Acceleration inside an event horizon

[jaketodd]

Consider an electron traveling through space at 99% of the speed of light. It passes within the event horizon of a black hole. We know that it can not escape. This implies something to me and I am wondering if it is correct: Since it does not come out, that means that the acceleration by the black hole within the event horizon is greater than 99% the speed of light, per second. I am guessing that most black hole event horizons are a lot less, in diameter, than 99% of 299,792,458 meters, which is how far the electron would travel in one second if its course is not altered by the black hole. Therefore, since most event horizon's diameters are less than 99% of 299,792,458 meters, the acceleration by the black hole, inside the event horizon, must be a lot larger than 99% of 299,792,458 meters per second, per second. Is this correct?

Thanks,

Jake

 

[tiny-tim]

Hi Jake!

Consider an electron traveling through space at 99% of the speed of light. It passes within the event horizon of a black hole.

Everything (including light) crosses the event horizon at the same speed (which we may as well call the speed of light).

Since it does not come out, that means that the acceleration by the black hole within the event horizon is greater than 99% the speed of light, per second. I am guessing that most black hole event horizons are a lot less, in diameter, than 99% of 299,792,458 meters, which is how far the electron would travel in one second if its course is not altered by the black hole. Therefore, since most event horizon's diameters are less than 99% of 299,792,458 meters, the acceleration by the black hole, inside the event horizon, must be a lot larger than 99% of 299,792,458 meters per second, per second.

i] c is a speed limit, not an acceleration limit

ii] the electron will hit the singularity and cease to exist, within a fraction of a second

 

[jaketodd]

So the electron never goes faster than the speed of light inside the event horizon? I know the speed of light is a speed limit and not an acceleration limit. Since the electron is already traveling so close to the speed of light, and once it enters the event horizon it is accelerated by a huge amount, wouldn't it go faster than light inside the event horizon? Or, is the size of the event horizon and the size of the mass at the center of the black hole always in a balance that may accelerate particles at different rates, but never accelerates them long enough for the speed of light to be broken inside the event horizon? (The particle reaches the mass at the center of the black hole before it can be accelerated beyond the speed of light always?)

You said everything including light crosses the event horizon at the same speed, which must be the speed of light. Since there is distance left to be traveled from the event horizon to the mass at the center of the black hole, then there is time for further acceleration. Therefore, things exceed the speed of light within the event horizon. Is this correct?

Thanks,

Jake

 

[tiny-tim]

Hi Jake!

Therefore, things exceed the speed of light within the event horizon. Is this correct?

This is a more specific question than in your original post, and really deserves a separate thread.

My interpretation of the Schwarzschild metric is that, inside an event horizon, the speed of light is still a limit, but it is a lower limit (a minimum), so that everything else travels faster than light.

But I've had arguments with other members about this in the past, and they disagree with my interpretation, and say that I'm misunderstanding the nature of time and space inside an event horizon.

 

[tom.stoer]

Inside the event horizon pace and time change roles. Instead of always moving towards the future you always move towards the singularity. All directions are "singularity-directed".

 

[DrGreg]

In general relativity, the local speed of light, measured by an observer right next to the light, is always c (299,792,458 m/s), and nothing can overtake light. This is true for any observer wherever they are, outside, inside, or falling through the event horizon.

However, if an observer tries to extend their coordinate system to measure light some distance away, he or she may get a very different answer. In some cases you can even measure the "remote speed of light" to be zero. This is due to spacetime curvature -- you can't extend a locally-square-shaped grid to cover a curved surface without the squares distorting.

So an observer outside the event horizon might conclude that something inside the event horizon is falling faster than c, but light itself would be falling even faster. A local observer falling inside the event horizon wouldn't notice anything unusual.

It's actually even weirder than that. The "Schwarzschild" coordinate system that is often used is really two separate coordinate systems, one outside & one inside the event horizon. They just happen to have the same equation that describes their metric, but for the outside coordinates, t_outside is (distorted) time and r_outside is (distorted) distance, whereas for the inside coordinates t_inside is (distorted) distance and r_inside is (distorted) time. It doesn't really make any physical sense for an outside observer to measure something inside using these coordinates because they aren't the same coordinates. (And indeed an observer outside the horizon can never detect anything inside the horizon except by falling through it himself.) There are other choices of coordinate system which are more appropriate for tracking something as it falls through the horizon.

