Does Gravity Deflect Light according to Newtonian Mechanics?

[Logic314]

I have read in several popular physics texts that general relativity predicts that gravity deflects light, but that Newtonian mechanics, in contrast, predicts that the trajectory of light is not affected by gravity. However, I am very skeptical and confused about this result.

We of course have the common famous result of Newtonian mechanics, which states that objects of different masses all accelerate the same way under the same local gravitational field (this is the basis for the experimental fact that a feather and a rock fall at nearly the same acceleration inside a vacuum chamber).

But if this is indeed what Newtonian mechanics predicts (that is, if the acceleration in a local gravitational field really is independent of the mass of the particle, according to Newtonian mechanics), then Newtonian mechanics should predict that light has the same acceleration as a rock does within the same local gravitational field, despite the fact that they have different masses (one being nonzero and the other being 0), and thus, Newtonian mechanics should predict that gravity bends light.

So, I'm very confused about why popular physics textbooks seem to emphasize that Newtonian mechanics predicts that light is unaffected by gravity, even though this seems to contradict the general principle from Newtonian gravity theory (that the way objects accelerate under the influence of gravity does not depend on their masses).

 

[Jonathan Scott]

The primary difference between Newtonian Mechanics and the predictions of General Relativity when it comes to light paths is the curvature of the shape of space. Specifically, light from a star is deflected twice as much by passing close to the sun than would be predicted by extending Newtonian mechanics to assume that light was accelerated by the same amount as material objects. This can be accounted for by noting that in GR the shape of a ruler or any locally rigid object would be slightly curved relative to a practical coordinate system suitable for describing orbits. Relative to local rulers, material objects and light are both accelerated at the same rate, but when the curvature of space is taken into account, there is an additional coordinate acceleration due to the effect of the curvature of space on anything moving rapidly, roughly equal to $g v^2/c^2$, where $g$ is the Newtonian acceleration, so for light moving tangentially at $c$ relative to the direction of the source the effective acceleration relative to the coordinates is twice the Newtonian acceleration.

 

[Dale]

thus, Newtonian mechanics should predict that gravity bends light.

That is my opinion also, and your rationale is the same as mine too.

I don’t think there is any official governing board to settle the issue, so I would not recommend arguing about it or pushing it. But you have company.

 

[Janus]

I have read in several popular physics texts that general relativity predicts that gravity deflects light, but that Newtonian mechanics, in contrast, predicts that the trajectory of light is not affected by gravity. However, I am very skeptical and confused about this result.

We of course have the common famous result of Newtonian mechanics, which states that objects of different masses all accelerate the same way under the same local gravitational field (this is the basis for the experimental fact that a feather and a rock fall at nearly the same acceleration inside a vacuum chamber).

The caveat here is the the the force causing the acceleration is
$$ F_g = \frac{GMm}{r^2}$$
thus we can argue that since acceleration is F/m, the acceleration of m is
$$ a_g = \frac{GM}{r^2} $$
and A_g is independent of the value of m.
However, if m=0 then

$$ F_g = \frac{GM(0)}{r^2} = 0 $$
and A_g = 0/0.

So the actual rule is, the acceleration of m is independent of the value of m,as long as m does not equal zero.

Of course, there is no rule that light must have a mass of zero in Newtonian physics, so gravity could have an effect if you assume a non-zero mass.
If you do a calculation using Newtonian orbital mechanics for an object following a hyperbolic orbit around the Sun, just skimming the surface at perihelion and with an initial velocity of c, you end up with a deflection angle just 1/2 that predicted by Relativity (using a quick and dirty method*, I got an estimate of 0.875 arcsec, which is just a tad more than 1/2 the 1.7 arcsec predicted by Relativity.)

* I used the perihelion distance instead of b ( the impact parameter) in an equation. The impact parameter is the closest approach the object would have made to the Sun if its trajectory hadn't been altered by gravity. Thus is will be more than the perihelion distance. Substituting perihelion for b will give a larger answer for the deflection, but for an object moving at c, the difference between impact parameter and perihelion will be small. Small enough that it won't badly effect an "order of magnitude" estimate like this.

 

[256bits]

I have read ..

You might have a look at this for a mathematical treatment of particles passing by a large object, such as a star, or even a black hole.
https://mathpages.com/rr/s6-03/6-03.htm

So, I'm very confused about why popular physics textbooks seem to emphasize that Newtonian mechanics predicts that light is unaffected by gravity, even though this seems to contradict the general principle from Newtonian gravity theory (that the way objects accelerate under the influence of gravity does not depend on their masses).

A confusing part is that light speed is a constant in all reference frames.

We of course have the common famous result of Newtonian mechanics, which states that objects of different masses all accelerate the same way under the same local gravitational field (this is the basis for the experimental fact that a feather and a rock fall at nearly the same acceleration inside a vacuum chamber).

Measuring the speed of light from the rock or feather, or from the release point, or from the surface of the earth, should yield c in all cases. If light accelerates according to Newtonian mechanics, and if light is to be treated as a particle, we should measure something greater than c as light falls to the surface. Instead, the energy loss or gain of light moving in a gravitational field is attributed to a change in frequency, rather than a change in kinetic energy which the particles with mass experience ( particles with mas can never travel at c ). And we thus keep our cherished light always travels a c.