Deborah’s Grandmother

In a public briefing on 25 March of 2020 the White House coordinator for COVID-19 response, Dr. Deborah Birx, told a personal story about how, in 1918, her grandmother at the age of 11 had come home from school with the Spanish Influenza, and had passed it on to her mother (Deborah’s great-grandmother) who had recently given birth to another child. Deborah’s grandmother soon recovered but her great-grandmother died of the disease. Deborah spoke movingly about how, for the remainder of her grandmother’s long life, she bore the guilt of having been responsible for her mother’s tragic early death.

She never forgot that she was the child that was in school that innocently brought that flu home.  This is why we keep saying to every American: You have a role to protect each and every person that you interact with. We have a role to protect one another.  If you inadvertently brought this virus home to someone with a pre-existing condition, I can tell you, my grandmother lived with that for 88 years.

It’s remarkable that one of the key architects of the national response to COVID-19 happened to have this experience in her family background, and it’s interesting to speculate on whether the policy advice given by Dr. Birx, advocating for school closures and stay-home directives, etc., may have been influenced by this traumatic “fulcrum” event in her family’s past.

It’s also interesting to compare the public reaction to the Great Influenza of 1918-1919 with the public reaction to COVID-19. For over a year from March 2020 through 2021 it seems that almost every conversation or interaction between people was affected by the pandemic, and it was at least partly the subject of almost all discourse. In contrast, just to take one admittedly anecdotal example, the collected correspondence of Albert Einstein for the years 1916 through 1920, which is quite extensive and covers a wide variety of topics, both professional and personal, contains not a single mention of the Great Influenza. From reading that correspondence, one would never suspect that there was any such pandemic. Granted, the first world war and its aftermath were the focus of much attention, and Einstein may have been somewhat isolated in his quarters in Berlin, but it still seems remarkable that a global pandemic, estimated to have killed 50 million people (equivalent to almost 200 million in proportion to today’s population), many of them young and healthy, was never mentioned once in any of the numerous letters written by, or to, him during that period. Perhaps there were more health hazards in those days, lacking modern medicine, so the presence of one more potentially lethal disease might have seemed less abnormal to them.  In particular, it was less unusual in those days for women to die from complications of childbirth, even without the Influenza.

It should also be noted that Einstein was deeply engrossed in scientific ideas during those years, and often seemed to exhibit a somewhat impersonal attitude toward life. One example of the kind of thing that occupied his mind in 1918 was how to resolve the so-called “twins paradox” in the context of his recently-formulated general theory of relativity. In November of that year he published a paper entitled “Dialogue about Objections to the Theory of Relativity”, in which he first notes the usual explanation of the “twins paradox” (which he casts as clocks rather than people) purely in terms of special relativity, noting that the symmetry is broken by the fact that one twin follows a inertial path whereas the other undergoes acceleration. The hypothetical critic then challenges this by pointing out that the general theory supposedly treats all coordinate systems equally, including accelerating systems. The critic thinks this resurrects the paradox.

To this Einstein replies that, although the general theory formally treats all coordinate systems (in the same diffeomorphism class) on an equal footing, it requires us to account for gravitational effects – by which Einstein meant any effect of non-zero Christoffel symbols, which can occur even in flat spacetime (i.e., even in the absence of what might be called “actual” gravity). He understood the “equivalence principle” to signify that local physics in a stationary coordinate system in an actual uniform gravitational field is the same as local physics in an accelerating system of coordinates. This implies that it is necessary to account for gravitational time dilation between two locations with different gravitational potentials.

Consider the paths through time and space of two twins, one moving inertially from A to D, and the other moving inertially from A to E, then undergoing constant proper acceleration from E to F, and then moving inertially from F to D.

We can make the segments of constant speed as long as we like, and the turn-around segment as brief as we like (provided we make the acceleration rate correspondingly large), so from the stay-home twin’s inertial coordinates his brother is essentially moving at constant speed v (which Einstein takes to be small compared with 1, the speed of light, for simplicity) for virtually the entire trip, so the relative rate of the traveler’s elapsed proper time will be

Consequently if T is the total elapsed time for the stay-home twin from A to D, we find that his brother will arrive at D younger by about (1/2)v²T.

