It is of course true that many physicists don't have as much formal education in differential geometry as mathematicians who have taken graduate courses in that subject and its prerequisites, but that three-page letter reveals Rabounski's own misunderstandings of some extremely basic facts of differential geometry and general relativity.
Some of those twelve sections are more wrong than others...
Rabounski should have known right from the start that something was terribly wrong with Crothers analysis.
Rabounski is a self-proclaimed expert at General Relativity, and his journal, Progress in Physics, is supposedly peer-reviewed; so it’s inexcusable that Crothers was ever allowed to stick his nose under the tent; let alone become the journal’s associate editor. But Crothers managed to hoodwink Rabounski by riding in on Abrams' coattails:
Dmitri Rabounski (January 2008) said:
Leonard S. Abrams’ professional reputation is beyond doubt. As a result, it is particularly noteworthy to observe that Stephen J. Crothers [2], building upon the work of Abrams, was able to deduce solutions for the gravitational field in a Schwarzschild metric space produced in terms of a physical observable (proper) radius. Crothers’ solutions fully verify the initial arguments of Abrams. Therefore, the claim that the correct solution for the gravitational field in a Schwarzschild space does not lead to a black hole as a physical object requires serious attention.
Eventually Rabounski woke up from his slumber and realized that Crothers was single-handedly turning the journal into a laughing stock.
The Rabounski/Crothers collaboration came to an abrupt halt in 2010. That was the year Rabounski published the Jason J. Sharples article; which essentially drove a stake through the heart of Crothers work. And to then add further insult, Rabounski refused to let Crothers respond to Sharples criticism:
Stephen Crothers (November 2010) said:
You say that Rabounski rejected my reply to Sharples because it is too long. This is a deception by Rabounski because the editorial policy of Progress in Physics, of which Rabounski is the Editor-in-Chief, is to publish special reports and other important papers no matter what length… The real reason Rabounski rejected my paper is because my detailed reply to Sharples includes material that impinges adversely upon his own research in General Relativity and that of his colleague L. Borissova.
Incidentally, Crothers has been forced to post his articles at viXra ever since this comical episode; and those once heady days when he could boast that his work had been published in a peer-reviewed journal are now long gone and forgotten.
Rabounski should have known right from the start that something was terribly wrong with Crothers analysis.
Rabounski is a self-proclaimed expert at General Relativity, and his journal, Progress in Physics, is supposedly peer-reviewed; so it’s inexcusable that Crothers was ever allowed to stick his nose under the tent; let alone become the journal’s associate editor.
It’s interesting and very telling that Clinger continues to make ad hominem references to me and other demonstrably erroneous remarks about me and my mathematics after he backed out of the public debate with me that you invited him to. It seems he prefers a one-sided ‘debate'; of course! I can’t help but laugh at the inanity of all their childish and spiteful comments. It’s painfully evident that Clinger and his cronies don’t even know what an infinite equivalence class is. I also noted with amusement that I have been accused of being unaware of Schwarzschild’s second paper, even though years ago I had a paper published on Schwarzschild’s second paper in the journal Progress in Physics. Clinger and his cronies ridicule Progress in Physics and its editors in their futile attempts to discredit me, as if their referring pejoratively to the publisher carries scientific weight. I also note that I have been repeatedly called a liar by Clinger’s cronies. What folly! Furthermore, the likes of Sharples and Corda have been cited against me by Clinger et al, yet I have included the former in my short and simple Summary, the very Summary that Clinger backed out of debating with me: My Malicious, Gormless Critics
If I had a penny for every time such egregious nonsense as Clinger’s has been levelled at me by cosmologists and their ilk I’d be very well off. That Clinger and his cronies have to resort to such drivel attests that they have nothing of any scientific merit to say. Here is another big nail in the coffin of Big Bang cosmology:
Robitaille P.-M., Crothers S. J.
“The Theory of Heat Radiation” Revisited: A Commentary on the Validity of Kirchhoff’s Law of Thermal Emission and Max Planck’s Claim of Universality, Progress in Physics, v. 11, p.120-132, (2015), http://www.ptep-online.com/index_files/2015/PP-41-04.PDF
No doubt Clinger and his cronies will again try to assert that the publisher is sufficient to invalidate the science in the paper. In the hands of the likes of Clinger and the cosmologists, it is no wonder that science is in such a parlous state of intellectual decrepitude.
