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## Entries for items 21-30 on the Home page

## (22) Event Horizon Telescope Time Synchronization Requirements

The time synchronization accuracy requirements among the Event Horizon Telescope's 8 radio telescope sites distributed across the face of the Earth which were necessary to obtain the EHT's 2019 (pseudo, synthesized from microwave data) pictures of the black hole at the center of the M87 galaxy were so severe that I couldn't think of any way in which they could be met. The M87 black hole (M87*) is a 6,500,000,000 solar mass black hole, whose diameter is about 36,000,000,000 km, about 4 times the diameter of the solar system (without Pluto), at the center of M87, which is 53,000,000 light years from Earth. After researching the problem in other ways, I emailed Andrew Novick, the public question answerer of the Time and Frequency Division of NIST (National Institute of Standards and Technology), asking about NIST's knowledge concerning this. Before I received an answer from Andrew, I ran across a recent technical article discussing the techniques used to obtain the black hole picture, and answering, in general, my question of how the time synchronization requirements could be met. I emailed this to Andrew, and next day got a reply from him. My email to him is posted below. (He said that he would prefer that I quote from the references he sent me links to, rather than quoting his emails to me, so those are not shown below.)

The answer given in the technical article, and also by Andrew and the NIST references he sent me links to, is basically that the time synchronization required for the BH picture, to within a few pS (picoseconds, trillionths of a second), could *not* be met by any existing timing system at the time, in 2017, that the M87* microwave radio telescope data was recorded, so (as revealed by the article), the EHT organization cheated (a little). What they did was synchronize their data as well as they could using the best existing systems- the Global Positioning System and atomic clocks- and next computer-searched the BH data using all the remaining possible time relationships (those allowed by the synchronization's not perfect accuracy) among the 8 EHT radio telescope sites' data for a combination that would give a picture resembling what it was thought the M87 black hole would look like, based on what was already believed to be known about it, chiefly its distance and mass, and what Einstein's general relativity implied about a BH with that mass. This procedure was not, as some might believe, guaranteed to generate a picture meeting those (prior to the EHT radio telescope BH data) expectations, regardless of the EHT M87* data, since if the true, exactly correct timing relationships among the data from the 8 world-wide EHT radio telescope sites would not have produced M87* pictures very similar to the actually published ones, almost certainly no possible combination of allowed (by the GPS & atomic clock timing constraints) timings that EHT computer-searched would have resulted in such pictures. Thus, the procedure used by the EHT organization to produce the M87* pictures, and those pictures, were only slightly less convincing than ones based on independently obtained timing synchronization to within, say, 2 pS (if that had been possible) would have been.

The procedure actually used to obtain the published M87* pictures and the relative data time-stamps for the 8 radio telescope sites is the most accurate mutual time synchronization procedure for a system of such sites 1000s of miles apart that is now available, accurate to within a few pS (at the time the M87* data was taken), but it is not very practical for general use, having required months of computation, and thus good only for a time months before the computed best relative time stamps were available; these relative time synchronization stamps for the 8 radio telescope sites could not be made to apply to the time the computation was finished, to obtain pS synchronization among the sites at that time, by using them to synchronize the atomic clocks at the 8 sites then, since even the best atomic clocks then (and probably now) wander in their mutual time relationships by several nS (nanoseconds, 1000 pS) over a period of months, so their time relationships at one time would likely not be the same as their relationship months later.

About angular units: One microarcsecond (angle, not time) is 1/1,000,000 of a second of angle (a second of angle is called an "arcsecond"), an arcsecond is 1/3,600 of a degree, and a degree is 1/360 of a complete revolution. Thus, 20 microarcseconds, which is about the angle M87* subtends as seen from Earth, is 1/180,000,000 of a degree, which is 1/64,800,000,000 of a complete revolution. Also, 20 microarcseconds is about 1/10,313,240,312 of a radian (1 radian = 180/pi, about 57, degrees), which is very nearly (the diameter of M87*)/(the distance of M87* from Earth).

