Solving for Speed of Light Inside a Black Hole

[Kilian]

What is the mass of a supermassive black hole in solar units? And in theory could you accelerate past the speed of light inside the Swarzchild's radius of a black hole? I used the equation g=GM/r2 then plugged it into the kinematic equation Vf=Vi+(a)(x). I also used the mass of my theoretical black hole as 1.9691x10x37kg this being 10 million solar units with an almost infinite density of 1 meter as the radius, from 1610m away, starting at 0m/s. If this initial velocity could be obtained would it not be possible? I would appreciate help with my problem. Thanks

 

[cristo]

What is the mass of a supermassive black hole in solar units?

According to wiki a SMBH has mass between 10^5 and 10^10 solar masses.

And in theory could you accelerate past the speed of light inside the Swarzchild's radius of a black hole?

The physical velocity* of the object will not exceed the speed of light, no.

I used the equation g=GM/r2 then plugged it into the kinematic equation Vf=Vi+(a)(x). I also used the mass of my theoretical black hole as 1.9691x10x37kg this being 10 million solar units with an almost infinite density of 1 meter as the radius, from 1610m away, starting at 0m/s. If this initial velocity could be obtained would it not be possible? I would appreciate help with my problem. Thanks

I don't really know what you're doing here. Any mathematical analysis of what goes on near black holes requires knowledge of the general theory of relativity. Finally, this is not a question on Quantum Mechanics.


*I imagine that the coordinate velocity of the object will, however, exceed the speed of light. (That is dr/dt using usual notation). However, since the object is inside the Schwarzschild radius, no observer will ever see this velocity, and so it is not a physical velocity.

 

[blechman]

*I imagine that the coordinate velocity of the object will, however, exceed the speed of light. (That is dr/dt using usual notation). However, since the object is inside the Schwarzschild radius, no observer will ever see this velocity, and so it is not a physical velocity.

This isn't really right. No velocity ever exceeds the speed of light in vacuum in any rest frame, even within the schwarzschild radius; saying otherwise only leads to misunderstanding. I know what you are trying to say, and from a naive point of view the words are more-or-less true. But it should be emphasized very loudly that what you are talking about isn't physically happening: from the point of view of the person falling into the black hole, they're still moving slower than the speed of light, and from the point of view of the person outside the black hole, they've actually slowed down - it takes an infinite amount of time to fall into a black hole! I acknowledge your last sentence about it not being a "physical velocity" but I just think it should be said even louder to avoid misunderstanding.

<Feel free to move this to the GR section.>

 

[Kilian]

why?

Even though for a supermassive black hole this is true, would it not be possible for some black hole with a mass close to that of our sun be able to accelerate objects past the speed of light before they hit the Schwarzschild's radius?

 

[blechman]

Even though for a supermassive black hole this is true, would it not be possible for some black hole with a mass close to that of our sun be able to accelerate objects past the speed of light before they hit the Schwarzschild's radius?

absolutely not! nothing ever goes faster than the speed of light. You can see this by studying the geodesic equations (the equation of motion for a massive particle falling into a black hole). To accelerate a massive particle to the speed of light requires an infinite energy transfer, and no object in the universe, not even a black hole, can do it. But in order to see this explicitly, you need GR. Newton won't get it right, as you pointed out earlier.

 

[cristo]

This isn't really right. No velocity ever exceeds the speed of light in vacuum in any rest frame, even within the schwarzschild radius; saying otherwise only leads to misunderstanding.

Which is why I used the term "coordinate velocity." Perhaps I should have instead said the rate of change of the coordinate r wrt the coordinate t.

 

[blechman]

Which is why I used the term "coordinate velocity." Perhaps I should have instead said the rate of change of the coordinate r wrt the coordinate t.

Except "coordinate velocity" is not well-defined inside the schwarzschild radius: there is no "r"; there is no "t"; so you can't talk about dr/dt - it doesn't make sense! That's all I'm saying. You say it too, but I just wanted to emphasize it. I've been asked many questions from people about things like this, so I just thought it would help others to really hammer in the point. I hope you agree.

 

[cristo]

Except "coordinate velocity" is not well-defined inside the schwarzschild radius: there is no "r"; there is no "t"; so you can't talk about dr/dt

Well, I disagree with this. But, to prevent this thread from going offtopic, I'll leave it at that.

 

[Chris Hillman]

It depends

No velocity ever exceeds the speed of light in vacuum in any rest frame, even within the schwarzschild radius; saying otherwise only leads to misunderstanding.

To be precise: the forward tangent vectors to any timelike curve in any Lorentzian manifold always remain within the forward light cone at each event along the curve.

"Physical velocities", the ones which would be measured by an observer tracking a nearby object, always refer to a frame field (aka "orthonormal basis of vector fields", "local Lorentz frame"), not to a "coordinate basis", and are taken wrt the lapse of proper time as measured by an ideal clock carried by a specific observer, not wrt a coordinate called "the time coordinate". However, sometimes coordinates can be given fairly immediate operational meaning (e.g Coll canonical chart, aka "emission coordinates", or the time coordinate in a comoving chart for an FRW model), or have a clear geometric interpretation (e.g. Schwarzschild radius), and then it may be useful to compute "coordinate speeds", which are simply partials wrt coordinate values. In particular, in order to discuss a timelike congruence, one typically writes the generating timelike unit vector field in some chart as a linear combination of the coordinate basis, and then the ratios of components of the tangent vector give coordinate speeds.

