objects appearing to slow at event horizon

[freefall]

How is it possible for an object to appear to slow down as it approaches the event horizon of a black hole? Gravity redshifts the light to lower energy so the object appears more red to the observer, but the light is always travelling at the same velocity right? Each photon emitted by the object would still reach the observer in the same amount of time as usual (no redshifting), so how could it look like it's slowing?

[Daecon]

Um... I think it may have to do with the intense gravity causing time dilation, instead of having anything to do with light itself.

 

I'm not certain though.

[timo]

Ok, first off: Forget redshift. Sure, there is redshift but that´s not the cause of the apparent slowdown of particles near the event horizont. Neither does the time the photon takes to reach an observer play any role. You´d have the same effect even with "instananeous" transformation of the position-information.

 

Then, before I twist my brain around a suitable answer, I´d like to ask you the following question first: Do you already know the math involved and ask for how the rather surprising result can be understood or haven´t you seen the math yet? Sadly, neither your post nor the information in your profile give a hint on this.

[freefall]

I do not know the math. I am currently studying newtonian physics but read an article in scientific american which breifly mentioned the phenomenon. I looked around a bit online but could not find a suitable answer without delving into the subject. I posted here hoping for an answer that could temporarily satisfy me until I begin studying this subject.

[timo]

You´re right. It´s surprisingly hard to find any info on the internet. I was browsing around a bit hoping to find a good website explaining the effect but sadly found nothing suitable.

Ok, I´ll start by giving you the plain math so that we have a starting point. For simplification, let´s reduce spacetime to a two-dimensional one with only the coordinates t and r being the time measured by an infinitely distant observer and the distance to the singularity measured by that observer, respectively. Also note that I´ll use fractions of lightspeed as unit for velocity.

 

There is a quantity called "invariant line element" ds² which plays a central role in relativity. When it´s integrated over a trajectory (a path taken through spacetime) of a particle, it gives the time experienced by that particle. This value is the same in any coordinate system. Massless particles take paths which have ds²=0, massive particles take paths with ds²=1.

 

In the case of Special Relativity, the line element is calculated by ds² = dt² - dr² where dt and dr are ... well, let´s hope no mathematican sees this post ... small displacements in time and radius. At this point, you can see that massless particles move at lightspeed while massive ones nessecarily move slower: For massless particles you get 0 = dt² - dr² => dt=dr => v := dr/dt = 1 and with a similar reasoning v= dr/dt < 1 for massive particles.

In the case of General Relativity however, ds²=dt²-dr² is not nessecarily the case. In fact, the actual rule for calculating this line element usually depends on the point in spacetime you´re at.

Sidenote before I´ll proceed further: This is not only some weird effect of GR - the fact that the calculation rule (called "metric") depends on coordinate system AND point in spacetime is absolutely fundamental and the whole inner structure of spacetime is encoded in this calculation rule.

 

In the case of a nonrotating Black Hole, one often uses the Schwarzschild Coordinate system. The Schwarzschild coordinates have the nice property, that a displacement (dt=1, dr=0) actually is what a distant obverver experiences as time and a displacement (dt=0, dr=1) actually is what this observer would call a displacement in space - this is not nessecarily true for an observer that is not far away from the singulariy (which would have been my explanation if you had already known the math).

In these coordinates, the line element has the form

[math] ds®^2 = \left(1-\frac{R}{r}\right) dt^2 - \frac{dr^2}{1- \frac{R}{r}} [/math]

with R being the Schwarzschild Radius (Event Horizon) and r>R. This relation has an explicit dependence on the position as I already anticipated.

 

We´re almost there and there´s two possibilities to see that any objects coming closer to the Event Horizon appear to slow down:

a) Chose a fixed dr (one, for example). Hopefully, you´ll see that the dt needed to make ds²=0 or ds²=1 must increase as r decreases. Therefore, v=dr/dt decreases.

b) explicitly solving for v=dr/dt. For the sake of simplicity, I´ll only do this for a massless particle (ds²=0):

[math] \begin{array}{lrcl}& 0 &=& \left(1-\frac{R}{r}\right) dt^2 - \frac{dr^2}{1- \frac{R}{r}} \\ \Leftrightarrow & \frac{dr^2}{1- \frac{R}{r}} &=& \left(1-\frac{R}{r}\right) dt^2 \\ \Leftrightarrow & \frac{dr^2}{dt^2 \left(1- \frac{R}{r}\right)^2} &=& 1 \\ \Rightarrow & v = \frac{dr}{dt} &=& \left(1- \frac{R}{r}\right) \end{array}[/math]

 

For decreasing r>1 (object getting closer to the Event Horizon), even a massless particle like a photon seems to become slower and slower and ultimately stop at the Event Horizon. This result might seems rather surprising as you´d expect massless particles to travel at a velocity of 1. The reason for this behaviour is that the chosen coordiantes are only physically meaningfull for positions very distant from the singularity at r=0.

[freefall]

interesting. thanks for the response, athiest. I see what you're talking about on the surface (I'm missing the fundamentals). I'm just going to go read about relativity; I think that's the best thing to do.