general relativity - Why doesn't gravity speed up light?

We know that gravity speeds up a body; for instance, a meteor which enters the earth gets constantly accelerated up by earth's gravity. And from relativity we know that light bends near a massive body, because Newton's law of gravitation is just an approximation and actually gravity depends on energy and momentum. So my question is: If a ray of light is aimed exactly at the center of a body, then will it get accelerated like a meteor? And if does get accelerated, then won't it surpass the universal speed limit of 3,00,000 km/s (approx.)?

Answer

If a ray of light is aimed exactly at the center of a body, then will it get accelerated like a meteor?

Short answer: no. However, when falling in a gravity field, the momentum of light increases.

Some background...

In Newtonian mechanics, the rate of change of momentum of a (massive) particle is proportional to the acceleration:

$$\frac{d\vec p}{dt} = m \vec a $$

In Relativistic mechanics, these quantities are not proportional. In fact, an accelerating massive particle can never reach speed $c$ but the momentum can reach arbitrarily large values.

This is because relativistic momentum is a non-linear function of velocity

$$\vec p = \frac{m \vec v}{\sqrt{1 - \frac{v^2}{c^2}}} $$

which diverges as $v \rightarrow c$.

In the special case of a massless particle, which must travel at speed $c$ in all frames, the numerator and denominator in the above are zero so, by this formula, the momentum of a massless particle is indeterminate.

However, the relativistic energy-momentum relation

$$E^2 = (pc)^2 + (mc^2)^2 $$

gives the momentum of a massless particle:

$$p = \frac{E}{c} $$

Thus, the momentum can change even though the speed does not. In falling from a higher potential to a lower potential, the massless particle gains energy and thus momentum but not additional speed.

For light, the momentum and frequency are proportional:

$$p = \frac{h\nu}{c} $$

so, while the speed of light does not increase as it falls, the frequency of light increases. From the Wikipedia article "Blueshift":

Photons climbing out of a gravitating object become less energetic. This loss of energy is known as a "redshifting", as photons in the visible spectrum would appear more red. Similarly, photons falling into a gravitational field become more energetic and exhibit a blueshifting