Why You Don’t Understand Relativity

A popular internet personality – let’s call her Roma – recently posted a video in which she attempts to explain time dilation in special and general relativity.  Roma recalls that she could never understand the explanations of relativity she read as a child, and now that she has a PhD in physics, it’s her turn to try to explain the subject.  The sub-title of the video is “Why You Don’t Understand Relativity”, but unfortunately the real reason people don’t understand relativity is videos like this, which is overflowing with errors and fundamental misunderstandings.

For example, Dr. Roma compares the clock rates (tacitly in terms of stationary global coordinates) for a person standing at sea level on the earth’s surface and a person standing on the top of a mountain, each holding a mass suspended by a spring. She notes that the spring for the person at sea level will be slightly more stretched than the spring at the mountain top, because the local acceleration of gravity is greater at sea level, and she says the clock at sea level runs slower because of this. She overlooks the fact that if someone were located at the center of the earth, their spring would not be stretched at all… they would be essentially weightless (zero local acceleration of gravity)… and yet their clock would be even slower than the clock on the earth’s surface. It’s well known that gravitational time dilation does not correlate with the local acceleration of gravity (i.e., with the amount of spring stretching), but rather with the difference in gravitational potential. The person at the earth’s center is deeper in the gravitational potential of the earth, even though the local acceleration of gravity there is zero.

Dr. Roma’s attempt to explain time dilation due to motion in special relativity suffers from essentially the same misunderstanding. She says time dilation (in flat spacetime) is due to acceleration, overlooking the well-known counter-examples that disprove her claim.  If, in terms of a standard inertial coordinate system S, one clock is at rest and another clock moves away at speed v until reaching some distance L, then undergoes an acceleration that changes its speed from +v to –v uniformly over a time interval Δt, and then maintains that speed until arriving back at the first clock, we will of course find the clocks have different elapsed times. However, if we repeat this experiment, but replace L with 2L, with the traveling clock undergoing precisely the same profile of acceleration during the middle of its journey as in the first experiment, changing its speed from +v to –v in time Δt, the difference in total elapsed times for the clocks will nevertheless be different.  In fact, by replacing L with some arbitrarily great distance, we can make the difference in total elapsed times at large as we like, and whatever takes place during the acceleration can be made negligibly small in comparison with the total difference. Thus the difference in total elapsed times is entirely dependent on the difference in velocity during the “coasting” periods of constant speed. 

Of course, we can express the effect in terms of the product of the acceleration and the distance… so the further apart the clocks are located when the acceleration takes place, the greater is the difference in elapsed times.  Just as with the gravitational discussion, the time dilation is due to the difference in potential between the clocks, and we can treat the accelerations analogously to a homogeneous gravitational field, in which the difference in potential is proportional to the product of the (uniform) “acceleration of gravity” multiplied by the “elevation”.  Indeed this is how Einstein described it in his 1918 paper reconciling the twins paradox with the general theory. Contrary to Einstein, Born, Pauli, etc., Dr. Roma blithely dismisses the idea that the resolution of the twins paradox in the context of general relativity has anything to do with general relativity. It’s perfectly true that we can described a twins scenario in flat spacetime, but general relativity can be applied in flat spacetime as well, and we must be able to reconcile the descriptions of the twins in terms of accelerating coordinate systems and the equivalence principle. It’s worth remembering that Einstein did not regard flat spacetime as the absence of a gravitational field, but as a particularly simple gravitational field, i.e., a particular solution of the field equations.  In a sense, he identified the gravitational field with the inertial field.