It turns out that Stephen Crothers is fully aware of (but doesn't understand) Schwarzschild’s 2nd paper, and even wrote an article about it in 2005. I assume the full responsibility for the initiating and spreading of the above mentioned inaccuracy, and now set the record straight. Here is Crothers error-ridden article published in 2005:
We now know that Stephen Crothers is closely watching this thread. So why doesn’t he just comment like everyone else?
The fact is, Crothers is scared **** less. He has never faced an audience like this one; well prepared and eager to pounce, with long knives out and sharpened.
So instead of joining the debate that’s now in progress, Crothers just posts his usual bluster from the comfort and safety of Myron Evans’ website; where no one is allowed to respond to his comments.
Edited breach of rule 10. Please type out all curse words in full with no masking or replaced characters, and allow the autocensor to take care of the word.
Wow - Crothers has retained the delusion that this paper with the crank Robitaille has anything to do with cosmology !
The authors inanely think that Planck had to "properly justify Kirchhoff’s Law". Kirchhoff's law of thermal radiation was first stated in 1859 and applies to perfect black bodies. Applying Kirchhoff's law of thermal radiation for perfect black bodies to perfect black bodies is perfectly justified!
Before Kirchhoff's law was recognized, it had been experimentally established that a good absorber is a good emitter, and a poor absorber is a poor emitter. Naturally, a good reflector must be a poor absorber.
The paper is basically a rant about real materials not being perfect black bodies as if everyone in the scientific world was not aware of this basic fact .
The paper concentrates on Planck's "The Theory of Heat Radiation" book published in 1915. The derivation of Planck's law in 1900 is only cited in conjunction with the book.
ETA: Their citation is actually to
"On an Improvement of Wien's Equation for the Spectrum".
Planck M. ¨Uber das Gesetz der Energieverteilung im Normalspektrum. Annalen der Physik, 1901, v. 4, 553–563.
This is the paper where Planck proposes an empirical law to fit experimental data from real materials !
The paper that derives his law is a few weeks later
"On the Theory of the Energy Distribution Law of the Normal Spectrum"
Planck, M. "Zur Theorie des Gesetzes der Energieverteilung im Normalspektrum". Verhandlungen der Deutschen Physikalischen Gesellschaft 2: 237
In order to stay in the game, Crothers had to jump from Abrams’ to Robitaille’s coattails. And as a result, Rabounski has allowed Crothers to weasel his way back in the door; with the provision that he keeps his mouth shut about black holes and General Relativity.
My apologies to “Strawman” for revealing his identity; I should have paid closer attention to the forum rules. But once his name appeared at the Myron Evans website, well, the cat was out of the bag, so to speak. If “Strawman” wants to retain his anonymity, he is strongly advised to never confide in Stephen Crothers and Myron Evans – the two biggest blabbermouths on the planet.
And what is the story with this juvenile and childish debate negotiation? Crothers requires a spokesperson to arrange special rules and guidelines before he will agree to comment? Yeah, right.
Here is a recommendation for Mr. Crothers: at age 58 you’re a big boy now, so man up, grow a pair, and just post your comments like everyone else at this forum.
Here is a recommendation for Mr. Crothers: at age 58 you’re a big boy now, so man up, grow a pair, and just post your comments like everyone else at this forum.
This post is not entirely relevant to this thread, or at least not directly. However, as it arose out of a discussion of a couple of Crothers' papers ...
There's a comment thread in PO, Universe's hidden supermassive black holes revealed, in which someone introduced 't Hooft's criticism of Crothers' work, and that led - after some back-and-forth - to this, by mytwocts:
mytwocts said:
@JT
Take a Euclidean space and use spherical coordinates about the origin. Now make the substitution R^3=(r+a)^3, a la Schwarzschild. The third power is actually not necessary for the argument. What was a single point in r,theta,phi at r=0 now is a sphere in R,theta,phi at R=a. Next ask a mathematicians to extend the space R,theta,phi with the domain R less than a.
Math can create space out of nothing but physics should restrict itself to what exists.