The actual (synthesized) pictures of M87* released by EHT on April 10, 2019 each consist of a central, roughly disc-shaped dark area surrounded by an orange, glowing ring, which itself is surrounded by darkness. The central dark area is the so-called "shadow" of the black hole, which for various General Relativistic and other astrophysical reasons is several times the diameter of the event horizon (which would be considered the diameter of the black hole itself), with the shadow having an angular diameter of around 50 microarcseconds as seen from Earth. The glowing ring is caused by incandescent matter in the accretion disc surrounding the black hole.

After my below emails to Andrew Novick, in reverse chronological order, and a few more paragraphs, there are my computations showing that transportation of atomic clocks from a central site to places 1000s of miles apart could not be used to obtain time synchronizations accurate to a few picoseconds among those places, which computations are based solely on special and general relativistic considerations together with limitations on the accuracies with which the speeds of transportation and the gravitational potentials over the transportation routes could be known.

To: novick@nist.gov Sun, Nov 8, 2020 at 4:50 AM

Andrew Novick,

I should have included: "... thousands of miles apart, **except by using the following very laborious technique**. EHT, ...".

Michael Fox

---------- Forwarded message ---------

From: Michael Fox <cmichaelfox@gmail.com> Date: Sun, Nov 8, 2020 at 3:54 AM Subject: Fwd: A question about the very precise time synchronization required by the Event Horizon Telescope To: <novick@nist.gov>

Andrew Novick,

I have found a book chapter, Very-Long-Baseline-Interferometry, Chapter 9 in Interferometry and Synthesis in Radio Astronomy, chapter available on www.researchgate.net, that more-or-less answers my question, which is stated at the end of the first multi-sentence paragraph in my Oct. 30 email to you. The answer is that, using current technology, times of events can't be compared (synchronized) to within a few pS at radio telescope sites thousands of miles apart. EHT, to create the M87 black hole image, synchronized time as well as they could between their 8 radio telescope sites, then computer-searched through all the possible remaining combinations of time and signal arrival-time rate-of-change, as the Earth turned, relationships among the 8 sites for the best (somewhat unspecified in the chapter) correlations of the M87 signals at the 8 sites, which time relationships were then used to create the EHT M87 black hole synthesized picture. I have attached the Research Gate chapter 9 and its RG URL in case you are interested.

Michael Fox

---------- Forwarded message ---------

From: Michael Fox <cmichaelfox@gmail.com> Date: Fri, Oct 30, 2020 at 5:27 AM Subject: A question about the very precise time synchronization required by the Event Horizon Telescope To: <novick@nist.gov>

Andrew Novick,

We've talked before, about NIST telephone & Internet time signals, & other things. This time, I have a question about the extremely precise timing required by the Event Horizon Telescope (EHT), whose organization last year released the first-ever pictures (sort of) of a black hole, the giant one at the center of the M87 galaxy, about 50,000,000 light years from Earth. I know that NIST wasn't the main organization involved in taking these pictures, but since doing that required knowing the time relationship of radio telescope signals from sites thousands of miles apart to within a few picoseconds (pS, 10^-12 seconds, 0.000,000,000,001 seconds), and NIST has expertise in such things, I thought that you might know something about this problem, or at least could refer me to, or send this email to, people in NIST who would either have been consulted by EHT about this, or were nevertheless knowledgeable about it. Below is a longer, more detailed description of the problem; this description turned out, however, to be much too long, but there is no obvious way to much shorten it while still including all essential details, so if you understandably don't want to read all of it for my explanation of what I think are the difficulties involved, my question basically is just this: how could the times of events at radio telescope sites thousands of miles apart be compared to within a few pS.

EHT has a website, but it just describes the Event Horizon Telescope, the EHT organization, and the M87 black hole pictures in non-technical terms. There are various other Internet articles, including a Wikipedia one, about the EHT & the BH pictures, but they also don't go into technical details. The most promising source for the information I want, the series of 6 semi-technical papers written by EHT collaboration members about EHT & the BH pictures, which papers are available thru Google Scholar, told quite a bit about various technical aspects of EHT & taking the BH pictures, but almost nothing in much detail about what I am most interested in, and which I find most puzzling, which is what was the method used for world-wide distribution, to the 8 EHT radio-telescope sites, of accurate enough time marks. Atomic clocks and the Global Positioning System were mentioned, but no details given.