The brings us to Dr. Roma’s misleading comments related to Newton’s bucket and Mach’s principle. She dismisses the idea that inertial effects are due to the relations with all the entities in the entire universe, by simply saying (along with Herbert Dingle) that inertial effects are due to inertia, represented by the absoluteness of acceleration. But it goes without saying (or so one would have thought) the whole point of Newton’s bucket and Einstein’s spinning globes, etc., is to understand the origin of inertia and the absoluteness of acceleration, not simply taking them for granted. Why is the universe not purely kinematic and relational? If there were nothing in the universe but a single spinning globe, would it even make sense to say it is spinning?  Relative to what would it be spinning?  Newton decided it must be “absolute space”, whereas Mach argued that we don’t really know if a single object in an otherwise empty universe would exhibit ordinary inertia. Einstein was very sympathetic to, and even guided by, what he called Mach’s Principle for many years, although in later life he realized that Mach’s principle could not be easily reconciled with general relativity, because the latter makes spacetime itself a dynamical entity, so it’s no longer possible to assert that matter is the only ontologically “real” constituent. Hence Einstein was driven toward the view that the inertial field is determined by boundary conditions on the field equations. As he noted, we can do this on an ad hoc basis for any limited region, but trying to establish suitable boundary conditions for the entire universe in a relativistic way is not trivial. (This is similar to the conundrums that arise when trying to define a wave function of the entire universe in quantum cosmology.)

Another fundamental misunderstanding in Dr. Roma’s exposition is her reliance (albeit unwitting) on standard inertial coordinates while at the same time claiming that such coordinate systems have no physical significance. For example, she announces that the invariant proper time for the (inertial) path segment with coordinate increments dt and dx is √(dt² – dx²), but of course this is false for almost all coordinate systems. It is true only for a very special class of coordinate systems, called standard inertial coordinate systems, which correspond to the readings on a grid of standard rulers and clocks all at rest and inertially synchronized in a given frame. These are the unique systems of coordinates in terms of which the equations of physics (including the equations of Newtonian mechanics in the low speed limit) take their simple homogeneous and isotropic form. Not only does Dr. Roma fail to explain this crucial fact, she denies it by dismissing coordinate systems as physically meaningless, so her “explanation” is devoid of rational content.

Dr. Roma also perpetuates the common misunderstanding about what she calls “pseudo time dilation”. She notes that the proper time for a clock moving at speed v = dx/dt in terms (tacitly) of a standard system of inertial coordinates x,t is dτ/dt = √(1 − v²), and she calls this pseudo time dilation because it depends on the coordinate system and is reciprocal (i.e., each of two clocks runs slow in terms of the standard inertial coordinates in which the other is at rest). However, she says that the “real” time dilation corresponds to the elapsed time along an interval with components dt,dx is dτ = √(dt² – dx²). Strangely, she overlooks the fact that if we divide through this last equation by dt we get dτ/dt = √(1 − v²), which of course is the relation that she just labeled as “pseudo time dilation”. The failure to recognize these as the same thing is due (again) to the failure to clearly grasp the physical significance of the standard inertial coordinate systems for the expressions of the Minkowski metric, which, ironically, is actually a pseudometric, since it violates the triangle inequality, etc., so it is really just an analogy to actual metrical spaces.

Another problem with Roma’s video is that it recites the familiar trope “gravity is not a force”. This is admittedly a common statement among sophomores, but it is a misconception, based on the notion of a test particle passively following a geodesic path through a stationary gravitational field. This is not actually a gravitational interaction, because the test particle is assumed to have no effect on the field or the body producing that (presumed stationary) field, it is just studying the geodesics in the one-body problem. Actual gravitational interactions cannot be described in those simple terms. Consider the interaction between two massive bodies whose oblique paths come near to each other, causing them to be “scattered” and emerge from the interaction on different inertial trajectories through the flat background. These (extended) bodies are not just passively following geodesics in a stationary manifold, they are each dynamically producing the gravitational fields that accompany them, and they interact to result in the deflection of the other body and the exchange of momentum… which is essentially the definition of a “force” interaction. (A similar distinction of contexts exists in Newtonian gravity, as Galileo’s demonstration from the tower applies only in the context of test particles in the assumed-stationary field of the earth; in a dynamical interaction between comparable masses the time for two objects to fall together is obviously not independent of the masses of the objects.) Furthermore, gravitationally interacting bodies in general radiate gravitational waves during the interaction, which is not an aspect of the simplistic model of a test particle passively following a geodesic in a stationary field. This is all perfectly consistent with the equivalence principle, and does not imply that gravitation is “not a force”.

