point. Here we have simply decomposed the relative velocity vector into two components. The first, (ωabηa), is orthogonal to ηa since ωab is anti-symmetric. It is naturally understood as the instantaneous rotational velocity of N with respect to O at p.

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The angular velocity (or twist) vector ωa. It points in the direction of the instantaneous axis of rotation of the fluid. Its magnitude ∥ωa∥ is the instantaneous angular speed of the fluid about that axis. Here ηa connects the fluid element O to the “infinitesimally close” fluid element N. The rotational velocity of N relative to O is given by ωbaηb. The latter is orthogonal to ηa

In support of this interpretation, consider the instantaneous rate of change of the squared length (−ηbηb) of ηa at p. It follows from equation (4) that

ξaa(−ηbηb) = −2θabηaηb —– (5)

Thus the rate of change depends solely on θab. Suppose θab = 0. Then the instantaneous velocity of N with respect to O at p has a vanishing radial component. If ωab ≠ 0, N can still have non-zero velocity there with respect to O. But it can only be a rotational velocity. The two conditions (θab = 0 and ωab ≠ 0) jointly characterize “rigid rotation.”

The rotation tensor ωab at a point p determines both an (instantaneous) axis of rotation there, and an (instantaneous) speed of rotation. As we shall see, both pieces of information are built into the angular velocity (or twist) vector

ωa = 1/2 εabcd ξbωcd —– (6)

at p. (Here εabcd is a volume element defined on some open set containing p. Clearly, if we switched from the volume element εabcd to its negation, the result would be to replace ωa with −ωa.)

If follows from equation (6) (and the anti-symmetry of εabcd) that ωa is orthogonal to ξa. It further follows that

ωa = 1/2 εabcd ξbcξd —– (7)

ωab = εabcd ξcωd —– (8)

Hence, ωab = 0 iff ωa = 0.

a = εabcd ξbωcd = εabcd ξb h[crhd]srξs = εabcd ξbhcrhdsrξ

= εabcd ξbgcr gdsrξs = εabcd ξbcξd

The second equality follows from the anti-symmetry of εabcd, and the third from the fact that εabcdξb is orthogonal to ξa in all indices.) The equation (6) has exactly the same form as the definition of the magnetic field vector Ba determined relative to a Maxwell field Fab and four-velocity vector ξa (Ba = 1/2 εabcd ξb Fcd). It is for this reason that the magnetic field is sometimes described as the “rotational component of the electromagnetic field.”

Dynamics of Point Particles: Orthogonality and Proportionality

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Let γ be a smooth, future-directed, timelike curve with unit tangent field ξa in our background spacetime (M, gab). We suppose that some massive point particle O has (the image of) this curve as its worldline. Further, let p be a point on the image of γ and let λa be a vector at p. Then there is a natural decomposition of λa into components proportional to, and orthogonal to, ξa:

λa = (λbξba + (λa −(λbξba) —– (1)

Here, the first part of the sum is proportional to ξa, whereas the second one is orthogonal to ξa.

These are standardly interpreted, respectively, as the “temporal” and “spatial” components of λa relative to ξa (or relative to O). In particular, the three-dimensional vector space of vectors at p orthogonal to ξa is interpreted as the “infinitesimal” simultaneity slice of O at p. If we introduce the tangent and orthogonal projection operators

kab = ξa ξb —– (2)

hab = gab − ξa ξb —– (3)

then the decomposition can be expressed in the form

λa = kab λb + hab λb —– (4)

We can think of kab and hab as the relative temporal and spatial metrics determined by ξa. They are symmetric and satisfy

kabkbc = kac —– (5)

habhbc = hac —– (6)

Many standard textbook assertions concerning the kinematics and dynamics of point particles can be recovered using these decomposition formulas. For example, suppose that the worldline of a second particle O′ also passes through p and that its four-velocity at p is ξ′a. (Since ξa and ξ′a are both future-directed, they are co-oriented; i.e., ξa ξ′a > 0.) We compute the speed of O′ as determined by O. To do so, we take the spatial magnitude of ξ′a relative to O and divide by its temporal magnitude relative to O:

v = speed of O′ relative to O = ∥hab ξ′b∥ / ∥kab ξ′b∥ —– (7)

For any vector μa, ∥μa∥ is (μaμa)1/2 if μ is causal, and it is (−μaμa)1/2 otherwise.