The EHT is a Very Long Baseline Interferometer which imaged the M87 BH by recording several about 230 GHz (where there is an atmospheric E-M signal window) microwave signals from it (these are said to be from synchrotron oscillations associated with the BH) on one day in 2017. The BH in M87 subtends about 20 microseconds of arc as viewed from Earth. The angular resolution of an interferometer with receiving dishes the diameter DE (= 8,000 miles) of the Earth apart, which EHT has, operating at 230 GHz, which can measure the time of arrival of parts of the signal emitted at the same time and arriving at the 8 receiving sites to within the period of one cycle, 4.3 pS, which signal has a wavelength WL of 1.3 mm, is (in radians) WL/DE = 1.3 mm/8,000 miles, which happens to be just about 20 microseconds (of arc, i.e., angle), the BH angular diameter as seen from Earth, and is the approximate resolution of the published BH photographs (which were, of course, not very detailed. You probably have seen them.) (This achievable resolution is assuming that the relative positions of the 8 radio telescopes are known exactly.) The pictures were, naturally, in fake color, being synthesized from the received microwave signal data.

The problem with this is that I don't know any way in which events at sites thousands of miles apart could be timed with a common time (which doesn't have to be a standard time such as UTC) to within 4.3 pS (actually less, since there were also relative positioning errors of the EHT telescope sites.) NIST has a system, which they rent for $750/month, that will give, at any place in the world (at which certain necessary facilities are available), UTC to within 15 nS (15 nanoseconds, 15 x 10^-9 seconds). I have also seen advertised by NIST a $1000/month system which will give UTC to within 10 nS. Both use the Global Positioning System satellites, together with special NIST equipment, and are said by NIST to be more accurate than the standard GPS time signals. However, those accuracy bounds, 10 & 15 nS, are more than 3 orders of magnitude (1000 times) larger (worse) than those required for the EHT. Also relevant, just a few years ago the NIST website said that the NIST single-shot timing resolution (not accuracy) was probably about 30 pS.

I can think of 3 ways in which time synchronization within a few pS between distant points on Earth might be obtained: By transported atomic clocks, by direct coaxial or fiber optic cables, or by satellite. (There is also synchronization by pulsar signals, but there are several problems with this method, which I won't go into, which I believe make it much too inaccurate.) The problems with the 3 are:

Even atomic clocks vary in frequency, enough that ground transport would take so long that the accumulated error would be >> 4 pS if they were transported to distant parts of the world that way. (I think; I know the fractional standard frequency deviation of NIST's primary frequency standard, F1, which is about, 3 x 10^-16 - - portable atomic clocks are not that good -- but I am not certain of the frequency variation statistics of such cesium beam (or hydrogen maser, which are used in EHT, and have somewhat better short-term stability than cesium beam) atomic clocks. Better optical atomic clocks are under development, but they were probably not available in 2017.) Air transport is probably not possible, since some of the sites can't be reached by air. Also, the gravitational potentials over either a ground, sea, or air route 1000s of mile long would not be well enough known to estimate the general relativistic gravitational time dilation (or expansion) effect during such transport to within a few pS, and also the speed of transport could not be measured well enough to estimate the special relativistic time dilation to within a few pS.

Time synchronization can be done accurately between two distant sites A & B if facilities are available to send a time mark from A to B, send (reflect) it back immediately, and the return trip from B to A is known to take the same time T as the trip from A to B (and the time interval between the mark's leaving A and its arrival again at A after being reflected at B can be measured accurately), since then the time of arrival at B is (the time of leaving A) + T, and T = 1/2 the round trip time from A back to A. This could be done using coaxial or fiber optic cables if the cables were short enough to not require the time mark signal to be amplified, but for cables 1000s of miles long, amplifiers would be required, either 2-way ones with identical time delays (within say 1 pS) in both directions, which I think are not available, or matched (within say 1 pS) pairs with some sort of directional router for the signals going different ways, which I suspect also are not available. Besides, such cables to each of the 7 other EHT sites from one central EHT site have, almost certainly, not been laid (for example, not to the Antarctic site).

