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PeterDonis submitted a new PF Insights post
The Schwarzschild Geometry: Part 1
Continue reading the Original PF Insights Post.
The Schwarzschild Geometry: Part 1
Continue reading the Original PF Insights Post.
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.HyperStrings said:"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?
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."the vector field ∂/∂t in Schwarzschild coordinates, is no longer timelike at r=2M "
How do you have any sort of 'dynamics', without time?
Nugatory said:If you look at the sign of the metric coefficients, you'll see that ##\frac{\partial}{\partial{r}}## is timelike in that region.
HyperStrings said: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.
PeterDonis said:Can you restate it using math instead of vague ordinary language?
HyperStrings said:Perhaps we'll go into variations on SR
I mean di sitter-Schwarzschild metrics, or questions about Schwarzschild relativity.PeterDonis said:by "variations on SR"
HyperStrings said:I mean di sitter-Schwarzschild metrics
HyperStrings said:questions about Schwarzschild relativity.
PeterDonis said:Those are fine for the topic of a new thread. (I'm not sure why you would describe them as "variations on SR", though.).
I'm sorry. I knew that and made a mistake. I will spell everything out from now on.George Jones said:Everyone else on the Special and General Relativity forum uses the acronym SR to "Special Relativity".
vanhees71 said:Will there be a discussion of the complete extensions of the Schwarzschild solution like Kruskal coordinates?
vanhees71 said:QFT in the Schwarzschild spacetime
tomdodd4598 said:How do you go about this - directly from the metric or from the geodesic equations?
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##.Android Neox said:For one thing, the radius is defined from the center of mass out to the surface.
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:event horizons are impossible because they cannot form in finite time.
Nugatory said:The Schwarzschild radial coordinate is defined (outside the horizon) by ##A=4\pi{r}^2##,
Android Neox said:Einstein was right that event horizons are impossible because they cannot form in finite time.
More precisely, observers who are free-falling inward from rest at infinity.cianfa72 said: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).
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 said: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##.
ah ok, so zero inward velocity (i.e. start form rest) applies only or observers at infinity.PeterDonis said: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##.
Yes.cianfa72 said:zero inward velocity (i.e. start form rest) applies only at infinity.
The Schwarzschild geometry is a mathematical model that describes the curvature of space and time around a non-rotating, uncharged spherical mass. It is a solution to Einstein's field equations in general relativity and is often used to study the behavior of objects in the vicinity of massive objects like black holes.
The Schwarzschild geometry is named after German physicist Karl Schwarzschild, who first derived the solution in 1916. He was one of the first scientists to use Einstein's theory of general relativity to describe the curvature of space and time caused by massive objects.
The Schwarzschild geometry predicts that objects near a massive body will follow curved paths in space and time due to the curvature of spacetime. This effect is known as gravitational lensing and has been observed in the bending of light from distant stars and galaxies around massive objects like black holes.
Yes, the Schwarzschild geometry can be applied to any spherical mass, not just black holes. This includes stars, planets, and even small objects like asteroids. However, the effects of the geometry are most noticeable when dealing with extremely massive objects.
Like any mathematical model, the Schwarzschild geometry has its limitations. It assumes that the massive object is non-rotating and uncharged, which may not always be the case in reality. Additionally, it does not take into account quantum effects, which may play a role in extremely dense objects like black holes.