The Schwarzschild geometry: part 1

[PeterDonis]

PeterDonis submitted a new PF Insights post

The Schwarzschild Geometry: Part 1

Continue reading the Original PF Insights Post.

 

[HyperStrings]

"the infalling object slows down as it approaches r=2M"
Would that infalling velocity slow down, if we were to magnify the hyper Planckian singularity as we move into the center? Just as if we spatially zoom into Earth using relativistic functions, while moving quickly towards Earth, from very, very, faraway, our spatial awareness observes the removal of the drag in velocity in the observation of the schwarzchild singularity as we move towards it.

"the vector field ∂/∂t in Schwarzschild coordinates, is no longer timelike at r=2M "

How do you have any sort of 'dynamics', without time?

 

[haushofer]

Nice one, Peter. I learned from Zee's book that independently from Schwarzschild, Johannes Droste also derived the same solution, a few months after Schwarzschild send his solution to Einstein.

 

[Nugatory]

"the infalling object slows down as it approaches r=2M"
Would that infalling velocity slow down, if we were to magnify the hyper Planckian singularity as we move into the center?

We are approaching $r=2M$ so we're still outside the event horizon. What's going on around the singularity is, by Birkhoff's theorem, completely irrelevant.

"the vector field ∂/∂t in Schwarzschild coordinates, is no longer timelike at r=2M "
How do you have any sort of 'dynamics', without time?

At $r=2M$, the Schwarzschild coordinates have a coordinate singularity, so they cannot be used to describe the dynamics there. However, that's just an artifact of having chosen that coordinate system (like trying to determine the longitude of the Earth's north pole); Painleve or EF or Kruskal coordinates all work just fine for describing the dyamics there.

At $r<2M$ the vector field $\frac{\partial}{\partial{t}}$ is not timelike, but that doesn't mean that there's no time; the manifold is still pseudo-Riemannian.

If you look at the sign of the metric coefficients, you'll see that $\frac{\partial}{\partial{r}}$ is timelike in that region.

 

[HyperStrings]

Thank you, Nugatory and PeterDonis. I actually have been intrigued recently by schwarzchild, de sitter and other subatomic geometries/topologies, so I really appreciate the subject matter.

 

[PeterDonis]

If you look at the sign of the metric coefficients, you'll see that $\frac{\partial}{\partial{r}}$ is timelike in that region.

This is true in Schwarzschild coordinates, but not in Painleve coordinates. In Painleve coordinates, all four of the coordinate basis vectors are spacelike in the region $r < 2M$! This is an excellent illustration of why you have to be careful basing anything on properties of coordinate charts.

I'll leave the question of how we can in fact confirm that there are still timelike vectors in the region $r < 2M$ in a chart like Painleve as an exercise for the reader, at least for now. There are more posts coming in this series and I might address this issue in one of them.

 

[PeterDonis]

Would that infalling velocity slow down, if we were to magnify the hyper Planckian singularity as we move into the center? Just as if we spatially zoom into Earth using relativistic functions, while moving quickly towards Earth, from very, very, faraway, our spatial awareness observes the removal of the drag in velocity in the observation of the schwarzchild singularity as we move towards it.

I don't understand what any of this means. Can you restate it using math instead of vague ordinary language? (If you can't, you might want to rethink the question.)

 

[Greg Bernhardt]

Great Insight Peter!

 

[HyperStrings]

Can you restate it using math instead of vague ordinary language?

.
Yes, but I am not trying to distract you into tangents during your paper. It is quite good. Perhaps we'll go into variations on SR some time down the road in a different thread?

 

[PeterDonis]

Perhaps we'll go into variations on SR

Not here on PF, since PF does not allow discussion of speculative or personal theories. Unless by "variations on SR" you simply mean "questions about SR", in which case you can start a separate thread asking (a more precise version of) whatever question you were trying to ask about SR in the post I responded to.

 

[HyperStrings]

by "variations on SR"

I mean di sitter-Schwarzschild metrics, or questions about Schwarzschild relativity.

 

[PeterDonis]

I mean di sitter-Schwarzschild metrics

Those are fine for the topic of a new thread. (I'm not sure why you would describe them as "variations on SR", though.)

questions about Schwarzschild relativity.

If you mean questions about the Schwarzschild geometry, if it's something discussed in the article, you can ask it here. Or you can start a new thread. Just bear in mind that answers will be based on that specific model--i.e., that specific solution of the Einstein Field Equation in classical GR.

 

[George Jones]

Those are fine for the topic of a new thread. (I'm not sure why you would describe them as "variations on SR", though.).

My guess is that is Hyperstrings has used very non-standard terminology, and this has resulted in miscommunication.

Hyperstrings by SR, do you mean "Schwarzschild Relativity"? Everyone else on the Special and General Relativity forum uses the acronym SR for "Special Relativity".

 

[HyperStrings]

Everyone else on the Special and General Relativity forum uses the acronym SR to "Special Relativity".

I'm sorry. I knew that and made a mistake. I will spell everything out from now on.

 

[kent davidge]

Great insight. When will the second part come in?

 

[vanhees71]

Will there be a discussion of the complete extensions of the Schwarzschild solution like Kruskal coordinates? I'd really finally like to understand it's meaning. Recently 't Hooft visited our university and gave a talk about QFT in the Schwarzschild spacetime and the problem, what happens to a particle which "goes through a wormhole". The main problem seemed to be to get a proper physical interpretation of the asymptotic states, and he said there's still controversies in the community.

