Regular readers of this column were warned that the Road Runner–Coyote picture of lumping1 would return in an analysis of black holes. However, you're here anyway, so you have only yourself to blame. We'll warm up by estimating how much light bends in a (weak) gravitational field as it passes by the sun (Fig. 1). The starting point is the Einstein field equations
$G μ ν = 8 π T μ ν ,$
(1)
ten nonlinear coupled partial-differential equations. Each of the three adjectives adds roughly 2 years to the mathematical study required for finding a solution. The adverb of number makes even these labors seem hopeless.
Fig. 1.

Light deflected slightly as it passes near the sun. Relative to the asymptotic incoming path, the asymptotic outgoing path is deflected by θ.

Fig. 1.

Light deflected slightly as it passes near the sun. Relative to the asymptotic incoming path, the asymptotic outgoing path is deflected by θ.

Close modal

We need a quicker method—perhaps dimensional analysis. However, it is a constraint argument, so it cannot give us a physical explanation. Instead, we'll use a lumping model (a mild form of the Road Runner–Coyote picture to come).

In this model, a blob of light moves in a straight line unaware of the sun except when near the sun (Fig. 2)—with “near” meaning at a distance comparable to r, the distance of closest approach.

Fig. 2.

Lumping approximation for the gravitational deflection. When the blob of light is far from the sun, gravity is weak and directed mostly along the path, so the path is (mostly) straight. When the blob is near the sun, gravity is strong and pointed (mostly) perpendicular to the path, resulting in deflection.

Fig. 2.

Lumping approximation for the gravitational deflection. When the blob of light is far from the sun, gravity is weak and directed mostly along the path, so the path is (mostly) straight. When the blob is near the sun, gravity is strong and pointed (mostly) perpendicular to the path, resulting in deflection.

Close modal
In a further simplification, when light is near the sun, the gravitational force is declared to have the same magnitude and direction (directly downward) as it has at the point of closest approach. Then, no matter the blob's mass (irrelevant when the only forces are gravitational), the blob's downward acceleration in the near zone is
$a z ∼ G m sun r 2 .$
(2)
We next need to know how long the blob spends in the near zone, for which we need to estimate the zone's length. Because the downward component of the acceleration falls off rapidly with x (the distance along the trajectory, with x =0 at the point of closest approach), let's say, that the near zone extends from x ∼ −r to x ∼ +r (a 45°-angle in each direction). Then the near zone has length ∼2r, and the transit time is
$t zone ∼ 2 r c .$
(3)
In that time, the blob acquires a downward velocity component
$v z ∼ a z t zone ∼ 2 G m sun r c .$
(4)
The blob's forward velocity component is c, so its path deflects by an angle
$θ ≈ v z c ∼ 2 G m sun r c 2 .$
(5)
(As a check on our algebra, both sides are dimensionless.)

The dimensionless factor of 2 in Eq. (5) cannot be taken too seriously. But it's not nonsense. In the full Newtonian analysis of small deflections, a rock moving at speed c in an extreme hyperbolic orbit deflects by just this amount including the factor of 2.

The general-relativity calculation—solving the field equations in Eq. (1)—gives instead a dimensionless factor of 4. Why is it a factor of 2 greater than the Newtonian value? The general-relativity solution includes not only time curvature, which reproduces Newtonian gravity for weak fields, but also space curvature of the same magnitude. Everyday objects, whose speeds are small compared to c, move in spacetime much faster in time than in space, so they see little of the space curvature. However, light moves at c and sees time and space curvature equally—wherefore the factor of 2 relative to the Newtonian prediction.2

The first interesting deflection is $θ ∼ 1$, which requires a strong gravitational field (near a massive sun, for example). To understand the path that light takes in this situation, we need the Road Runner–Coyote lumping model.

In this model, all the deflection happens at a point. Coyote—the blob of light—moves in a straight line, unconcerned about the strong gravitational field. But as Coyote seems to be escaping the sun, Road Runner suddenly holds up a “Gravity downward” sign. Coyote acknowledges his physics sins and performs the neglected deflection of roughly 1 rad or 60° (Fig. 3).

Fig. 3.

The blob's lumped path in the Road Runner–Coyote model. When $θ ∼ 1$, the blob is trapped in orbit around a black hole.

Fig. 3.

The blob's lumped path in the Road Runner–Coyote model. When $θ ∼ 1$, the blob is trapped in orbit around a black hole.

Close modal

Then Coyote cruises straight ahead unmindful of the gravitational field until he seems to be escaping again, when Road Runner again holds up the sign. Coyote does his physics duty, deflecting by 60°, and then continues straight ahead—until the dreaded sign reappears. The result is a hexagonal orbit around the sun, a closed path from which the blob cannot escape. The blob is (barely) trapped in orbit around a black hole.

This trapping happened with $θ ∼ 1$ or, from Eq. (5),
$G m sun r c 2 ∼ 1.$
(6)
However, this analysis holds only when the blob's closest approach, at a distance r from the center, lies outside the sun. (Otherwise the shell theorem tells us that the gravitational force is smaller than what was incorporated into az in Eq. (2).) Thus, $r ≥ R sun$ (the radius of the sun), so Eq. (6) becomes
$G m sun R sun c 2 ≳ 1$
(7)
or
$R sun ≲ G m sun c 2 .$
(8)
If we'd carried forward the factor of 2 in Eq. (5) that gives the correct Newtonian deflection, then the condition would be
$R sun ≤ 2 G m sun c 2 ,$
(9)
where the length on the right is the Schwarzschild radius. However, carrying forward that factor of 2 and claiming triumph would tempt the gods of approximation into revenge.

In summary, a lumping analysis inspired by the Saturday-morning-cartoon characters Road Runner and Coyote helps us understand the condition required to form a great wonder of the universe, a black hole. Who said that cartoons are only useful for hooking children on sugary cereal, fizzy drinks, and other novel engineered food?

1.
Sanjoy
Mahajan
, “
Energy cost of flight
,”
Am. J. Phys.
88
,
903
905
(
2020
), footnote 6.
2.
The Newtonian analysis was never made by Newton. However, it figured prominently in the famous eclipse expedition led by Eddington in 1919; see, for example,
Matthew
Stanley
,
Einstein's War: How Relativity Conquered Nationalism and Shook the World
(
Penguin
,
London
,
2020
). Some years later, it became notorious when Nazi scientists claimed that it was first calculated by the German scientist Johann von Soldner in 1803 and illustrated Aryan rather than Jewish science;
see
Clifford M.
Will
, “
Henry Cavendish, Johann von Soldner, and the deflection of light
,”
Am. J. Phys.
56
,
413
415
(
1988
).

Sanjoy Mahajan is interested in the art of approximation and physics education and has taught varying subsets of physics, mathematics, electrical engineering, and mechanical engineering at MIT, the African Institute for Mathematical Sciences, and the University of Cambridge. He is the author of Street-Fighting Mathematics (MIT Press, 2010), The Art of Insight in Science and Engineering (MIT Press, 2014), and A Student's Guide to Newton's Laws of Motion (Cambridge University Press, 2020).

Published open access through an agreement with Massachusetts Institute of Technology