We have from equations 2, 3, 5 and 6

∥kab ξ′b∥ = (kab ξ′b kac ξ′c)1/2 = (kbc ξ′bξ′c)1/2 = (ξ′bξb)

and

∥hab ξ′b∥ = (−hab ξ′b hac ξ′c)1/2 = (−hbc ξ′bξ′c)1/2 = ((ξ′bξb)2 − 1)1/2

so

v = ((ξ’bξb)2 − 1)1/2 / (ξ′bξb) < 1 —– (8)

Thus, as measured by O, no massive particle can ever attain the maximal speed 1. We note that equation (8) implies that

(ξ′bξb) = 1/√(1 – v2) —– (9)

It is a basic fact of relativistic life that there is associated with every point particle, at every event on its worldline, a four-momentum (or energy-momentum) vector Pa that is tangent to its worldline there. The length ∥Pa∥ of this vector is what we would otherwise call the mass (or inertial mass or rest mass) of the particle. So, in particular, if Pa is timelike, we can write it in the form Pa =mξa, where m = ∥Pa∥ > 0 and ξa is the four-velocity of the particle. No such decomposition is possible when Pa is null and m = ∥Pa∥ = 0.

Suppose a particle O with positive mass has four-velocity ξa at a point, and another particle O′ has four-momentum Pa there. The latter can either be a particle with positive mass or mass 0. We can recover the usual expressions for the energy and three-momentum of the second particle relative to O if we decompose Pa in terms of ξa. By equations (4) and (2), we have

Pa = (Pbξb) ξa + habPb —– (10)

the first part of the sum is the energy component, while the second is the three-momentum. The energy relative to O is the coefficient in the first term: E = Pbξb. If O′ has positive mass and Pa = mξ′a, this yields, by equation (9),

E = m (ξ′bξb) = m/√(1 − v2) —– (11)

(If we had not chosen units in which c = 1, the numerator in the final expression would have been mc2 and the denominator √(1 − (v2/c2)). The three−momentum relative to O is the second term habPb in the decomposition of Pa, i.e., the component of Pa orthogonal to ξa. It follows from equations (8) and (9) that it has magnitude

p = ∥hab mξ′b∥ = m((ξ′bξb)2 − 1)1/2 = mv/√(1 − v2) —– (12)

Interpretive principle asserts that the worldlines of free particles with positive mass are the images of timelike geodesics. It can be thought of as a relativistic version of Newton’s first law of motion. Now we consider acceleration and a relativistic version of the second law. Once again, let γ : I → M be a smooth, future-directed, timelike curve with unit tangent field ξa. Just as we understand ξa to be the four-velocity field of a massive point particle (that has the image of γ as its worldline), so we understand ξnnξa – the directional derivative of ξa in the direction ξa – to be its four-acceleration field (or just acceleration) field). The four-acceleration vector at any point is orthogonal to ξa. (This is, since ξannξa) = 1/2 ξnnaξa) = 1/2 ξnn (1) = 0). The magnitude ∥ξnnξa∥ of the four-acceleration vector at a point is just what we would otherwise describe as the curvature of γ there. It is a measure of the rate at which γ “changes direction.” (And γ is a geodesic precisely if its curvature vanishes everywhere).

The notion of spacetime acceleration requires attention. Consider an example. Suppose you decide to end it all and jump off the tower. What would your acceleration history be like during your final moments? One is accustomed in such cases to think in terms of acceleration relative to the earth. So one would say that you undergo acceleration between the time of your jump and your calamitous arrival. But on the present account, that description has things backwards. Between jump and arrival, you are not accelerating. You are in a state of free fall and moving (approximately) along a spacetime geodesic. But before the jump, and after the arrival, you are accelerating. The floor of the observation deck, and then later the sidewalk, push you away from a geodesic path. The all-important idea here is that we are incorporating the “gravitational field” into the geometric structure of spacetime, and particles traverse geodesics iff they are acted on by no forces “except gravity.”

