Black Holes, Mercury, and Einstein: The Role of Dimensional Analysis

In last week’s posts we looked at basic astronomy and Einstein’s famous E=mc² through the lens of the secret weapon of theoretical physicists, “dimensional analysis”, which imposes a simple consistency check on any known or proposed physics equation.  For instance, E=mc² (with E being some kind of energy, m some kind of mass, and c the cosmic speed limit [also the speed of light]) passes this consistency condition.

But what about E=mc or E=mc⁴ or E=m²c³ ? These equations are obviously impossible! Energy has dimensions of mass * length² / time². If an equation sets energy equal to something, that something has to have the same dimensions as energy. That rules out m²c³, which has dimensions of mass² * length³ / time³. In fact it rules out anything other than E = # mc² (where # represents an ordinary number, which is not necessarily 1). All other relations fail to be consistent.

That’s why physicists were thinking about equations like E = # mc² even before Einstein was born. 

The same kind of reasoning can teach us (as it did Einstein) about his theory of gravity, “general relativity”, and one of its children, black holes.  But again, Einstein’s era wasn’t first to ask the question.   It goes back to the late 18th century. And why not? It’s just a matter of dimensional analysis.

Escape Velocity and Dimensional Analysis

Newtonian physics teaches us that if you fire a projectile from the surface of a planet or star, it has to move at a minimum speed, called the “escape velocity” (let’s write it “v_e”) for it to overpower that object’s gravity and head out into space.  Since v_e has to be related G (Newton’s constant, ever-present in gravity), the mass M of the object, and the radius R of the object, we can estimate it using dimensional analysis. We saw in this post that 

• G has units of length³ / mass / time², and so 
• G M has units of length³ / time², and thus
• G M / R has units of length² / time². 

Since velocity has units of length / time, the last relation implies

• v_e² = # G M / R

is consistent; and it’s the only possible equation that is.  As usual, # is an unknown number that we expect (from experience with many calculations) is not too, too different from 1.  (It turns out # = 2.)

This relation implies that if you crush an object down, making R smaller but M the same, then its escape velocity increases. To escape an Earth one-fourth the size (with the same mass) would require a projectile with twice the speed. 

Can Light Escape?

Here’s an obvious question, even in the 18th century.  How small would the Earth have to be in order that you would have to fire the projectile at the speed of light for it to escape?  The answer is estimated by  setting v_e equal to the speed of light c:

• c² = # G M / R

Reshuffling this equation would give us the required radius:

• R = # G M/ c²

For the Earth, that’s roughly the size of a pebble or golfball. For the Sun, it’s close to a mile (a couple of km). 

In other words, as John Michell (1784) and Pierre-Simon Laplace (1796) argued, if the Earth were crushed down to the size of a small stone, or smaller, then light emitted from its surface would never escape.  It would slow down, stop, and fall back to Earth.  And this would make the Earth invisible to someone looking in its direction from far out in space: it would be a black rock.

More generally, any star with mass M that’s smaller than # GM/c² would be invisible, no matter how hot and bright.  None of its light would escape into space; it would still be a star, perhaps, but it would be invisible from far away. 

Would these black rocks and black stars be black holes?  No.  You could go visit them, and if you were close enough you could see them, take photos, and come back home. You’d have to travel faster than light’s speed to do that, but in the 18th century, who was sure that was a problem?

So the statement that “black holes were invented/discovered in the 18th century” really isn’t true.  The conceptions of light and of its speed were very different, as was the expectation of what such objects would be like. We now know

• Light doesn’t slow down and fall back like a projectile;
• You can’t visit a black hole beyond light’s point of no return (the black hole’s “horizon”), and hope to make it home;
• A black hole does not contain a stable object with a surface; what’s inside must collapse completely.

But the 18th century correctly identified that for an object of mass M, whether rock or star or anything else, GM/c² is an interesting radius that combines light with gravity in some way. [Let’s call it the object’s “gravitational radius” R_G.]

What wasn’t known is that the speed of light c isn’t just the speed of light; it’s a universal speed limit, a feature of the space and time that form the substance of the universe itself.  When Einstein proposed that c is deeply integrated into space and time, and that gravity involves the curvature of space and time, it was pretty obvious that this radius was far more important in nature than Michell and Laplace could have realized.  It’s not just an astrophysical curiosity.

Changes to Gravity’s Force Law

An object’s gravitational radius isn’t just about whether it looks black. It’s a measure of the degree to which its gravity might differ from Newton’s original proposal of a simple inverse square law. 

The force law for gravity between two objects of mass M and m (with m very small and M very large) separated by a distance R might actually not be

• F_Newton = GMm / R²

Instead it might be corrected by a term that looks like R_G/R:

• F  = F_Newton * (1 + # R_G/R + …) = GMm / R² + # G²M²m / (c²R³) + …

where R_G = GM/c² is the gravitational radius of the object of larger mass M, the # is an unknown number not far from 1, and the dots represent effects that are even smaller at large R.  When R is very large, the R_G/R term is tiny, and so Newton’s law still works wonderfully unless you measure with extreme care.  But for R comparable to R_G, it would be obvious that Newton’s law isn’t quite right; gravity’s not an inverse square law anymore.