There's a comment thread in PO, Universe's hidden supermassive black holes revealed, in which someone introduced 't Hooft's criticism of Crothers' work, and that led - after some back-and-forth - to this, by mytwocts:
mytwocts said:
@JT
Take a Euclidean space and use spherical coordinates about the origin. Now make the substitution R^3=(r+a)^3, a la Schwarzschild. The third power is actually not necessary for the argument. What was a single point in r,theta,phi at r=0 now is a sphere in R,theta,phi at R=a. Next ask a mathematicians to extend the space R,theta,phi with the domain R less than a.
Math can create space out of nothing but physics should restrict itself to what exists.
This is yet another example of someone talking nonsense because they either don't know the mathematical definition of manifold or don't understand how quickly their discussion of manifolds will devolve into nonsense if they don't take the definitions seriously. It was once a common mistake among physicists as well as crackpots, but the physicists are now aware that not all coordinate singularities represent physical singularities.
A spacetime manifold is a special case of a pseudo-Riemannian manifold, which consists of three things:
a sufficiently well-behaved topological space
an atlas of sufficiently well-behaved charts
a sufficiently well-behaved pseudo-metric
"Sufficiently well-behaved" abbreviates a number of important details I'm omitting here.
Farsight, Crothers, and (if you have quoted him correctly) mytwocts have made the common rookie mistake of believing one particular chart describes the entire spacetime manifold. They have then compounded that mistake by not understanding the limitations of that particular chart.
A coordinate chart is a function that maps some open subset of a manifold to some open subset of the Euclidean space Rn (where n=4 in this context), subject to additional conditions that require overlapping charts to induce sufficiently smooth diffeomorphisms on open subsets of Rn.
The particular chart assumed by mytwocts does not map any point of the manifold to a tuple of coordinates with r=0. When he says "What was a single point in r,theta,phi at r=0", he is referring to something that doesn't even exist in his preferred chart. When he refers to that nonexistent thing as a point, he is talking about his own mental baggage, not mathematics. Mental baggage in, mental baggage out.
As I have explained in some detail, Leonard S Abrams made this mistake. Mr Crothers read Abrams, didn't spot the error, proposed to make this error the foundation of his PhD thesis, and didn't listen when his proposed thesis advisor, having done due diligence by asking some genuine experts about it, advised Crothers he was making a common mistake. That should have been the end of it, but Crothers promoted his error by publishing 17 papers in a crackpot-friendly journal. As you have discovered, several anti-relativity crackpots now cite Crothers and copy his fundamental mistake.
I read your post with great interest, W.D.Clinger, and am looking forward to mytwocts joining the discussion*.
Not really surprising, given the severe limitations on meaningful discussion in PO comments, but I made a mistake ... what mytwocts posted was not original to him/her! Rather it is - per her/his later comment (see below) - from "David Hilbert and the origin of the "Schwarzschild solution"", by Salvatore Antoci (see http://arxiv.org/abs/physics/0310104 - link is to the abstract). Here are two, more recent, comments:
mytwocts said:
It was Prof. Salvator Antoci of Pavia that pointed out the interesting problem of the R^3=(r+a)^3.
@JT
Starting here, by clicking on his name you will find all of his contributions: http://arxiv.org/physics/0310104** including his translations of Schwarzschild's work.
*in another comment, she/he said that the application to join ISF has (had) not yet been approved. Myself, I'd rather wait until that happens before commenting further ...
The key point is that Stephen Crothers has based his work upon a foundation laid by Leonard S. Abrams. That’s what this is all about! And here are the papers written by Abrams:
So Crothers has put himself in the precarious position of having 10 years work rest entirely upon the validity of Abrams papers. Big mistake! And that’s Stephen Crothers’ Achilles heel.
Unfortunately for Mr. Crothers, Abrams was wrong; which means Crothers now has to toss his 82 papers and videos into the trash can. Good job, Crothers. Well done!
To understand exactly why Abrams and Crothers are wrong, just read William Clinger’s superb mathematical analysis; which I facetiously call “Black Hole Denialism: The Legacy of Crothers’ Buffoonery”: Here
With so many names bandied about, it is very easy to lose sight of the actual crux of the argument. Here it is in Crothers own words:
The key point is that Stephen Crothers has based his work upon a foundation laid by Leonard S. Abrams. That’s what this is all about! And here are the papers written by Abrams:
So Crothers has put himself in the precarious position of having 10 years work rest entirely upon the validity of Abrams papers. Big mistake! And that’s Stephen Crothers’ Achilles heel.