An additional wrinkle: even in flat spacetime, there are multiple distinct operationally significant notions of "distance in the large". Failure to recognize this invariably leads to much unneccessary confusion.

Except "coordinate velocity" is not well-defined inside the schwarzschild radius: there is no "r"; there is no "t"; so you can't talk about dr/dt - it doesn't make sense! That's all I'm saying.

I think you mean to say that the "exterior Schwarzschild chart" is not defined inside the horizon, so we can't use it to discuss coordinate speeds there. But there are "interior Schwarzschild charts" which are not defined in the (left or right) exterior but which are defined in the (past or future) interior regions. Using these charts, one can discuss coordinate speeds of families of observers defined in these regions, such as the Frolov observers. See for example https://www.physicsforums.com/showthread.php?t=146912&page=5

 

[AlphaNumeric2]

Except "coordinate velocity" is not well-defined inside the schwarzschild radius: there is no "r"; there is no "t"; so you can't talk about dr/dt - it doesn't make sense! That's all I'm saying. You say it too, but I just wanted to emphasize it. I've been asked many questions from people about things like this, so I just thought it would help others to really hammer in the point. I hope you agree.

There is an r and a t within the event horizon, the Schwarzschild metric is valid within the event horizon, it's just not valid crossing the event horizon.

Since the event horizon is a coordinate singularity (ie an artifact of our inappropriate choice of coordinates), such issues can be removed. Kruskal coordinates or Eddington-Finklestein advanced coordinates allow a description of objects crossing the event horizon in a completely valid manner.

 

[Chris Hillman]

The METRIC is valid on the horizon, but some charts are not

There is an r and a t within the event horizon, the Schwarzschild metric is valid within the event horizon, it's just not valid crossing the event horizon.

No, the metric tensor is perfectly valid on the horizon, as can be seen by changing to some coordinate chart whose domain straddles the horizon, such as the ingoing Eddington chart (aka "advanced Eddington-Finkelstein" chart) or the Kruskal-Szekeres chart or...

I think you meant to say that

• the exterior Schwarzschild chart is valid outside but not on the horizon (more precisely on one of the two exterior regions in case of the "eternal Schwarzschild vacuum"),
• the future interior Schwarzschild chart is valid on the future interior but not on the horizon,
• the leftward ingoing Eddington chart is valid on the right exterior and future interior ("left" and "right" referring to the usual Carter-Penrose diagram depicting the global conformal structure of the "eternal Schwarzschild vacuum",
• the Kruskal-Szekeres chart is a valid on the entire manifold (apart from the usual "seams" along the "International Date Lines" on our nested spheres)
• etc.

A picture would be worth a few thousand words, but in lieu of a picture here are a few thousand words https://www.physicsforums.com/showthread.php?t=146912&page=5

 

[blechman]

I think you mean to say that the "exterior Schwarzschild chart" is not defined inside the horizon, so we can't use it to discuss coordinate speeds there. But there are "interior Schwarzschild charts" which are not defined in the (left or right) exterior but which are defined in the (past or future) interior regions. Using these charts, one can discuss coordinate speeds of families of observers defined in these regions, such as the Frolov observers.

Yes, this is exactly what I was getting at. You should not use Schwarzschild coordinates defined outside the event horizon (Region I) when describing events inside the horizon (Region 2), or else you start to get some funny results. If you really want to talk about events in both regions, you must use Kruskal extension, for example. I think you said it much better than I did.

Let me just point out that students have asked me funny questions in the past about the strangeness of black holes, and many of their (and my) contradictory intuition came from using Schwarzschild coordinates in Region 2. This is why I warned so loudly that it's a dangerous thing to do. However, I am not a GR expert, and I hope that I didn't say anything wrong or misleading. If I did, I apologize.

 

[cristo]

Yes, this is exactly what I was getting at. You should not use Schwarzschild coordinates defined outside the event horizon (Region I) when describing events inside the horizon (Region 2), or else you start to get some funny results. If you really want to talk about events in both regions, you must use Kruskal extension, for example. I think you said it much better than I did.

But the point is that one doesn't need to use a Kruskal extension, unless passing through the Schwarzschild radius. The original question, which I answered (and I'm somewhat wishing I didn't now) was regarding motion inside the Schwarzschild radius.

 

[blechman]

But the point is that one doesn't need to use a Kruskal extension, unless passing through the Schwarzschild radius.

Except isn't that what you're doing? You're talking about dr/dt as measured by a person outside the event horizon for someone that's INSIDE the horizon, and that quantity being larger than c:

*I imagine that the coordinate velocity of the object will, however, exceed the speed of light. (That is dr/dt using usual notation). However, since the object is inside the Schwarzschild radius, no observer will ever see this velocity, and so it is not a physical velocity.

this sounds like a no-no to me!

