105 105105 105972010-05-13 00:43:32 / rev b667c9e4c1f1+5.4 Bending of light by gravityRocks, birds, and people feel the effect of gravity. So why not light?The analysis of that question is a triumph of Einstein’s theory of generalrelativity. We could calculate how gravity bends light by solving theso-called geodesic equations from general relativity:d2xβdλ2+ Γβµνdxµdλdxνdλ= 0,wwhere Γβµνare the Christoffel symbols, whose evaluation requires solv-ing for the metric tensor gµν, whose evaluation requires solving the general-relativity curvature equations Rµν= 0.The curvature equations are themselves a shorthand for ten partial-differentialequations. The equations are rich in mathematical interest but are a night-mare to solve. The equations are numerous; worse, they are nonlinear.Therefore, the usual method for handling linear equations – guessing ageneral form for the solution and making new solutions by combininginstances of the general form – does not work. One can spend a decadelearning advanced mathematics to solve the equations exactly. Instead,apply a familiar principle: When the going gets tough, lower your stan-dards. By sacrificing some accuracy, we can explain light bending in fewerthan one thousand pages – using mathematics and physics that you (andI!) already know.The simpler method is dimensional analysis, in the usual three steps:1. Find the relevant parameters.2. Find dimensionless groups.3. Use the groups to make the most general dimensionless statement.4. Add physical knowledge to narrow the possibilities.These steps are done in the following sections.5.4.1 Finding parametersThe first step in a dimensional analysis is to decide what physical para-meters the bending angle can depend on. For that purpose I often startwith an unlabeled diagram, for it prods me into thinking of labels; andmany of the labels are parameters of the problem.106 106106 106982010-05-13 00:43:32 / rev b667c9e4c1f1+sunHere various parameters and reasons to include them:1. The list has to include the quantity to solve for. So the angle θ is thefirst item in the list.2. The mass of the sun, m, has to affect the angle. Black holes greatlydeflect light, probably because of their huge mass.3. A faraway sun or black hole cannot strongly affect the path (near theearth light seems to travel straight, in spite of black holes all over theuniverse); therefore r, the distance from the center of the mass, is arelevant parameter. The phrase ‘distance from the center’ is ambigu-ous, since the light is at various distances from the center. Let r bethe distance of closest approach.4. The dimensional analysis needs to know that gravity produces thebending. The parameters listed so far do not create any forces. Soinclude Newton’s gravitational constant G.Here is the diagram with important parameters labeled:sunmass mθrHere is a table of the parameters and their dimensions:Parameter Meaning Dimensionsθ angle –m mass of sun MG Newton’s constant L3T−2M−1r distance from center of sun L107 107107 107992010-05-13 00:43:32 / rev b667c9e4c1f1+where L, M, and T represent the dimensions of length, mass, and time,respectively.5.4.2 Dimensionless groupsThe second step is to form dimensionless groups. One group is easy:The parameter θ is an angle, which is already dimensionless. The othervariables, G, m, and r, cannot form a second dimensionless group. To seewhy, following the dimensions of mass. It appears only in G and m, so adimensionless group would contain the product Gm, which has no massdimensions in it. But Gm and r cannot get rid of the time dimensions. Sothere is only one independent dimensionless group, for which θ is thesimplest choice.Without a second dimensionless group, the analysis seems like nonsense.With only one dimensionless group, it must be a constant. In slow motion:θ = function of other dimensionless groups,but there are no other dimensionless groups, soθ = constant.This conclusion is crazy! The angle must depend on at least one of mand r. Let’s therefore make another dimensionless group on which θ candepend. Therefore, return to Step 1: Finding parameters. The list lacks acrucial parameter.What physics has been neglected? Free associating often suggests themissing parameter. Unlike rocks, light is difficult to deflect, otherwisehumanity would not have waited until the 1800s to study the deflection,whereas the path of rocks was studied at least as far back as Aristotleand probably for millions of years beforehand. Light travels much fasterthan rocks, which may explain why light is so difficult to deflect: Thegravitational field gets hold of it only for a short time. But none of theparameters distinguish between light and rocks. Therefore let’s includethe speed of light c. It introduces the fact that we are studying light, andit does so with a useful distinguishing parameter, the speed.Here is the latest table of parameters and dimensions:108 108108 1081002010-05-13 00:43:32 / rev b667c9e4c1f1+Parameter Meaning Dimensionsθ angle –m mass of sun MG Newton’s constant L3T−2M−1r distance from center of sun Lc speed of light LT−1Length is strewn all over the parameters (it’s in G, r, and c). Mass, how-ever, appears in only G and m, so the combination Gm cancels out mass.Time also appears in only two parameters: G and c. To cancel out time,form Gm/c2. This combination contains one length, so a dimensionlessgroup is Gm/rc2.5.4.3 Drawing conclusionsThe most general relation between the two dimensionless groups isθ = fGmrc2.Dimensional analysis cannot determine the function f , but it has told usthat f is a function only of Gm/rc2and not of the variables separately.Physical reasoning and symmetry narrow the possibilities. First, stronggravity – from a large G or m – should increase the angle. So f shouldbe an increasing function. Now apply symmetry. Imagine a world wheregravity is repulsive or, equivalently, where the gravitational constant isnegative. Then the bending angle should be negative; to make that hap-pen, f must be an odd function: namely, f (−x) = −f (x). This symmetryargument eliminates choices like f(x) ∼ x2.The simplest guess is that f is the identity function: f (x) ∼ x. Then thebending angle isθ =Gmrc2.But there is probably a dimensionless constant in f . For example,θ = 7Gmrc2and109 109109 1091012010-05-13 00:43:32 / rev b667c9e4c1f1+θ = 0.3Gmrc2are also possible. This freedom means that we should use a