This leaves synchronization by satellite. This would most accurately be done using the 1/2 round trip time method described above and the GPS satellites, but just as for the cable method, would require extremely closely matched (within a few pS) amplifier/repeaters, and the GPS system wasn't, I think, designed to have this accuracy. The satellite method has the additional problem that the satellites are moving rapidly wrt the EHT sites, so the time signal path one way would be a different length than for the signal's return trip, by an amount that would have to be known to within less than 1 mm (which however might be possible, since the GPS satellite orbits are known very accurately.) Communications between each of the EHT sites and the GPS satellites would be possible, and their use as described would be, I think, the only time mark distribution method which would give the required accuracy, but would require modification of at least some of the GPS satellites to provide the required pS precision, which as far as I know hadn't been done by the 2017 M87 BH measurement date.

Michael Fox

The following 2 computations show that time synchronization among the 8 EHT radio-telescope sites by (e.g., atomic) clock transport from a central location, at an average speed of v<<c, where c = the speed of light, could not distribute time marks with the accuracy required to image the M87 BH with a resolution less than or equal to the diameter of its event horizon as seen from Earth, (which would require time synchronization among the 8 sites within about 4 pS, if there were no other errors than time synchronization errors, but there are others, including significant relative position errors, so the time synch errors are needed to be less than about, say, 2 pS), due solely to the combination of (1), the inaccuracies resulting from uncertainties in the speed of transport v together with the special relativistic time dilation formula, according to which the SR time dilation depends on v, and (2), the inaccuracies resulting from uncertainties in the gravitational potential p over the transportation route together with the general relativistic time dilation formula, according to which the GR time dilation depends on p. (These computations ignore the fact that Earth's surface is not exactly an inertial coordinate system.)

(1) If a clock is transported a distance D at speed v wrt Earth in proper time (its own time, the time it indicates) T & Earth time Te, by special relativity Te = T/ √ (1-v^2/c^2), so dTe/dv = Tv/[(c^2-v^2) √ (1-v^2/c^2)] ≈ D/[(c^2-v^2) √ (1-v^2/c^2)]. Thus, for v<<c, dTe/dv ≈ D/c^2 ≈ D/10^17(m^2/s^2), so Δv ≈ 10^17(m^2/s^2)( ΔTe/D) for small Δv. Thus, for D = 5,000 km & ΔTe = 2pS, Δv ≈ 4 cm/s ≈ 1/10 mph. This is the maximum average uncertainty there can be in the speed of transport of the clock for there to be no more than 2 pS uncertainty in the clock's transmitted time stamp when it is transported over a distance of 5,000 km, a typical distance clocks used for time synchronization for EHT's calculation of the M87 black hole image would need to be transported. However, the actual uncertainty in speed would be much greater than that unless the speed of transport v was much less than typical jet plane speeds, 600 mi/hr, which would pose the problem indicated in the last sentence of the next paragraph.

(2) According to general relativity, the time T1 passed at point X1 in a constant gravitational field is related to the time T2 passed at point X2 which is at gravitational potential p wrt X1 by T2 = T1e^(p/c^2). Thus, for p<<c^2, T2 ≈ (1+p/c^2)T1, so ΔT ≡ T2 - T1 ≈ (p/c^2)T1. For points near Earth's surface, p ≈ h(10m/s^2, where h is the difference between the altitude of X2 & that of X1, so ΔT ≈ hT1/[(10^16)m]. Thus, for the difference ΔT to be 2 pS, h ≈ 20,000(m-s)/T1, so for T1 = 5 hrs., a typical trip time for jet plane travel between 2 places 5,000 km apart, the average uncertainty Δh in altitude h must be no more than about 1.1 m ≈ 3.6 ft. However, the actual uncertainty in the gravitational potential during a 5,000 km plane trip or a land trip over Earth of somewhat unknown density would be much greater than that due to 1.1 m uncertainty in altitude. In addition, to make the time uncertainty due to speed uncertainty less than 2 pS by making the speed uncertainty less than 4 cm/s, the speed of transport of the clock would have to be much less than 600 mi/hr, as noted in the paragraph above, in which case the time of transport would be much greater than 5 hrs., so the uncertainty in ΔT due to uncertainty in the gravitational potential would be proportionally bigger than the already-bigger-than 2 pS uncertainty which is due to gravitational potential uncertainty during a 5 hr. trip.