The acceleration of our massive point particle – i.e., its deviation from a geodesic trajectory – is determined by the forces acting on it (other than “gravity”). If it has mass m, and if the vector field Fa on I represents the vector sum of the various (non-gravitational) forces acting on it, then the particle’s four-acceleration ξnnξa satisfies

Fa = mξnnξa —– (13)

This is Newton’s second law of motion. Consider an example. Electromagnetic fields are represented by smooth, anti-symmetric fields Fab. If a particle with mass m > 0, charge q, and four-velocity field ξa is present, the force exerted by the field on the particle at a point is given by qFabξb. If we use this expression for the left side of equation (13), we arrive at the Lorentz law of motion for charged particles in the presence of an electromagnetic field:

qFabξb = mξbbξa —– (14)

This equation makes geometric sense. The acceleration field on the right is orthogonal to ξa. But so is the force field on the left, since ξa(Fabξb) = ξaξbFab = ξaξbF(ab), and F(ab) = 0 by the anti-symmetry of Fab.

Harmonies of the Orphic Mystery: Emanation of Music

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As the Buddhist sage Nagarjuna states in his Seventy Verses on Sunyata, “Being does not arise, since it exists . . .” In similar fashion it can be said that mind exists, and if we human beings manifest its qualities, then the essence and characteristics of mind must be a component of our cosmic source. David Bohm’s theory of the “implicate order” within the operations of nature suggests that observed phenomena do not operate only when they become objective to our senses. Rather, they emerge out of a subjective state or condition that contains the potentials in a latent yet really existent state that is just awaiting the necessary conditions to manifest. Thus within the explicate order of things and beings in our familiar world there is the implicate order out of which all of these emerge in their own time.

Clearly, sun and its family of planets function in accordance with natural laws. The precision of the orbital and other electromagnetic processes is awesome, drawing into one operation the functions of the smallest subparticles and the largest families of sun-stars in their galaxies, and beyond even them. These individual entities are bonded together in an evident unity that we may compare with the oceans of our planet: uncountable numbers of water molecules appear to us as a single mass of substance. In seeking the ultimate particle, the building block of the cosmos, some researchers have found themselves confronted with the mystery of what it is that holds units together in an organism — any organism!

As in music where a harmony consists of many tones bearing an inherent relationship, so must there be harmony embracing all the children of cosmos. Longing for the Harmonies: Themes and Variations from Modern Physics is a book by Frank Wilczek, an eminent physicist, and his wife Betsy Devine, an engineering scientist and freelance writer. The theme of their book is set out in their first paragraph:

From Pythagoras measuring harmonies on a lyre string to R. P. Feynman beating out salsa on his bongos, many a scientist has fallen in love with music. This love is not always rewarded with perfect mastery. Albert Einstein, an ardent amateur of the violin, provoked a more competent player to bellow at him, “Einstein, can’t you count?”

Both music and scientific research, Einstein wrote, “are nourished by the same source of longing, and they complement one another in the release they offer.” It seems to us, too, that the mysterious longing behind a scientist’s search for meaning is the same that inspires creativity in music, art, or any other enterprise of the restless human spirit. And the release they offer is to inhabit, if only for a moment, some point of union between the lonely world of subjectivity and the shared universe of external reality.

In a very lucid text, Wilczek and Devine show us that the laws of nature, and the structure of the universe and all its contributing parts, can be presented in such a way that the whole compares with a musical composition comprising themes that are fused together. One of the early chapters begins with the famous lines of the great astronomer Johannes Kepler, who in 1619 referred to the music of the spheres:

The heavenly motions are nothing but a continuous song for several voices (perceived by the intellect, not by the ear); a music which, through discordant tensions, through sincopes [sic] and cadenzas, as it were (as men employ them in imitation of those natural discords) progresses towards certain pre-designed quasi six-voiced clausuras, and thereby sets landmarks in the immeasurable flow of time. — The Harmony of the World (Harmonice mundi)

Discarding the then current superstitions and misinformed speculation, through the cloud of which Kepler had to work for his insights, Wilczek and Devine note that Kepler’s obsession with the idea of the harmony of the world is actually rooted in Pythagoras’s theory that the universe is built upon number, a concept of the Orphic mystery-religions of Greece. The idea is that “the workings of the world are governed by relations of harmony and, in particular, that music is associated with the motion of the planets — the music of the spheres” (Wilczek and Devine). Arthur Koestler, in writing of Kepler and his work, claimed that the astronomer attempted

to bare the ultimate secret of the universe in an all-embracing synthesis of geometry, music, astrology, astronomy and epistemology. The Sleepwalkers