When viewed in the context of the four fundamental forces or interactions of nature (the other three being electromagnetism and the strong and weak nuclear forces), gravity is explicitly regarded as a force. (Also, attempts to model gravitation in terms of a spin-2 “graviton” analogous to the other force-carrying particles in quantum field theory is based on the idea that gravity can be represented as a force.) Telling people that “gravity is not a force” is fun, but the description of gravitation in terms of test particles passively following geodesics in a stationary field, with no actual exchange of momentum between the source of gravitation and the test particle, and no gravitational radiation, really doesn’t give a physically meaningful or accurate representation of actual non-trivial gravitational interactions in general relativity.

On a historical point, the video states that the most important part of “Einstein’s theory” is that it combines space and time into one common entity, space-time, and that this idea came not from Einstein but from Minkowski, although “Einstein was the one to understand what it means”. This contortionist statement is quite garbled, both historically and conceptually (not to mention rhetorically). Einstein’s view was that physics, going back to Galileo and Newton, had always involved four dimensions, three of space and one of time, and it was always possible to draw pictures with a time axis depicted as if it was another space dimension, so the depiction of spatio-temporal relations in this combined way was not a novelty introduced by special or general relativity. Amusingly, a scholar recently claimed that when Einstein said his theory of relativity did not geometrize physics he was saying that physics is not geometrical, whereas Einstein was actually just saying that although spatio-temporal relations are geometrical and four-dimensional, his theory of relativity isn’t what geometrized them… they were always geometrical and four-dimensional. Indeed Galileo pointed out that the vertical trajectory of a thrown object could be depicted as a parabolic path in the x,t “plane”. Moreover, the Galilean transformation between relatively moving systems of inertial coordinates includes the relation x′ = x − vt, so the transformed space coordinate is a linear combination of both the space and the time coordinates. Thus an interval that has only a time component and no space component in terms of one system of coordinates can have both time and space components in terms of another. Space and time have always been “unified” in this sense. This does not imply that space and time no longer have independent significance, nor that time is “the same as” or “has been combined with” space. Time and space are obviously quite different concepts, and this remains true in the modern theory of relativity.

What people mean when they say modern relativity “combines space and time” more than Galilean relativity did is just that the time part of the Galilean transformation is just t′ = t, whereas in special relativity the full transformation is x′ = (x − vt)γ and t′ = (t – vx)γ where γ = 1/√(1 − v²), which is to say, t′ is (like x′ was always known to be) a linear combination of x and t. But it’s mathematically trivial that the relationship between standard inertial coordinate systems (assuming such things exist) must be of the form x′ = (x − vt)γ and t′ = (t – kvx)γ where γ = 1/√(1 − kv²) for some constant k, and with k = −1 this would imply that time is just like space (Euclidean transformation, ordinary rotation), whereas with k = 0 we have the Galilean transformation, and with k = 1 (or 1/c² in conventional units) we have the Lorentz transformation (hyperbolic rotation). In accord with these relations, the quantity of energy E has inertia kE, and certain quantities involving both space and time intervals are invariant under these transformations. It is on this basis that people say “space and time are combined” in special relativity. The most explicit early exposition of this, including the crucial fact that the subject coordinate systems related by Lorentz transformations are none other than the standard inertial coordinate systems, was Einstein’s 1905 June paper.

But in physical terms, the “most important part of Einstein’s theory” is not meaningfully expressed by saying space and time have been combined into a single entity, it is rather the fact that all energy E has inertia corresponding to E/c², which essentially follows from the conservation of energy. This is the key realization that was absent from Galilean relativity, and this is what requires that the temporal foliations of relatively moving standard systems of inertial coordinates are mutually skewed. All the special relativistic effects are a direct consequence of this fundamental fact, i.e., the inertia of energy, as Einstein emphasized in his September 1905 paper. It was just a historical accident that Einstein arrived at these realizations in reverse order. (Note well that the prior writings of Poincare and others on the relation between electromagnetic energy and momentum implied by Maxwell’s equations pertained specifically to electromagnetism, and did not entail the general equivalence of mass and energy.)

