# How does the velocity of light change in the vicinity of a black hole?

1. May 3, 2010

### Starwanderer1

Hi everyone,
This may be a really simple one for most of you but I got to start somewhere..
When a ray of light approaches a black hole, is it really slowed down for it to curve from its previous path and enter it.. or does it speed up according to general gravity considerations (of two entities coming closer..).
(I am not skipping the fact that "general gravity considerations" cant be made on stellar bodies and photons.. I actually mean taking into account the mass of the black hole and the factor hf/c^2 where f is the frequency,for the "photon wave").

If I stick to the latter possibility, wont it make it difficult for light to be "absorbed" into the black hole?

2. May 3, 2010

### mgb_phys

The speed of light doesn't change - only it's direction changes.

3. May 3, 2010

### OrionVTOL

Do light photons have weight to them? I've never really understood the theory of "black holes affecting light, to where not even it could escape".

4. May 3, 2010

### mgb_phys

No but they can be affected by gravity - thats the point of general relativity.

Newton's law says that masses attract each other - so only masses create gravity and only masses are affected by it.
Einstein says that masses bend space - and anything moving through bent space is bent - whether it has mass or not.

5. May 3, 2010

### Jonathan Scott

I disagree - that's not helpful at best, and in general I think it's probably wrong.

The problem is one of choice of coordinate systems. The speed of light is indeed always c to a local observer, but the viewpoint of a local observer is not very useful in this case, as it cannot be extended consistently to cover more than a very small region in the vicinity of a large mass, and it certainly cannot be extended to cover such concepts as orbits, because on that scale space-time is curved.

In order to describe what is happening around a central mass, one has to choose a coordinate system that can cover the region of interest. This is like choosing how to map a curved surface on to paper. Obviously, the choice of mapping method does not affect the thing that is being mapped, but it does mean there are alternative ways of describing it.

The most practical coordinate system for most astronomical purposes involving central dominant masses, for example in the solar system, is the isotropic coordinate system, where the scale factor mapping between local lengths and coordinate lengths is the same in all directions, but varies with potential. In this system, the speed of light relative to the coordinate system is the same in all directions, and light gets slower as you approach the object. For small changes in potential, as in the solar system, the fractional decrease in the size of a local ruler relative to the coordinate space and the fractional decrease in the rate of a local clock relative to coordinate clocks are both approximately equal to the Newtonian potential energy per unit energy, typically -Gm/rc2. This means that the speed of light relative to the coordinate system is decreased by both these fractions, so it decreases by twice the relative potential. For deeper gravitational potentials, as in the vicinity of a black hole, it is necessary to use a more complicated expression for the speed of light, but it continues to get slower as you approach the central object. This is as if space is "thicker" the closer you get to the central object.

For calculations in the vicinity of a black hole, it is mathematically simpler to use the Schwarzschild coordinate system. In this system, the radial coordinate is defined in a simple way, but this means that the scale factor for radial distances is not the same as that for tangential distance, so the speed of light relative to the coordinate system is different in radial and tangential directions and just talking about the "speed of light" without specifying the direction isn't very meaningful. However, the speed of light relative to the coordinate system, in any direction, still gets slower as you get closer to the central object.

The "thickness" of space in this sense can be used like a refractive index to calculate the curvature with respect to space of a light beam and hence its acceleration, but for the full picture you also need to take into account the curvature of space-time with respect to time, which is normally around the same strength as the curvature with respect to space and which accounts for the gravitational acceleration of objects at rest or with non-relativistic velocities. For objects moving at or near the speed of light (including light itself), the resulting acceleration (at least for weak fields, as in the solar system) is exactly twice that predicted by Newtonian theory.

6. May 3, 2010

### IcedEcliptic

I thought that light never changed its velocity independent of a specific medium, with vacuum not being a medium at all, as the maximum: 'c'. Before it crosses the event horizon, ignoring observer effects, is it not degrees of freedom which are restricted, and not a change in velocity?

7. May 3, 2010

### OrionVTOL

Help me out in trying to understand how something with no mass is affected by something that attracts mass.