In Longing for the Harmonies the authors refer to the “music of the spheres” as a notion that in time past was “vague, mystical, and elastic.” As the foundations of music are rhythm and harmony, they remind us that Kepler saw the planets moving around the sun “to a single cosmic rhythm.” There is some evidence that he had association with a “neo-Pythagorean” movement and that, owing to the religious-fomented opposition to unorthodox beliefs, he kept his ideas hidden under allegory and metaphor.

Shakespeare, too, phrases the thought of tonal vibrations emitted by the planets and stars as the “music of the spheres,” the notes likened to those of the “heavenly choir” of cherubim. This calls to mind that Plato’s Cratylus terms the planets theoi, from theein meaning “to run, to move.” Motion does suggest animation, or beings imbued with life, and indeed the planets are living entities so much grander than human beings that the Greeks and other peoples called them “gods.” Not the physical bodies were meant, but the essence within them, in the same way that a human being is known by the inner qualities expressed through the personality.

When classical writers spoke of planets and starry entities as “animals” they did not refer to animals such as we know on Earth, but to the fact that the celestial bodies are “animated,” embodying energies received from the sun and cosmos and transmitted with their own inherent qualities added.

Many avenues open up for our reflection upon the nature of the cosmos and ourselves, and our interrelationship, as we consider the structure of natural laws as Wilczek and Devine present them. For example, the study of particles, their interactions, their harmonizing with those laws, is illuminating intrinsically and, additionally, because of their universal application. The processes involved occur here on earth, and evidently also within the solar system and beyond, explaining certain phenomena that had been awaiting clarification.

The study of atoms here on earth and their many particles and subparticles has enabled researchers to deduce how stars are born, how and why they shine, and how they die. Now some researchers are looking at what it is, whether a process or an energy, that unites the immeasurably small with the very large cosmic bodies we now know. If nature is infinite, it must be so in a qualitative sense, not merely a quantitative.

One of the questions occupying the minds of cosmologists is whether the universal energy is running down like the mechanism of an unwinding Swiss watch, or whether there is enough mass to slow the outward thrust caused by the big bang that has been assumed to have started our cosmos going. In other words, is our universe experiencing entropy — dying as its energy is being used up — or will there be a “brake” put upon the expansion that could, conceivably, result in a return to the source of the initial explosion billions of years ago? Cosmologists have been looking for enough “dark mass” to serve as such a brake.

Among the topics treated by Wilczek and Devine in threading their way through many themes and variations in modern physics, is what is known as the mass-generating Higgs field. This is a proposition formulated by Peter Higgs, a Scottish physicist, who suggests there is an electromagnetic field that pervades the cosmos and universally provides the electron particles with mass.

The background Higgs field must have very accurately the same value throughout the universe. After all, we know — from the fact that the light from distant galaxies contains the same spectral lines we find on Earth — that electrons have the same mass throughout the universe. So if electrons are getting their mass from the Higgs field, this field had better have the same strength everywhere. What is the meaning of this all-pervasive field, which exists with no apparent source? Why is it there? (Wilczek and Devine).

What is the meaning? Why is it there? These are among the most important questions that can be asked. Though physicists may provide profound mathematical equations, they will thereby offer only more precise detail as to what is happening. We shall not receive an answer to the “What” and the “Why” without recourse to meta-physics, beyond the realm of brain-devised definitions.

The human mind is limited in its present stage of evolution. It may see the logical necessity of infinity referent to space and time; for if not infinity, what then is on the other side of the “fence” that is our outermost limit? But, being able to perceive the logical necessity of infinity, the finite mind still cannot span the limitless ranges of space, time, and substance.

If we human beings are manifold in our composition, and since we draw our very existence and sustenance from the universe at large, our conjoint nature must be drawn from the sources of life, substance, and energy, in which our and all other cosmic lives are immersed.