Now, Einstein evaluates this situation in terms of coordinates in which the traveling twin is at rest – something which the general theory of relativity assures us we can do. Bear in mind that he is considering the case of small v, so the total elapsed times for the two twins differ by only a very small amount. During the legs from A to E and from F to D, the stay-home twin is moving at speed v in terms of the traveling twin’s co-moving inertial coordinate system, so in the time elapsed on those two legs for the traveling twin the stay-home twin has elapsed

Consequently we would conclude that the stay-home twin is younger by (1/2)v²T, exactly the opposite conclusion that we reached previously. However, for this analysis we cannot neglect the turn-around, which occupies a span of time Δt, and during that time it must reverse the speed from –v to +v, so we must have acceleration “a” such that aΔt = 2v.  During this turn-around there is a gravitational potential between the twins of aL where L=vT/2 is the maximum distance, and according to the gravitational time dilation of general relativity the traveling twin is deep in a gravitational “well”, so he loses relative to his brother an amount of time equal to approximately aLΔt = v²T. Therefore, taking this into account, the traveling twin expects to be younger than his brother by about (1/2)v²T when they re-unite, in agreement with the result derived in terms of the stay-home inertial coordinates.

After outlining this explanation, Einstein’s hypothetical critic says

I see you extricated yourself very skillfully, but I would be lying if I declared myself completely satisfied… You solved the paradox by taking into account the influence on clocks of the gravitational field, but isn’t this gravitational field only fictitious?  It’s existence is, I should say, only simulated by the choice of coordinates. After all, real gravitational fields are always generated by masses and cannot be made to vanish by a suitable choice of coordinates. How should one believe that a merely fictitious field could influence the rate of clocks?

In this the “critic” is actually expressing what is probably the consensus view among modern physicists, meaning that invoking the theory of gravity to resolve the paradox in flat spacetime is considered unnecessary and even potentially misleading. As can be seen in the above figure, the “gravitational time dilation” during the turn-around is simply due to using the tilting simultaneity mapping of the co-moving inertial coordinate systems of the accelerating twin.

Furthermore, a difficulty with the correspondence between this effect and gravitational time dilation near a massive body arises if we consider the possibility that, after the traveling twin has accelerated back toward his brother, he re-accelerates in the outward direction, so that he is again moving away from his brother at speed v. This reveals another restriction that must be placed on the parameters to allow the correspondence, because if the traveling twin is sufficiently distant during the turn-around, even a small outward acceleration will result in the mapped simultaneous age of the stay-home brother to be reduced, rather than just slowing the rate of increase, as shown in the figure below.

Since the stay-home twin is to the right of the fulcrum event J and the traveling twin accelerates to the left from F to H, the mapping given by the momentarily co-moving systems of inertial coordinates proceeds from event C to event K of the stay-home twin, i.e., from later to earlier proper times. In a stationary gravitational field produced by masses, it is never the case that the stationary mapping involves time going backwards at different potentials. The rate of proper time (in terms of the stationary time coordinate near a gravitating body (e.g., the Schwarzschild time coordinate) can be reduced to an arbitrarily small value, but it never goes negative (neglecting conditions that reach across event horizons).

The reason for the difficulty in the case of outward acceleration can be seen in the figure above. For constant proper acceleration from E to F the event G serves as a fulcrum, meaning the temporal foliations for that worldline “rotate” about that event.  Also the proper distance between each event on that accelerating worldline to the event G is the same. Using these foliations to provide the mapping, the rate of proper time for any worldline to the right of G is simply slowed. However, for an outward acceleration the fulcrum event J is between the twins, and beyond the fulcrum event (i.e., to the right of J) this mapping gives negative rate of proper time. The fulcrum events serves as a kind of event horizon, always behind the accelerating worldline, so if the traveler accelerates away from his brother, and if his brother is beyond the fulcrum, the correspondence with simple gravitational time dilation is not applicable.

None of this conflicts with the equivalence principle, because this scenario pertains to relations over extended distances, whereas the equivalence principle applies only in the infinitesimal region around any event. The difficulty in applying the equivalence principle over extended regions is fundamentally due to the fact that there is no such thing as a perfectly uniform stationary gravitational field over extended regions, as discussed in the note on an infinite wall.

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