Gravitational Lensing: Deriving the Light Deflection Formula Using Classical Mechanics

Causes of Gravitational Lensing

How is gravitational lensing produced? Naturally, it is because light deflects in a gravitational field. But why would a light ray, which normally travels in a straight line, take a curved path? According to Fermat’s Principle, light always takes the path that requires the least time to reach its destination¹. If it travels along a curve, doesn’t that violate this principle? Usually, we answer this using General Relativity: mass causes the curvature of spacetime dimensions, making the shortest distance between two points in space no longer a straight line (just as the shortest distance between two points on the Earth’s surface is an arc). However, the explanation of General Relativity is quite professional and not very intuitive. We might consider this problem from a different perspective using classical theory.

Normally, when light passes through a lens, refraction occurs because the lens has a refractive index greater than 1. What does this mean? In fact, it indicates that light travels slower in the lens (glass). According to Fermat’s Principle in optics, it must travel along a bent path to spend the shortest time, which leads to the law of refraction. Does “gravitational lensing” have a similar principle? Does the speed of light slow down in a gravitational field? In a sense, it can be thought of this way, because light travels along a curve in curved high-dimensional spacetime, while in our three-dimensional space, it still appears to travel in a straight line, but the time taken becomes longer. Thus, it appears as if the speed has slowed down, which is equivalent to it having a spacetime “refractive index,” and therefore it must take a curved path². Conversely, if we affirm the principle of relativity that “the speed of light in a vacuum is constant,” then we can conclude that “mass warps spacetime,” because only this view can maintain the validity of Fermat’s Principle. This is also why gravitational lensing is used to test General Relativity.

On the other hand, from the perspective of classical mechanics, according to the photon energy E=h \nu and the mass-energy equivalence E=mc^2, we can derive m=\frac{h\nu}{c^2}, meaning light can be viewed as a particle with mass m=\frac{h\nu}{c^2}. Since this is the case, a particle passing through a gravitational field will naturally deflect due to gravitational force. This is the “two-body problem” in celestial mechanics³. Surprisingly, any enthusiast with some foundation in celestial mechanics can start from this perspective and use some laws of classical celestial mechanics, combined with some loose assumptions, to derive the formula for the deflection angle of light in a gravitational field. And most incredibly, the result of this non-rigorous derivation is exactly half of the result from General Relativity! Of course, this is just a miraculous coincidence; it is not rigorous, but it has certain appreciative value. I share it here with fellow enthusiasts as a way to invite further discussion.

Classical Mechanics: Light Deflection Angle in a Gravitational Field

The speed of a photon is c, which is already far greater than the escape velocity of many stars. According to celestial mechanics, a photon passing near these stars follows a hyperbolic path, with the star (the Sun) located at one focus of the hyperbola. From analytical geometry, for any hyperbola, the equation is: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

How do we represent the deflection angle? We can consider the two asymptotes of the hyperbola: \frac{y}{x} = \pm \frac{b}{a}.

As shown in the figure, the size of angle \theta is the size of the deflection angle, and \theta = 2\varphi = 2 \arctan\frac{a}{b}. Assuming the eccentricity e of the hyperbola is very large (which is exactly the case for light in a gravitational field because its initial velocity c is very large), then we have a \ll b: \frac{a}{b} = \frac{a}{a\sqrt{e^2-1}} \approx \frac{1}{e} \approx 0

For a very small angle \theta, we have \tan\theta \approx \theta, so: \theta \approx 2 \arctan \frac{1}{e} \approx \frac{2}{e}

The next goal is to find e. There is an important “vis-viva equation” in celestial mechanics (which seems to be a must-know for Olympiad friends?): v^2 = GM(\frac{2}{r} - \frac{1}{a}). For a hyperbola, this becomes v^2 = GM(\frac{2}{r} + \frac{1}{a}), which describes how the velocity v changes with distance r. Of course, the speed of light does not change, but here we ignore this situation. We assume that when the light reaches the “perihelion” (point B in the figure), its speed is c, so we have v=c and r = a(e-1) \approx ae. Substituting these into the vis-viva equation, we get: c^2 = GM\left(\frac{e}{r} + \frac{2}{r}\right) = \frac{GM(e+2)}{r} \approx \frac{GMe}{r}

That is, e \approx \frac{c^2 r}{GM}. Substituting this into \theta \approx \frac{2}{e}, we get: \theta \approx \frac{2GM}{c^2 r} This is half of the result given by General Relativity. It can be said that to initially understand gravitational lensing, one does not necessarily need the profound theory of General Relativity.

A Few More Words

Directly applying classical mechanics analogies to understand and estimate phenomena predicted by General Relativity can often yield reasonable results. Sometimes, like deriving the black hole radius r_g = \frac{2GM}{c^2} from the escape velocity formula v = \sqrt{\frac{2GM}{r}}, one gets the correct estimation formula. Other times, as in the estimation above, the form is correct, but it differs by a ratio. In short, based on the “particle nature” of light, it is possible for us to use classical mechanics to initially understand some relatively profound theoretical results. However, I do not know much about gravitational theory, so I won’t say more. I’ll just leave the process here for everyone’s reference.

The above is the result of BoJone’s unintentional calculation after reading the article “Gravitational Lensing: Leading the Scientific Tide Again.” I have always maintained the philosophy of “Accumulate much to release much, accumulate little to release little.” Even with a small amount of knowledge, one can use their own strength to study and analyze astrophysical phenomena and obtain some exciting results. Perhaps the process is not rigorous and the result is not entirely correct, but this does not hinder our interest in pursuing astronomy; on the contrary, it is one of the drivers and pathways for us to chase our astronomical dreams.

Addendum

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