Using classical mechanics, the formula for gravitational attraction is
$$F = G\frac{m_1m_2}{r^2}.$$
This formula does not work for photons, and we need to use Einstein's theory of gravity to account for that, as the photon is massless. What if, instead of using the mass of the photon, we rather used the inertia of it?
The momentum of a photon is
$$p=\frac{h}{\lambda}.$$
Thus, using that, we can calculate its supposed 'mass' (inertia) as momentum is also equal to mass times velocity. So 'mass' of a photon is
$$m=\frac{p}{v}=\frac{h}{\lambda c}.$$
If we use this mass as the mass of a photon, can we use Newton's equation for gravitational attraction?
Answer
In Newtonian gravity the acceleration due to gravity is independant of the mass of the object - a falling elephant accelerates downwards at the same speed as an accelerating gnat (ignoring air resistance). That means the orbit of an object does not depend on the object's mass (provided the object is much lighter than the star).
The deflection of an object by a massive body is just a hyperbolic orbit, and like any other orbit the trajectory doesn't depend on the mass of the orbiting object, just its velocity and initial position. This means that if we calculate the angular deflection of an object travelling at $c$ in a hyperbolic orbit the result turns out to be independant of the mass of the object:
$$ \theta \approx \frac{2GM}{r_0 c^2} $$
where $M$ is the mass of the star/planet/whatever and $r_0$ is the distance of closest approach.
The point is that assuming some hypothetical value for the photon mass doesn't affect the Newtonian prediction because the Newtonian prediction doesn't depend on mass. So the answer to your question is that no, using an effective photon mass of $m = h/\lambda c$ does not give the correct result.
The GR calculation gives the deflection of light as:
$$ \theta \approx \frac{4GM}{r_0 c^2} $$
which is twice the Newtonian result. But this isn't some special case that applies only to light because GR gives different results to Newtonian gravity for all objects regardless of mass - the deflection of light is just a limiting case.
No comments:
Post a Comment