8. May 3, 2010

### mgb_phys

That's the whole point, Newton was wrong, F != GMm/r^2
Gravity doesn't attract mass - it just happens that (for small masses at least) Newton's law is a good approximation.
Mass curves space (hard to picture but there you are) anything moving through space moves along what it thinks is a straight line - but only because the space it is moving through is curved.

Note, people did try and make this work by saying E=mc^2 so light has a 'sort of mass' m=E/c^2 and you can still use Newton's laws.

There was a test of this during an eclipse in 1919 to measure the amount light bends passing the sun. Using E=mc^2 would give a different amount of bending than Einstein's theory of General relativity - it turns out that Einstein was right

9. May 3, 2010

### IcedEcliptic

Light has energy and momentum, which is contributing to the SET. This in turn, effects the path of light, yes?

10. May 3, 2010

### Jonathan Scott

Einstein was right, but E=mc^2 has nothing to do with the different bending of light by the sun.

It doesn't matter what "mass" value you assign to a test object for gravitational purposes as the effective force is proportional to the mass.

The double bending is effectively due to the curvature of space, which is not predicted by Newtonian theory.

11. May 3, 2010

### mgb_phys

Yes, that's the point - the Eddington expedition was to test if the classical bending of light (ie E=mc^2) or Relativity was correct.

12. May 3, 2010

### Jonathan Scott

Note that if you express gravitational forces not in terms of velocity and acceleration but rather in terms of momentum and effective forces, then the equation of motion for free fall in a central gravitational field (in the weak field approximation, in isotropic coordinates) becomes nearly as simple as in Newtonian theory:

dp/dt = g E/c2 (1 + v2/c2).

where E is the energy of the falling object (equal to its rest energy plus kinetic energy) and p is the relativistic momentum, Ev/c2. The right hand is similar to the Newtonian mg except that the rest mass m is replaced with the mass E/c2 corresponding to the total energy and there is a factor of (1+v2/c2) which corrects for curved space, causing the effective force to be doubled as the speed approaches that of light.

(To be accurate, c in the above should be the coordinate speed of light, which varies very slightly from the standard value).

Note that this equation holds regardless of whether the motion is tangential, radial or somewhere in between. The rate of change of momentum always points to the central mass (at this level of approximation) so angular momentum is conserved.

For an object in free fall, E is constant (in that it includes potential energy due to the effect of the time rate on the rest mass and kinetic energy, and the sum is constant). One can therefore divide both sides by E and state the above rule as follows:

d(v/c2)/dt = g/c2 (1 + v2/c2)

This shows that it is not the velocity which changes in a simple way, but rather the velocity divided by c2. This works even for vertical motion of light, where as mentioned earlier the coordinate value of c decreases with lower potential so the value of 1/c changes with time at a rate of 2g/c2 (which is positive in the downwards direction, as usual for the gravitational field, regardless of whether the light is moving upwards or downwards).

13. May 3, 2010

### Jonathan Scott

I still disagree. The effective mass assigned to light is not relevant to how it is affected by gravity. An object with twice or half the energy but the same speed still follows the same free fall path.

The part of GR which causes the double deflection angle is the curvature of space.

14. May 3, 2010

### Jonathan Scott

"Tachyon" is simply a term meaning "particles which move faster than light". No such particles are known. If they existed and it were possible to communicate using them, then it would be possible to construct a device which would return such a signal to the sender before it was initially sent, violating causality, so it is normally assumed that such things cannot exist. However, some attempts to explain quantum mechanics make use of such weird ideas.

15. May 3, 2010

### IcedEcliptic

How does this change what mgb physics is saying? This is established in GR, is it not? If c were variable in a vacuum then there would be no relativity to begin with.

16. May 3, 2010

### Nabeshin

You're missing the point...

The "effective mass of light" theory was one created in order to preserve newton's law of gravitation while adhering to some parts of special relativity (let's not get into an argument about this. This is just what the theory is). The point is that this prediction is NOT correct. Every physicists knows this view of gravity is incorrect.

Of course we also know that in GR energy or mass of the particle is irrelevant in the test particle approximation.

The point is that you cannot simply doctor newton's gravity to try to accommodate light particles -- it doesn't work. mgb's note about e=mc^2 was merely pre-empting the question so many people have of why you cannot simply plug in an effective mass in the newtonian gravity calculation, not stating that this is correct!