 

[tom.stoer]

... but one can introduce coordinates which need not be patched at the horizon but are covering both pieces of spacetime w/o coordinate singularity.

The peculiar nature of the event horizon can be understood by anaylzing why all massive bodies cross the event horizon with speed of light. This is due to the nature of the horizon itself, it is a light-like surface.

 

[jaketodd]

I want to hear from someone who disagrees with the speed of something inside the event horizon being faster than light. It seems like everyone here agrees that it's faster than light. Should I make a new thread, as tiny-tim suggests?

Thanks all,

Jake

 

[djy]

Assuming the electron is free-falling, then, in the semantics of GR, there is no "acceleration": the electron is merely following a geodesic in spacetime.

However, you can ask what the proper acceleration is felt by a stationary observer -- one whose world line is constant in $$r, \theta, \phi$$. It is,

$$a = \left (1 - \frac{2M}{r} \right )^{-1/2} \frac{M}{r^2}.$$

Note that the acceleration diverges to infinity as $$r$$ approaches the event horizon, $$2M$$.

Even more importantly, note that the expression becomes imaginary inside the event horizon, when $$r < 2M$$. This is because a world line constant in $$r, \theta, \phi$$ is spacelike inside the horizon, so it is impossible to have a "stationary" observer to measure proper acceleration: all timelike world lines lead to the singularity.

 

[Passionflower]

My interpretation of the Schwarzschild metric is that, inside an event horizon, the speed of light is still a limit, but it is a lower limit (a minimum), so that everything else travels faster than light.

That is a factually wrong statement in GR, the local speed limit is c, also passed the event horizon.

 

[DrGreg]

I want to hear from someone who disagrees with the speed of something inside the event horizon being faster than light. It seems like everyone here agrees that it's faster than light.

As I tried to explain in post #6, it depends what you mean by "faster than light".

If you mean "faster than a photon at the same place in the same direction", nothing ever goes faster than light, anywhere.

If you mean "faster than 299,792,458 m/s measured by someone at the same place", nothing ever goes faster than light, anywhere.

If you mean "faster than 299,792,458 m/s measured by someone some distance away", things can go "faster than light" in that sense, but light itself goes even faster!

If you mean "faster than 299,792,458 m/s measured by someone at the same place hovering at a fixed height inside the horizon", the question makes no sense, because nothing can hover in that way.

If you mean "faster than 299,792,458 m/s inside the horizon measured by someone outside the horizon", the question makes no sense, because someone outside the horizon can't measure anything inside the horizon.

 

[starthaus]

Consider an electron traveling through space at 99% of the speed of light. It passes within the event horizon of a black hole. We know that it can not escape. This implies something to me and I am wondering if it is correct: Since it does not come out, that means that the acceleration by the black hole within the event horizon is greater than 99% the speed of light, per second.

You seem to have "solved" this problem in http://www.wbabin.net/ntham/todd2.pdf "paper".

 

[jaketodd]

Is learning here and applying that knowledge to my work against any rules?

 

[Dale]

In curved or non-inertial coordinate systems it is very easy for things to obtain a coordinate velocity greater than c. Simply look at a distant star while turning around at 1 revolution per second or so. In your rotating reference frame the star is going much faster than c.

To avoid this kind of non-physical "faster than c" it is better to talk about coordinate-independent concepts like the spacetime interval. In terms of the spacetime interval the electron's worldline outside the event horizon is timelike, meaning slower than c in a coordinate independent sense. Inside the event horizon the electron's worldline remains timelike, again meaning slower than c in a coordinate independent sense.

 

[stevebd1]

I want to hear from someone who disagrees with the speed of something inside the event horizon being faster than light. It seems like everyone here agrees that it's faster than light. Should I make a new thread, as tiny-tim suggests

You might compare the event horizon of the black hole with the cosmic light horizon of the observable universe. Beyond the CLH, the universe is expanding faster than c relative to us because of the expansion of space but if you consider an object just beyond the CLH, nothing is traveling faster than c locally. It's something like this for an object that falls past the event horizon of a BH.

 

[tom.stoer]

In GR you can quite easily observe speed greater speed of light, but never locally! You can e.g. observe (!) objects beyond the Hubble sphere having velocity greater than speed of light.

But this is always due to non-local effects. An observer doing a localized measurement observing particles just at the position of the observer will always find velocities smaller or equal to the speed of light. That means that locally the light cone is always valid just as in SR.