This proposal isn’t easy to check. For the Sun, R_G is almost a million times smaller than the Sun itself, so the R_G/R effect can never be large for any object outside the Sun.  Indeed, for the Earth, for which R = 150 million kilometers, R_G/R is less than a billionth!  That’s why Newton could never have noticed this effect in his predictions for the planets. 

But with precision measurements and for objects closer to the Sun, perhaps the small deviation from Newton’s law could be observable?

Mercury

It was well known, at Einstein’s time, that something was up with the orbit of the planet Mercury. Mercury’s somewhat elliptical orbit gradually rotates (“precesses”) [see the Figure for an exaggerated sketch]. This is mainly an effect of the other planets’ gravity on Mercury. The precession is often described as 570ish arcseconds per century, which is not illuminating to most people, including me; but in units I can understand, it represents a precession of one millionth of a circle per orbit.  That is, Mercury’s ellipse rotates around completely every million orbits, or about 250,000 Earth years.  

The astronomer Le Verrier found that the other known planets generate only 90% or so of Mercury’s precession.  Roughly 40 to 50 arcseconds were still unexplained, an effect of only 80 billionths of a circle per orbit! (That’s .00000008 of a circle per orbit.) Since Le Verrier had successfully predicted the planet Neptune through his work on Uranus’s orbit, he tried again with Mercury: he predicted a new planet, Vulcan, closer to the Sun.

Vulcan, unlike Neptune, didn’t turn up. (I recommend this popular book about the failed hunt. https://thomaslevenson.com/hunt-for-vulcan; or for technical details, see “Mercury’s Perihelion” by Roseveare).  Other proposals were made, such as rings of dust around the Sun that might be harder to detect than a planet.  But the matter remained unsettled.

Enter Einstein. Because of dimensional analysis, Einstein already knew, before he even had a real theory of gravity, that his approach to relativity might explain Mercury’s extra precession.  This is not quite as obvious as our previous cases. The dimensions of precession are the same as a number, so at first glance it might seem we should expect

• extra precession per orbit = #

where # is not too, too far from 1.  That’s obviously in contradiction to observation! So our naive dimensional analysis is dead on arrival.

But we can be cleverer than that, using what we may call “improved dimensional analysis”, where we try to build on our physics knowledge and intuition.  Here’s a logical requirement that our formula must satisfy, as we imagine varying the distance R_MS between Mercury and the Sun:

• When R_MS is very large, Newton’s force law should hold, so there ought to be no extra precession (i.e. nothing beyond what Le Verrier calculated) when R_MS becomes infinite.

Our naive dimensional analysis guess above fails to satisfy this requirement. But it is satisfied if we imagine another consistent equation:

• extra precession per orbit = # R_G^sun / R_MS
  • = # ( 1.5 km / 50 million km ) =  # (30 parts per billion) 

In other words, improved dimensional analysis suggests a precession not too, too far from the observed 80 parts per billion. That’s why Einstein suspected he might be on the right track.

The only way to know more was to find a consistent theory of gravity and calculate the unknown #.  The first time he did it, with his friend Michel Besso using the 1912 version of his theory, Einstein got about 30 parts per billion, well below the observed 80.  But in any case Einstein knew the 1912 theory wasn’t right; it was inconsistent. He kept tinkering with it, and in November 1915 he finally found a consistent theory. When he used it to repeat the precession calculation, he now found 80 parts per billion per orbit, or 43 seconds of arc per century. That was well within the range astronomers knew at the time. 

Today, with a more precise measurement of the precession than was available back then, we know Einstein’s gravity explains all but 1% of the extra precession, a remarkable achievement.  (The remainder is also quite well understood; the main effect is that the Sun is not a perfect sphere but is slightly squashed.)

Black Holes, Finally 

Just a month later, Karl Schwarzschild, deployed at the Russian front during the first World War, used Einstein’s new equations to calculate what we now think of as the spacetime geometry of a black hole.  It took 42 years to interpret Schwarzschild’s formulas correctly, but even back then, it was obvious something very striking happens to space and time near the radius R_G predicted by dimensional analysis.  What we describe today as the location of the black hole’s horizon — the point of no return — is known as the Schwarzschild radius

• R_Sch = 2 G M /c² = 2 R_G 

(For spinning black holes the location is slightly smaller, but dimensional analysis still works.)  Up to a factor of 2, it’s the gravitational radius, just as we, and 18th century physicists, would have guessed.

Again and again, dimensional analysis that combines gravity and relativity has pointed physicists toward good questions, and has told them, before they even start calculating, roughly what the answers may be.  That doesn’t make the calculation any less important.  Only knowing the # exactly could Einstein claim he’d explained Mercury’s perihelion.  Nevertheless, before setting off on a hard calculation, or designing a challenging experiment, scientists always put their dimensional analysis hats on, and do their famous “back of the envelope” pre-calculation estimates.  That’s how they know, before they start, that they’re embarking on a worthwhile task, one with a real chance of success.