As the authors conclude their fascinating work:

“The worlds opened to our view are graced with wonderful symmetry and uniformity. Learning to know them, to appreciate their many harmonies, is like deepening an acquaintance with some great and meaningful piece of music — surely one of the best things life has to offer.”

Weyl’s Lagrange Density of General Relativistic Maxwell Theory

Weyl pondered on the reasons why the structure group of the physical automorphisms still contained the “Euclidean rotation group” (respectively the Lorentz group) in such a prominent role:

The Euclidean group of rotations has survived even such radical changes of our concepts of the physical world as general relativity and quantum theory. What then are the peculiar merits of this group to which it owes its elevation to the basic group pattern of the universe? For what ‘sufficient reasons’ did the Creator choose this group and no other?”

He reminded that Helmholtz had characterized ∆o ≅ SO (3, ℜ) by the “fact that it gives to a rotating solid what we may call its just degrees of freedom” of a rotating solid body; but this method “breaks down for the Lorentz group that in the four-dimensional world takes the place of the orthogonal group in 3-space”. In the early 1920s he himself had given another characterization living up to the new demands of the theories of relativity in his mathematical analysis of the problem of space.

He mentioned the idea that the Lorentz group might play its prominent role for the physical automorphisms because it expresses deep lying matter structures; but he strongly qualified the idea immediately after having stated it:

Since we have the dualism of invariance with respect to two groups and Ω certainly refers to the manifold of space points, it is a tempting idea to ascribe ∆o to matter and see in it a characteristic of the localizable elementary particles of matter. I leave it undecided whether this idea, the very meaning of which is somewhat vague, has any real merits.

. . . But instead of analysing the structure of the orthogonal group of transformations ∆o, it may be wiser to look for a characterization of the group ∆o as an abstract group. Here we know that the homogeneous n-dimensional orthogonal groups form one of 3 great classes of simple Lie groups. This is at least a partial solution of the problem.

He left it open why it ought to be “wiser” to look for abstract structure properties in order to answer a natural philosophical question. Could it be that he wanted to indicate an open-mindedness toward the more structuralist perspective on automorphism groups, preferred by the young algebraists around him at Princetion in the 1930/40s? Today the classification of simple Lie groups distinguishes 4 series, Ak,Bk,Ck,Dk. Weyl apparently counted the two orthogonal series Bk and Dk as one. The special orthogonal groups in even complex space dimension form the series of simple Lie groups of type Dk, with complex form (SO 2k,C) and real compact form (SO 2k,ℜ). The special orthogonal group in odd space dimension form the series type Bk, with complex form SO(2k + 1, C) and compact real form SO(2k + 1, ℜ).

But even if one accepted such a general structuralist view as a starting point there remained a question for the specification of the space dimension of the group inside the series.

But the number of the dimensions of the world is 4 and not an indeterminate n. It is a fact that the structure of ∆o is quite different for the various dimensionalities n. Hence the group may serve as a clue by which to discover some cogent reason for the di- mensionality 4 of the world. What must be brought to light, is the distinctive character of one definite group, the four-dimensional Lorentz group, either as a group of linear transformations, or as an abstract group.

The remark that the “structure of ∆o is quite different for the various dimensionalities n” with regard to even or odd complex space dimensions (type Dk, resp. Bk) strongly qualifies the import of the general structuralist characterization. But already in the 1920s Weyl had used the fact that for the (real) space dimension n “4 the universal covering of the unity component of the Lorentz group SO (1, 3)o is the realification of SL (2, C). The latter belongs to the first of the Ak series (with complex form SL (k + 1,C). Because of the isomorphism of the initial terms of the series, A1 ≅ B1, this does not imply an exception of Weyl’s general statement. We even may tend to interpret Weyl’s otherwise cryptic remark that the structuralist perspective gives a “at least a partial solution of the problem” by the observation that the Lorentz group in dimension n “4 is, in a rather specific way, the realification of the complex form of one of the three most elementary non-commutative simple Lie groups of type A1 ≅ B1. Its compact real form is SO (3, ℜ), respectively the latter’s universal cover SU (2, C).

