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? $\endgroup$ – Keith Thompson Mar 19 '15 at 18:09
  • $\begingroup$ General Theory of Relativity. $\endgroup$ – PearsonArtPhoto Mar 19 '15 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 '15 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. $\endgroup$ – andy256 Mar 29 '15 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. $\endgroup$ – WetSavannaAnimal aka Rod Vance Mar 29 '15 at 5:17

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.

  • $\begingroup$ "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. $\endgroup$ – WetSavannaAnimal aka Rod Vance Mar 29 '15 at 0:55
  • $\begingroup$ 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). $\endgroup$ – WetSavannaAnimal aka Rod Vance Mar 29 '15 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) $\endgroup$ – David Hammen Mar 29 '15 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). $\endgroup$ – David Hammen Mar 29 '15 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. $\endgroup$ – David Hammen Aug 25 '15 at 8:14

In Relativity an important factor is the Lorentz factor

$$ \gamma = \sqrt{1 - v^2 / c^2} $$

Using the Lorentz factor the relativistic length contraction is

$$ L = L_0/\gamma $$

and the time dilation is

$$ T = T_0\gamma $$

Assuming the maximum velocity of an Apollo mission is around 14 km/s, and allowing the speed of light to be $3*10^8$ m/s, the Lorentz factor evaluates as

$$ \gamma = \sqrt{1 - \frac{{(1.4*10^4)}^2} {{(3*10^8)}^2}} $$

$$ = \sqrt{1 - \frac{1.96*10^8} {9*10^{16}}} $$

$$ = 0.999999999 $$

So any error due to ignoring relativity is 1 in $10^9$. This is at least 4 orders magnitude less than any of the other errors involved in the mission.

Such calculations are a basic part of any undergraduate physics Bachelor's program, and so were common knowledge among the NASA engineers.

  • $\begingroup$ -1 No, this is not correct. Whilst this is a plausibility argument, you are citing special relativity which only applies to a locally freefalling frame. The Earth-Moon system cannot be rigorously seen to admit a local treatment as the metric tensor manifestly varies throughout this system - you need to compute geodesics in a reasonable model of the metric, which is here the Schwarzschild metric. Have a look at the answers to the question on Physics SE to see how this is done. $\endgroup$ – WetSavannaAnimal aka Rod Vance Mar 29 '15 at 7:21
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    $\begingroup$ Since you know that, you know that the effect is completely negligible. So why even ask the original question? $\endgroup$ – andy256 Mar 29 '15 at 7:30
  • $\begingroup$ As I said, my emphasis is historical. I'd love to find documentation of the reasoning within the archives. It is not straightfoward as you imply and I think there would be an opinion sought from outside expertise. It's not hard, as the physics SE thread shows but it definitely was not part of an engineering education 40 years ago and certainly not part of an Australian education (as I assume yours, like mine, was from your user page - apologies if this is wrong) - I have direct experience of this last fact. $\endgroup$ – WetSavannaAnimal aka Rod Vance Mar 29 '15 at 7:40
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    $\begingroup$ @andy256 - Not the down voter, but this is wrong. What you wrote is special relativity. The one thing special relativity cannot handle is gravitation. That's the job of general relativity. $\endgroup$ – David Hammen Mar 29 '15 at 8:18
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    $\begingroup$ @andy256 - Special relativity most certainly does account for acceleration. Just not acceleration due to gravity. For example, the point of the twin paradox is that one of the twins underwent acceleration, the other did not. $\endgroup$ – David Hammen Mar 29 '15 at 12:42

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