Needless to say, Lorentz had referred to what we call the Lorentz transformations previously, and Poincare had commented and elaborated on Lorentz’s papers by 1905, but they were both still under the influence of the electromagnetic world view, according to which everything was to be reduced to electromagnetism, an idea that was in vogue around the turn of the century, and missed the crucial fact that the systems of coordinates related by Lorentz transformations are none other than the standard inertial coordinate systems, i.e., the systems of coordinates in terms of which, as Einstein said in the very first sentence of Section 1 of his paper, “the equations of Newtonian mechanics hold good” (to the first approximation, i.e., in the low speed limit). This is the crucial insight leading to the full realization that the Lorentz transformation transcends its connection with Maxwell’s equations, and pertains to the basic inertial measures of space and time. Admittedly Lorentz (and Poincare) by 1905 could have realized this, because Lorentz had explicitly noted that to assure null results for all attempts to detect the ether we need to assume that whatever forces were responsible for maintaining the structure of the elementary particles, and even inertia itself of electrically neutral bodies, must transform like electromagnetic forces… but neither he nor Poincare recognized the full significance of this fact.

Obviously the great mathematician Poincare was astute enough to note the formal properties of the Lorentz transformation, such as that they form a group and they represent hyperbolic rotations, and so on. He described this in his Palermo paper published in 1906 (though remarkably it was written around June of 1905).  It is unclear if Minkowski was familiar with Poincare’s paper, since it appeared in a somewhat obscure journal. In any case, many of the formal observations made by Minkowski had been anticipated by Poincare, but even Minkowski as late as 1907, after having read Einstein’s paper, was still wedded to the electromagnetic world view, as can be seen by his comment at the conclusion of his paper:

The validity without exception of the world-postulate, I like to think, is the true nucleus of an electromagnetic image of the world, which, discovered by Lorentz and further revealed by Einstein, now lies open in the full light of day.

Of course, Minkowski also famously began his paper by declaring that “henceforth only a kind of union of space and time will preserve an independent reality”, but these stirring words are more poetic than accurate. The Lorentz invariance of the laws of physics does not imply that the concepts of space and time no longer have distinct significance, even given the relativity of inertial simultaneity. We still distinguish between space and time, and we always will, so Minkowski’s prediction that those individual concepts would fade away, though memorable and inspiring, has not been fulfilled. Einstein and most other physicists held to a more sober view, as expressed in Einstein’s 1905 papers. Nevertheless, with the enthusiasm and naiveté of youth, successive generations of students are evangelized by Minkowski’s rhetorical flourishes, and imagine that they have glimpsed behind the curtain of nature and perceived a great hidden truth. Einstein’s 1949 appraisal of the significance of Minkowski’s paper was more accurate: 

Minkowski’s important contribution to the theory lies in the following: Before Minkowski’s investigation it was necessary to carry out a Lorentz transformation on a law in order to test its invariance under such transformations; but he succeeded in introducing a formalism so that the mathematical form of the law itself guarantees its invariance under Lorentz transformations.  By creating a four-dimensional tensor calculus, he achieved the same thing for the four-dimensional space that the ordinary vector calculus achieves for the three spatial dimensions.

Thus Minkowski’s contribution was a formal one, although (again) he was actually anticipated by Poincare in much of what he wrote. It isn’t clear if Einstein ever saw Poincare’s Palermo paper, at least not until Pais gave him a copy in the 1950s.

One fun aspect of the video is that Dr. Roma charmingly pronounces the word hyperbola as “hyperbolee”.  In other videos she has “explained” that special relativity implies 1+1=1, that special relativity is compatible with superluminal conveyance of mass-energy and/or information, that the “arrow of time” singles out a particular temporal foliation, and other such fundamental misconceptions.

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