 

[PeterDonis]

Will there be a discussion of the complete extensions of the Schwarzschild solution like Kruskal coordinates?

Yes, coming in further articles in the series.

 

[PeterDonis]

QFT in the Schwarzschild spacetime

I don't plan to talk about that in this series of articles; I'm only talking about the classical Schwarzschild solution.

 

[tomdodd4598]

I have read through your four articles and found them really interesting and very well written! When I originally read them, it was just because I was interested, but now I'm going back through the calculations - there's one part where you calculate the coordinate velocity of a radially free-falling body from rest at infinity (it comes out as the same as the proper velocity in Schwarzschild coordinates). How do you go about this - directly from the metric or from the geodesic equations? I tried finding the expressions for the conserved quantities, but the energy came out rather nasty.

 

[PeterDonis]

How do you go about this - directly from the metric or from the geodesic equations?

You can use the geodesic equation, yes. You can also use the conserved quantities and the effective potential equation to find expressions for $dr / d\tau$, where $r$ is the "areal radius" (i.e., a 2-sphere at $r$ has area $4 \pi r^2$) and for $dt / d\tau$, and then take their ratio to get $dr / dt$.

 

[Android Neox]

A problem with the Schwarzschild view of black holes is that it depends upon a Euclidean view of space. The radius is defined from the perspective of an observer at infinite distance. This eliminates the spacetime dilation of the gravitational field. For one thing, the radius is defined from the center of mass out to the surface. In Relativity, this region has no definition. The concept "radius" has no sensible interpretation in this case. By taking the view of the observer at infinity, the radius can be defined in terms of a distance to the Schwarzschild radius, but this description is meaningless when viewed "close to" the black hole. I put "close to" in quotes because the distance to any event horizon from any point in space will be infinite. Rather, the rules of Relativity require that this distance is infinite.

The Shapiro effect shows that space is dilated in a gravity well. The amount of dilation is proportional to the difference in gravitational potential energy between the observer and the observed. For an event horizon, the difference in gravitational potential energy between the EH and any point in space is infinite.

Einstein was right that event horizons are impossible because they cannot form in finite time.

 

[Nugatory]

For one thing, the radius is defined from the center of mass out to the surface.

It is not. The Schwarzschild radial coordinate is defined (outside the horizon) by $A=4\pi{r}^2$, where $A$ is the area of a sphere centered on the origin; that sphere defines a surface of constant $r$.

event horizons are impossible because they cannot form in finite time.

This is a very common misconception, usually the result of misinterpreting the Schwarzschild $t$ coordinate and/or forgetting that the Schwarzschild solution is a static solution so does not account for the change of mass of the black hole when something falls into it. If you want more explanation of how this works, you should start a new thread, but not until after you have:
1) Worked you way through the Oppenheimer-Snyder solution to the Einstein Field Equations.
2) Looked at the worldline of an infalling test particle in Kruskal coordinates, so you aren't misled by the coordinate singularity at the event horizon.
3) Looked at some of the many other threads on the subject here.

But in any case - this is a new topic, so don't continue the discussion in this thread.

 

[PeterDonis]

The Schwarzschild radial coordinate is defined (outside the horizon) by $A=4\pi{r}^2$,

It's also defined that way inside the horizon. And the definition of $r$ works fine even at the horizon, although Schwarzschild coordinates overall are singular there; there are other charts (such as Painleve and Eddington-Finkelstein) which define $r$ the same way and are not singular at the horizon.

Einstein was right that event horizons are impossible because they cannot form in finite time.

Einstein never made such a claim, as far as I know. He made two different claims about the minimum possible radius of a stationary system being larger than $2M$ (the event horizon radius corresponding to mass $M$); neither one had anything to do with the time required to form a horizon. Both claims were correct mathematically speaking, but his physical interpretation ignored the possibility of non-stationary collapse.

 

[cianfa72]

I'm keeping reading this interesting insights. As far as I can understand the coordinate time $T$ in Gullstrand-Painleve chart is actually the proper time of free-falling radially infalling observers (i.e. it is the proper time of the congruence of such infalling observers).

So if we consider the worldline of one of such infalling observers that start to fall from event $(r_1,T_1,\theta_1,\phi_1)$ when it reach the event $(r_2,T_2,\theta_1,\phi_1)$ the proper time elapsed on its own wristwatch is actually $T_2 - T_1$.

 

[PeterDonis]

As far as I can understand the coordinate time $T$ in Gullstrand-Painleve chart is actually the proper time of free-falling radially infalling observers (i.e. it is the proper time of the congruence of such infalling observers).

More precisely, observers who are free-falling inward from rest at infinity.

So if we consider the worldline of one of such infalling observers that start to fall from event $(r_1,T_1,\theta_1,\phi_1)$ when it reach the event $(r_2,T_2,\theta_1,\phi_1)$ the proper time elapsed on its own wristwatch is actually $T_2 - T_1$.

If "start to fall from" means "has zero velocity at", this is not correct. The congruence of observers for whom Painleve coordinate time is their proper time have nonzero inward velocity at any finite value of $r$.

 

[cianfa72]

If "start to fall from" means "has zero velocity at", this is not correct. The congruence of observers for whom Painleve coordinate time is their proper time have nonzero inward velocity at any finite value of $r$.

ah ok, so zero inward velocity (i.e. start form rest) applies only or observers at infinity.

 

[PeterDonis]

zero inward velocity (i.e. start form rest) applies only at infinity.

Yes.