Weyl stated clearly that the answer cannot be expected by structural considerations alone. The problem is only “partly one of pure mathematics”, the other part is “empirical”. But the question itself appeared of utmost importance to him

We can not claim to have understood Nature unless we can establish the uniqueness of the four-dimensional Lorentz group in this sense. It is a fact that many of the known laws of nature can at once be generalized to n dimensions. We must dig deep enough until we hit a layer where this is no longer the case.

In 1918 he had given an argument why, in the framework of his new scale gauge geometry, the “world” had to be of dimension 4. His argument had used the construction of the Lagrange density of general relativistic Maxwell theory Lf = fμν fμν √(|detg|), with fμν the components of curvature of his newly introduced scale/length connection, physically interpreted by him as the electromagnetic field. Lf is scale invariant only in spacetime dimension n = 4. The shift from scale gauge to phase gauge undermined the importance of this argument. Although it remained correct mathematically, it lost its convincing power once the scale gauge transformations were relegated from physics to the mathematical automorphism group of the theory only.

Weyl said:

Our question has this in common with most questions of philosophical nature: it depends on the vague distinction between essential and non-essential. Several competing solutions are thinkable; but it may also happen that, once a good solution of the problem is found, it will be of such cogency as to command general recognition.

Classical Metrics do not Provide an Unambiguous Inner Product Between Timelike and Spacelike Vectors

The unified theory of mass-ENERGY-Matter in motion

Similarly, in Newtonian gravitation, the acceleration of a timelike curve must always be spacelike, and so the total force on a particle at a point must be spacelike as well. A vector ξa at a point in a classical spacetime is timelike if ξata ≠ 0; otherwise it is spacelike. The required result thus follows by observing that given a curve with unit tangent vector ξa, tannξa) = ξnnata) = 0, again because ξa has constant (temporal) length along the curve. Note that one cannot say simply “orthogonal” (as in the relativistic case) because in general, the classical metrics do not provide an unambiguous inner product between timelike and spacelike vectors.

Local Gauge Transformations in Locally Gauge Invariant Relativistic Field Theory

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The question arises of whether local space-time symmetries – arbitrary co-ordinate transformations that leave the explicit form of the equations of motion unaffected – also have an active interpretation. As in the case of local gauge symmetry, it has been argued in the literature that the introduction of a force is required to ‘restore’ local symmetry.  In the case of arbitrary co-ordinate transformations, the force invoked is gravity. Once again, we believe that the arguments (though seductive) are wrong, and that it is important to see why. Kosso’s discussion of arbitrary coordinate transformations is analogous to his argument with respect to local gauge transformations. He writes:

Observing this symmetry requires comparing experimental outcomes between two reference frames that are in variable relative motion, frames that are relatively accelerating or rotating….One can, in principle, observe that this sort of transformation has occurred. … just look out of the window and you can see if you are speeding up or turning with respect to some object that defines a coordinate system in the reference frame of the ground…Now do the experiments to see if the invariance is true. Do the same experiments in the original reference frame that is stationary on the ground, and again in the accelerating reference frame of the train, and see if the physics is the same. One can run the same experiments, with mechanical forces or with light and electromagnetic forces, and observe the results, so the invariance should be observable…But when the experiments are done, the invariance is not directly observed. Spurious forces appear in the accelerating system, objects move spontaneously, light bends, and so on. … The physics is different.

In other words, if we place ourselves at rest first in an inertial reference frame, and then in a non-inertial reference frame, our observations will be distinguishable. For example, in the non-inertial reference frame objects that are seemingly force-free will appear to accelerate, and so we will have to introduce extra, ‘spurious’, forces to account for this accelerated motion. The transformation described by Kosso is clearly not a symmetry transformation. Despite that, his claim appears to be that if we move to General Relativity, this transformation becomes a symmetry transformation. In order to assess this claim, let’s begin by considering Kosso’s experiment from the point of view of classical physics.

Suppose that we describe these observations using Newtonian physics and Maxwell’s equations. We would not be surprised that our descriptions differ depending on the choice of coordinate system: arbitrary coordinate transformations are not symmetries of the Newtonian and Maxwell equations of motion as usually expressed. Nevertheless, we are free to re-write Newtonian and Maxwellian physics in generally covariant form. But notice: the arbitrary coordinate transformations now apply not just to the Newtonian particles and the Maxwellian electromagnetic fields, but also to the metric, and this is necessary for general covariance.