The original question, which I answered (and I'm somewhat wishing I didn't now) was regarding motion inside the Schwarzschild radius.

I'm sorry that you feel that this discussion isn't worth your time. Personally, I think it is very enlightening.

 

[Chris Hillman]

Woe is we

I think you both made some good points.

No doubt we three can all agree that when answering newbie queries in PF, we walk a fine line between trying to answering something close to the question which was asked--- in which case we risk getting balled up in a sisyphean attempt to make sense of nonsense--- or else trying to explain why it would be wiser to ask a substantially different question--- in which case we risk being accused of "arrogance" by someone who resents our thoughtful attention to the fine points of logical reasoning

 

[randa177]

Assuming that the black hole at the center of the Milky Way Galaxy (MWG) is Schwarzschild (non-rotating) black hole, how large is its Schwarzschild radius, in AU?

 

[Chris Hillman]

Careful!

What is the mass of a supermassive black hole in solar units?

Cristo already answered although his source (Wikipedia, a website which absolutely anyone can edit--- note that they are even about to remove the slight restriction on IP anons which was announced after the Siegenthaler defamation scandal) was unreliable. See this page from a course taught in the Astro department of the University of Tennesee:
http://csep10.phys.utk.edu/astr162/lect/active/smblack.html

Assuming that the black hole at the center of the Milky Way Galaxy (MWG) is Schwarzschild (non-rotating) black hole, how large is its Schwarzschild radius, in AU?

The supermassive black hole near the center of our own galaxy is known as Sag A*; it is about 27000 light years distant and has a estimated mass of somewhere around 2.6 million solar masses. See
http://chandra.harvard.edu/photo/2003/0203long/
http://science.nasa.gov/headlines/y2000/ast29feb_1m.htm

The mass of our Sun is about 1.5 km, so 2.6 million solar masses corresponds to about 4 million km. 1 AU is about $1.5 \times 10^{8}$ km. So the "Schwarzschild radius" of Sag A* would be about 0.03 AU.

And in theory could you accelerate past the speed of light inside the Swarzchild's radius of a black hole?

Black holes must be treated using gtr. In Newton's theory of gravitation, as you may know, the gradient of the "gravitational potential" function gives the gravitational acceleration of a small object ("test particle" in an ambient gravitational field. But in gtr, the gravitational field is treated quite differently, as the curvature of spacetime. In the geometric picture used in gtr, the kind of "acceleration" you have in mind corresponds to the path curvature of a world line, but the world line of a free falling particle is a "timelike geodesic" and thus has vanishing path curvature. In both gtr and Newton's theory, a freely falling test particle experiences no "body force" but does experience small "tidal forces" (if it is falling in radially, these forces cause a slight radial expansion and orthogonal compression). The "tidal tensor" which models this effect corresponds, in gtr, to part (not all) of the Riemann curvature tensor.

I used the equation g=GM/r2 then plugged it into the kinematic equation Vf=Vi+(a)(x).

That's incorrect; you are trying to use Newtonian concepts in a highly relativistic situation!

I imagine that the coordinate velocity of the object will, however, exceed the speed of light. (That is dr/dt using usual notation). However, since the object is inside the Schwarzschild radius, no observer will ever see this velocity, and so it is not a physical velocity

This must have been a brain blip on cristo's part. Coordinate speeds are indeed not in general physically meaningful (and one should point out that even in flat spacetime there exist multiple operationally distinct notions of "distance in the large" and thus "speed in the large"). But the interior Schwarzschild chart he apparently had in mind is only defined on the "future interior:" domain $0 < r < 2 \, m$, while the exterior chart is only defined on the "exterior" domain $r < r < \infty$

No velocity ever exceeds the speed of light in vacuum in any rest frame, even within the schwarzschild radius; saying otherwise only leads to misunderstanding.

Exactly!

from the point of view of the person falling into the black hole, they're still moving slower than the speed of light, and from the point of view of the person outside the black hole, they've actually slowed down - it takes an infinite amount of time to fall into a black hole!

Unfortunately, this is potentially misleading. In some of my previous posts (see my sig) I have given a very detailed analysis of the physical experience of various families of observers, including Lemaitre observers, who fall in freely and radially, Hagihara observers in stable circular orbits, and distant static observers.

would it not be possible for some black hole with a mass close to that of our sun be able to accelerate objects past the speed of light before they hit the Schwarzschild's radius?

No; to begin to appreciate what gtr says about black hole interiors you need to know something about what it models gravitation and the motion of particles. See the posts in my sig.

 

[randa177]

If matter were accreting spherically onto the MWG black hole, what is the maximum luminosity in units of Le it could achieve?

 

[Chris Hillman]

If matter were accreting spherically onto the MWG black hole, what is the maximum luminosity in units of Le it could achieve?

You mean the object known as Sag A*? Aka the supermassive black hole near the center of our own galaxy? What are "units of Le"?

 

[rubecuber]

yeah