This question is wholly seeking historical evidence and not about physics. It is a follow on from the Physics Stack Exchange question:

Could we send a rocket to the Moon without knowledge of general relativity?

The answer is a definitive yes. A simple back of the envelope calculation with the Schwarzschild metric shows that the order of magnitude deviation between corresponding points Earth-Moon transfer trajectories calculated with Newtonian and GTR physics is of the order of 0.1 meters (you read that right- the length of your tallman finger). A neat approach to the problem is John Rennie's answer here.

This conclusion MUST have been reached by NASA (or even NACA) in the leadup to the Apollo landings. I should like to know any of the following how was the conclusion reached, who first raised the question and when. A link to / citation of a report would be great. I suspect the question arose and was resolved in one of three ways:

  1. The general relativity / gravitation literature holds this calculation done in the first half of the 1900s in the early days of "dreaming" about reaching the Moon (although I can't seem to find anything), and the relevant papers were known to orbital mechanical scientists;

  2. The question was raised very early in the NASA programs, and quickly resolved by a back of the envelope calculation like John's. If so, I'd expect that there would be somewhere in the archives a short report of one or two pages comprising a calculation like John Rennie's with the endorsement of a prominent GR theorist of the day, like John Wheeler. This would be dated 1950s / early 1960s;

  3. Empirically. Once Mariner / Ranger data became available, there would be no noticeable error between Newtonian theory and observations, so the question was never raised. GTR effects would be utterly swamped by others. In particular, I learnt today that the error Frank Borman was referring to when he said that Apollo 8's final position error after lunar orbit insertion was "about a mile and a half from where we were supposed to be" was indeed owing to five high density "lumps" on the Moon's surface see "Bizzare Lunar Orbits" here (Thanks to user David Hammen for this knowledge) and that the calculation error was thereafter reduced to a 120 meter difference between calculated and actual landing position for Apollo 12. Still three orders of magnitude bigger than the GTR effects.

So, in summary, references / answers to whom the question was raise by, when and how was it answered?

  • $\begingroup$ I presume "GTR" refers to General Relativity, but what do the letters stand for? $\endgroup$ Mar 19, 2015 at 18:09
  • $\begingroup$ General Theory of Relativity. $\endgroup$
    – PearsonArtPhoto
    Mar 19, 2015 at 19:14
  • $\begingroup$ I strongly suspect the last one. When there wasn't any significant errors, then they just didn't bother looking more in to it. $\endgroup$
    – PearsonArtPhoto
    Mar 19, 2015 at 19:32
  • $\begingroup$ When I was taught physics 40 years ago I learned that relativistic effects were negligible below 1% of C. So there was no decision. It was common knowledge. $\endgroup$
    – user8406
    Mar 29, 2015 at 4:32
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    $\begingroup$ @andy256 But don't you think there must have been some quantification backing that statement up? In other words How negligible? NASA was doing something quite extraordinary here that had never been done before: aiming for a lunar injection window a few kilometres across at the most. Now, a closed loop strategy afforded by the Kalman filter working on sextant data puts a whole new slant on the problem as discussed in David Hammen's answer, but in the early days a clear idea of closed loop navigation hadn't yet been formed. $\endgroup$ Mar 29, 2015 at 5:17

2 Answers 2


This conclusion MUST have been reached by NASA (or even NACA) in the leadup to the Apollo landings.

Despite the name, getting people to the Moon is not rocket science. It's rocket engineering. Engineers know that effects that are orders of magnitude smaller than the sensitivity of their systems are essentially non-effects. General relativity is a non-effect. A memo to that effect was not needed.

Much of the Apollo trajectory planning work was done in the late 1950s / early 1960s.That early trajectory work used the patched conic approximation of the N-body problem. Near the Earth, and on the way to the Moon, only the gravitation from the Earth was considered. Once inside the Moon's sphere of influence, only the gravitation of the Moon was considered. Only one gravitating body is active at a time in the patched conic approximation.

The reason for using this low-level approximation was that they were solving a non-linear boundary value using extremely archaic computing machinery (by modern standards). Computers back then were slow. Very slow. The first Macintosh (1984) was ten times faster in terms of floating point operations per second. If you have a laptop bought within the last ten years, it dwarfs the supercomputers of the 1980s. The supercomputers of the early 1960s? They didn't exist. That's a mid-1960s development, and those first supercomputers weren't that super.

Even using that archaic machinery, those pioneers of the space program saw they had a problem, and it wasn't that they weren't using general relativity. The problem was the sensitivity of the transfer orbit to initial conditions. Thrust errors over 10% were quite common back then. In fact, that more or less remains the case today. There have been some improvements, but not a lot. Rockets remain a bit chaotic.

What used to surpass that challenge wasn't general relativity. It was the extended Kalman filter. NASA didn't need general relativity to send humans to the Moon, but they absolutely did need the Kalman filter.


