# Newtonian gravity vs. general relativity: exactly how wrong is Newton?

Is there a simple function I can use to describe the difference between simple Newtonian dynamics and the actual observed motion? Or maybe some ratios for common examples of, say, the motion of stars and planets?

I know for example that Newtonian physics is sufficient to fire a rocket at the moon, so the error there must be miniscule. What exactly is that ratio, and could a layperson calculate it generally for other bodies?

The context is this: in a debate regarding how drastically wrong science can be, I want to make the case that even though Newton was wrong regarding gravity in the bigger picture, he was only _% wrong regarding what he'd observed. I know science can often be wrong, but I want to emphasize the fact that our system of observation isn't fundamentally dysfunctional.

## 2 Answers

Let me skip to your third paragraph, because this highlights a very important point not commonly appreciated by non-scientists.

In Physics a "theory" is a mathematical model based on various assumptions and valid for a limited range of physical conditions. Newton's laws are a mathematical model that is limited to non-relativistic speeds and low gravitational fields, and within those limits it is exceedingly accurate. There is no sense in which Newton was proved wrong by Einstein. What relativity did is expand the range of physical conditions over which the theory applied. Special relativity extended the range to include high speeds, and general relativity extended it again to include high gravitational fields. Even GR is not applicable everywhere because it fails at singularities like the centre of black holes. We expect that some future theory (string theory?) will extend GR to describe places that are singular in GR.

Anyhow, rant over, and on to your real question. The classic difference is the precession of Mercury. This is probably the biggest effect and it's certainly the most easily observed. Because the orbit of Mercury is an ellipse it has a long axis that points in a particular direction. In Newtonian gravity the direction of this axis doesn't change, but GR predicts it changes by 43 arc-seconds per century. This is a tiny tiny amount. The angular resolution of the unaided human eye is about 1 arc-minute, so you would have to watch Mercury for 140 years before the change in the axis would be perceptible.

(Someone is going to point out this isn't strictly true because the precession of Mercury is about 500 arc-seconds/century, however only 43 arc-seconds of this are due to relativistic corrections. The rest is due to perturbations from other planets accurately predicted by Newton's laws.)

• There weren't really any hints that Newtonian gravity was limited until special relativity came along and raised problems. In the context of pre-SR mechanics Newtonian gravity was a complete theory. It would have been natural, but not justified, to assume then that Newtonian gravity was exactly correct. This is the sense in which GR "overthrew" Newton. These days physicists tend to think we know better where the limits of our theories are. Our "effective theories" have built-in breaking points which tell us where to look for new physics. Not so much for Newton pre-special relativity. – Michael Brown Jan 25 '13 at 12:12
• The anomalous precession of Mercury was first recognised in 1859, well before special relativity. However I take your point that at the time people didn't believe it was a limitation of Newton's laws, but rather that there was some other cause e.g. undiscovered planets (like Vulcan). – John Rennie Jan 25 '13 at 12:26
• @JohnRennie - the 43 arc-seconds is an angular measure related to Mercury's orbit around the sun, not an angular measure related to the position where we observe Mercury's position in the night sky. So your statements about human observers and their eye resolution is misleading because it suggests the latter is the case. In other words: the eye resolution comparison only make sense if you are talking about a hypothetical human observer located at the sun. – Johannes Oct 11 '14 at 3:41

The appropriate small parameter varies a bit depending on the circumstance. There is a general (more general even than GR) framework called the Parameterized Post-Newtonian framework, a set of ten numbers which specify the behaviour of a gravity theory in the weak field regime where Newton's law should be a close approximation. GR gives a specific prediction for all ten PPN parameters (through the weak field approximation), as do competing gravity theories, which allows experiments focusing on the PPN parameters to address many competing theories simultaneously.

In a specific setting the story is often simpler. For example, in the central body problem (a good approximation of the solar system), the small parameter is essentially $$\frac{r_S}{r} = \frac{2GM}{c^2 r}$$ where $r_S$ is the Schwarschild radius and $r$ is the distance of the planet or whatever from the central body. A more general expression that has the same role is $\Phi/c^2$ where $\Phi$ is the Newtonian gravitational potential.

## protected by Qmechanic♦Dec 27 '13 at 20:19

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