We don’t know what an astrophysical black hole really is. Semiclassical Gravity predicts that General Relativity fails at the horizon where critical, but not well understood quantum processes take place (e.g. a firewall) that affect information about matter. So your question cannot be answered without Quantum Gravity (other than there is no practical difference outside the horizon, as you stated) +1

@safesphere With suitable choice of coordinates, GR is perfectly fine at the horizon.

I thought that was the meaning of vacuum solution, $T_{\mu \nu}=0$

@RC_23 $T_{ab}=0$ only outside the collapsing matter, or any region not containing matter-field

A Schwarzschild BH contains no matter, but does have mass. The stress-energy tensor of the Schwarzschild BH is zero everywhere inside and outside.

@safesphere Then where did the collapsing matter go after formation of BH?

@KP99 A Schwarzschild BH does not form from a collapsing matter, but is eternal (has existed forever).

@safesphere Ok, I'm confused now, you mean gravitational collapse of stellar object into a schwarzschild BH is not possible? It will be helpful if you can elaborate your point as a separate answer to this question, or at least mention any source(s) which has explained this point.

@KP99 the question you just asked is essentially my question. "When a stellar object collapses and settles into a steady state condition, is there a meaningful difference between it and a Schwartzchild BH?" (Besides rotation)

@RC_23 Yup, which is why I think posting a separate answer will clarify our doubts. I think we are missing the context here and so far I have never come across this statement in any resources that a schwarzschild BH does not form from a collapsing matter

@KP99 A Schwarzschild Solution is for a BH that is the only thing that exists forever in an infinite universe. So indeed, a collapse to a Schwarzschild BH is impossible by definition. A collapse is possible to something that may look like a Schwarzschild BH. This question is about the degree and details of the similarity.

@RC_23 The Schwarzschild solution is a vacuum solution outside the source. It's the same thing as saying that the electrostatic field of an electron comes from solving Maxwell's equations outside the point charge: but there is obviously a point charge that sources the field. There is obviously a matter source for Schwarzschild as well. The origin is the singularity, however, and it is excluded from the domain of spacetime manifold under consideration. You can derive Schwarzschild by putting a non-zero, distributional source of some mass at the origin.

@Avantgarde "The Schwarzschild solution is a vacuum solution outside the source" - Incorrect. It is a solution without a source. (The source is added by the Brkhoff theorem.) - "It's the same thing as saying that the electrostatic field..." - It is not the same. - "There is obviously a matter source for Schwarzschild as well" - Incorrect. It is a vacuum solution. - "You can derive Schwarzschild by putting a non-zero, distributional source of some mass at the origin" - You cannot. It is a vacuum solution. All your statements are incorrect plus you are answering a question in comments.

@safesphere when we discuss an empty pipe that is filled with water (say right after a valve is opened), after an initial transient phase the flow reaches steady state where all time dependent terms vanish, and can be very well described by (say) Poisuelle eqn, with no $t$ dependence. Is the stellar collapse to a Schwartz BH essentially the same?

(As long as I live I will never spell that name correctly...)

@safesphere I think you consider schwarz. BH as the standard analytically extended schwarz BH with r=0 removed from the domain. However, you do realise that a schwarz. line element need not be globally defined throughout the space-time, like geometry outside a static spherical body. This is guaranteed by Birkhoff's theorem, so your response to Avantgarde is incorrect and needs reconsideration. This elementary result is presented in any literature on Schwarzschild space time so apart from confusion with definition of schwarz BH, I don't see why this confusion is arising in the first place

@safesphere Thanks for taking time to clarify the issues. I completely agree with your first comment. Isn't stitching the outer schwarz. soln with interior soln. of a spherical body, the same thing as saying solving for spherical body distribution which behaves like schwarzschild solution outside the distribution? I'm assuming stitching should be such that the metric is continuous overal I get the point in what sense matter source is not needed for SBH, but it doesn't mean I can't derive schwarz solution by starting from a spherical body, of course solution inside the body will be different.

@RC_23 "after an initial transient phase the flow reaches steady" - Not really (see my comment under the answer of Jan Gogolin). There is no practical difference outside, but different theoretical models for the inside. For example, compare a collapsed star (where yellow is the star matter): upload.wikimedia.org/wikipedia/commons/2/27/… - and the Schwarzschild BH: jila.colorado.edu/~ajsh/insidebh/penrose_schwpar.gif

@safesphere I'll respectfully disagree with this viewpoint, because I don't know what it means to derive vacuum solutions based on vacuum. The vac. soln. (as part of total soln.) will depend on the symmetry + boundary conditions given at the interface b/w vac. and matter part. Once I have the vac. part, I can extrapolate it by treating the source to be invisible. Same argument for SBH, except now due to spherical symmetry I can shrink the source down to a point without affecting the exterior soln. I can now remove the r=0 point (thus changing the topology), thereby making it a total vac. soln.

@KP99 The vacuum solution is not derived based on the boundary conditions around the source. It is derived assuming there is no source and the entire spacetime is empty. The origin is excluded not because it is a source, but because polar coordinates are undefined there. The boundary conditions are zero at infinity, not anything at the source - there is no source in the vacuum solution. Also, as I mentioned earlier, you cannot shrink the source to make it smaller than the horizon, because the horizon is in the causal past relative to its interior and therefore is the source to anything inside.

@KP99 The vacuum solution is stitched to the matter solution not by changing the initial conditions along the border, but by varying the parameter M. The initial conditions remain the same - zero field at infinity.