Kosso’s example is given in terms of passive transformations – transformations of the coordinate systems in which we re-coordinatise the fields. In the Kosso experiment, however, we re-coordinatise the matter fields without re-coordinatising the metric field. This is not achieved by a mere coordinate transformation in generally covariant classical theory: a passive arbitrary coordinate transformation induces a re-coordinatisation of not only the matter fields but also the metric. The two states described by Kosso are not related by an arbitrary coordinate transformation in generally covariant classical theory. Further, such a coordinate transformation applied to only the matter and electromagnetic fields is not a symmetry of the equations of Newtonian and Maxwellian physics, regardless of whether those equations are written in generally covariant form.

Suppose that we use General Relativity to describe the above observations. Kosso suggests that in General Relativity the observations made in an inertial reference frame will indeed be related by a symmetry transformation to those made in a non-inertial reference frame. He writes:

The invariance can be restored by revising the physics, by adding a specific dynamical principle. This is why the local symmetry is a dynamical symmetry. We can add to the physics a claim about a specific force that restores the invariance. It is a force that exactly compensates for the local transform. In the case of the general theory of relativity the dynamical principle is the principle of equivalence, and the force is gravity. … With gravity included in the physics and with the windows of the train shuttered, there is no way to tell if the transformation, the acceleration, has taken place. That is, there is now no difference in the outcome of experiments between the transformed and untransformed systems. The force pulling objects to the back of the train could just as well be gravity. Thus the physics, all things including gravity considered, is invariant from one locally transformed frame to the next. The symmetry is restored.

This analysis mixes together the equivalence principle with the meaning of invariance under arbitrary coordinate transformations in a way which seems to us to be confused, with the consequence that the account of local symmetry in General Relativity is mistaken.

Einstein’s field equations are covariant under arbitrary smooth coordinate transformations. However, as with generally covariant Newtonian physics, these symmetry transformations are transformations of the matter fields (such as particles and electromagnetic radiation) combined with transformations of the metric. Kosso’s example, as we have already emphasised, re-coordinatises the matter fields without re-coordinatising the metric field. So, the two states described by Kosso are not related by an arbitrary coordinate transformation even in General Relativity. We can put the point vividly by locating ourselves at the origin of the coordinate system: I will always be able to tell whether the train, myself, and its other contents are all freely falling together, or whether there is a relative acceleration of the other contents relative to the train and me (in which case the other contents would appear to be flung around). This is completely independent of what coordinate system I use – my conclusion is the same regardless of whether I use a coordinate system at rest with respect to the train or one that is accelerating arbitrarily. (This coordinate independence is, of course, the symmetry that Kosso sought in the opening quotation above, but his analysis is mistaken.)

What, then, of the equivalence principle? The Kosso transformation leads to a physically and observationally distinct scenario, and the principle of equivalence is not relevant to the difference between those scenarios. What the principle of equivalence tells us is that the effect in the second scenario, where the contents of the train appear to accelerate to the back of the train, may be due to acceleration of the train in the absence of a gravitational field, or due to the presence of a gravitational field in which the contents of the train are in free fall but the train is not. Mere coordinate transformations cannot be used to bring real physical forces in and out of existence.

It is perhaps worthwhile briefly indicating the analogy between this case and the gauge case. Active arbitrary coordinate transformations in General Relativity involve transformations of both the matter fields and the metric, and they are symmetry transformations having no observable consequences. Coordinate transformations applied to the matter fields alone are no more symmetry transformations in General Relativity than they are in Newtonian physics (whether written in generally covariant form or not). Such transformations do have observational consequences. Analogously, local gauge transformations in locally gauge invariant relativistic field theory are transformations of both the particle fields and the gauge fields, and they are symmetry transformations having no observable consequences. Local phase transformations alone (i.e. local gauge transformations of the matter fields alone) are no more symmetries of this theory than they are of the globally phase invariant theory of free particles. Neither an arbitrary coordinate transformation in General Relativity, nor a local gauge transformation in locally gauge invariant relativistic field theory, can bring forces in and out of existence: no generation of gravitational effects, and no changes to the interference pattern.