Here's a really nice site: http://www.ibiblio.org/apollo/index.html . There you can find the Apollo Guidance Computer (AGC) flight software, an AGC emulator, and lots of historic documentation. The AGC flight software modeled the following gravitational effects:

  • Earth gravity. Effects modeled were spherical gravity ($GM/r^2$) and the first four zonal harmonics ($J_2$, $J_3$, $J_4$, and $J_5$). No sectoral harmonics, no tesseral harmonics. Ignoring these (and ignoring higher order terms) is a much bigger breach of reality than ignoring general relativity. The general relativistic contribution to acceleration due to Earth's gravity is roughly equivalent to the 20th zonal harmonic.

  • Moon gravity. Effects modeled were spherical gravity, the first four zonal harmonics, and the first tesseral harmonic ($J_{22}$). This was not high enough fidelity to capture the effects of the lunar mascons. Then again, nobody knew about those mascons at the time the AGC flight software was written.

  • Optionally, Sun gravity. This apparently was disabled during lunar descent and ascent.

And that's it. Venus and Jupiter both have a greater perturbative impact on the trajectories of vehicles in the Earth-Moon system than does general relativity, and yet they weren't modeled.

  • 2
    $\begingroup$ "Engineers know that effects that are orders of smaller than the sensitivity of their systems are essentially non-effects" - I totally agree. But that is the point of my question: how did they know that these effects were smaller. Most engineers never study GTR. NASA was doing something never before done - find a lunar injection point a couple of kilometers wide. I find it hard to believe that someone in the very early days - before the engineering teams were built up - would not have formally asked and answered the question about the contribution of GTR, even though it is easily answered. $\endgroup$ Mar 29, 2015 at 0:55
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    $\begingroup$ Also, I re-emphasise that the main point of my question is historical - I think that somewhere in the archives, probably dated 1950s, there is a document - probably a one pager - that shows that henceforth we don't need to worry about GTR - and I'd love to find it. And I wouldn't be surprised if that the document involved correspondence with John Wheeler. Great answer BTW, especially the point about the Kalman filter. But again, the Kalman filter came - for NASA at least - later (Gauss used it to simplify hand processing of planetary orbit data in the early 19thC - published in 1809). $\endgroup$ Mar 29, 2015 at 1:16
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    $\begingroup$ @WetSavannaAnimalakaRodVance - Why would you think that? Engineers in general do not call in the scientists. Both scientists and engineers have rather patronizing views of one another. NASA engineers first response is to call in other NASA engineers when they are confronted with a problem they don't understand. Their next option is to call in some eggheaded NASA scientist. They only call in outside scientists only when forced to do so (e.g., Feynman for the Challenger disaster) or when truly boggled. With Apollo, they did have a big problem, and they were truly boggled. (continued) $\endgroup$ Mar 29, 2015 at 7:42
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    $\begingroup$ The problem was the lousy knowledge of the translunar injection burn. These errors swamped everything, including Newtonian mechanics. They needed to know how to correct that error. The outsider who helped wasn't a physicist. He was an engineer, Dr. Rudy Kalman. The Kalman filter eats errors as a pre-dinner snack, and nice accommodates new estimates of the vehicle state (e.g., a ping from the Deep Space Network). General relativity? That was just another unmodeled error. There were lots of unmodeled error sources in the 1960s; e.g., Venus and Jupiter (both much greater than GR). $\endgroup$ Mar 29, 2015 at 7:49
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    $\begingroup$ @WetSavannaAnimalakaRodVance - Another way to look at it: I did a lot of the work on the system the Johnson Space Center uses to simulate the environmental and orbital dynamics effects on a spacecraft. (I won some major awards for that work.) Every year, I asked to add GR to our orbital dynamics. Every year, that request was denied. (Maybe next year, David.) My education was in physics, but my career was in aerospace engineering. From a physics POV, I wanted to add GR. From an engineering POV, I knew how small those effects were over a short period of time; much smaller than sensor errors. $\endgroup$ Aug 25, 2015 at 8:14

I actually know the answer to this question, and I found this discussion because I was wondering if anything had been written about it.

In 1985 I volunteered for a political campaign in Hingham, Mass. I ended up doing a 4 hour shift 'poll checking' names of voters in a precinct. The rep for our opponent was seated right next to me and we spent the four hours in a great conversation.

He had worked on developing the inertial guidance system for the Apollo rockets and also ICBMs. He explained how the gyroscopes work in great detail. He said they were so accurate that his team decided to include the effects of relativity in their calculations. If I recall correctly, he said the difference was a few feet, and they indeed adjusted their flight plan to include the effect. I think I remember that he said this was just for fun, or just to show off how accurate their system was.

I've never seen anything else on this topic until the discussion above, so I don't think my memory is clouded by any subsequent info.

Unfortunately, by Googling, I can find a few NASA scientists who lived in Hingham at this time, so I'm not sure which one it was. It might be this man because he also told me not to major in Government (as was my intention) but to major in History instead. He said "majoring in Government is like majoring in the op-ed page of the New York Times." I followed his advice and love History to this day.


  • $\begingroup$ Welcome to Stack Exchange! While normally anecdotal, unsupported answers are discouraged, we have exceptions from time to time, see for example No, they didn't. (personal knowledge). It's also possible others here might be able to help you with narrowing this down, so I think that in this particular case this answer is fine. $\endgroup$
    – uhoh
    Feb 19 at 15:37

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