Unfortunately for Mr. Crothers, Abrams was wrong; which means Crothers now has to toss his 82 papers and videos into the trash can. Good job, Crothers. Well done!
To understand exactly why Abrams and Crothers are wrong, just read William Clinger’s superb mathematical analysis; which I facetiously call “Black Hole Denialism: The Legacy of Crothers’ Buffoonery”: Here
Ten years of ones life is a long time -- especially when one has a public investment in contrary positions for such a long time. People who have a long term commitment to a crackpot theory simply cannot bear to face reality; they cannot look at equations honestly or think clearly when considering possible errors in their reasoning. Their commitment is frozen and becomes a personal impenetrable dogma. That is why we will never see a dialog here with someone who has the tools to demonstrate Crothers' errors.
Einstein and his followers assert that material sources are both present and absent by the very same mathematical constraint (Tμν = 0). That's impossible!
Tμν is a tensor field, not a constant. When we say Tμν = 0 'outside a body such as a star', we do mean Tμν is zero outside the star or other massive body. Mr Crothers apparently does not realize Tμν can be zero outside a star but nonzero inside the star.
Let's back up a few sentences and look at the structure of his argument:
Stephen Crothers said:
Einstein and his followers assert that Rμν = 0 describes his gravitational field 'outside a body such as a star'. However, this reasoning is circular, and therefore false, since all material sources are removed mathematically by setting Tμν = 0, then a material source is immediately reinstated with the words 'outside a body such as a star'. Since Tμν = 0, there are no material sources present to produce any gravitational field.
For all his symbol-pushing, Mr Crothers apparently does not realize Rμν and Tμν are tensor fields rather than constants. They can be zero outside a star, but nonzero inside a star.
The conclusion Mr Crothers wants to draw is that black holes are inconsistent with general relativity. What he's actually saying, however, is that ordinary stars are inconsistent with general relativity.
Mr Crothers appears to have forgotten, or perhaps never learned, that the Schwarzschild/Droste/Hilbert solution, when restricted to the exterior of an isolated but perfectly ordinary star (hence outside the Schwarzschild radius), describes the geometry of spacetime outside the star. The Schwarzschild/Droste/Hilbert solution also describes the geometry of spacetime outside an isolated planet or other massive object.
That geometry implies observable bending of light as it passes near a star, which was one of the first consequences of general relativity to be tested by experiment. It also implies gravitational lensing, the Shapiro effect, and (part of) the relativistic correction used by the Global Positioning System. When Mr Crothers pretends Tμν = 0 outside a star is inconsistent with the very existence of the star whose gravitational field we're calculating, he is attacking a well-tested theory.
I'd like to know who did the peer review of Crothers' article? It's probably the same guy who does the peer review for Progress in Physics.
Note: Since Crothers doesn't have any credentials to speak of, he listed the "Alpha Institute for Advanced Studies" as his affiliation. That should make Myron Evans very happy!
I'd like to know who did the peer review of Crothers' article? It's probably the same guy who does the peer review for Progress in Physics.
Note: Since Crothers doesn't have any credentials to speak of, he listed the "Alpha Institute for Advanced Studies" as his affiliation. That should make Myron Evans very happy!
A little checking has the AJSS listed as one of the journals "which normally publish the received manuscripts with no peer reviewing, just in front of money"
A little checking has the AJSS listed as one of the journals "which normally publish the received manuscripts with no peer reviewing, just in front of money"
American Journal of Space Science is printed half-yearly and has a total of 3 issues! It looks like Crothers 's nonsense has been accepted for the online edition - probably without peer-review.
The papers include some truly abysmal ones, e.g. THREE FUNDAMENTAL MASSES DERIVED BY DIMENSIONAL ANALYSIS has an author has practicing numerology - mixing up the speed of light (c), the gravitational constant (G), the Plank constant (?) and the Hubble constant (H)" to get 3 masses that he guesses are significant. The author does not know that the mass of a graviton would be zero - no peer review there !
Click in the Most Cited link on their main page to get a blank list!
black holes: history versus mathematics, part 1 of 3
In post #78, Slings and Arrows quoted Dmitri Rabounski's criticism of "historical" approaches to general relativity:
Dmitri Rabounski (April 2008) said:
...a “historical” approach, which is very common in most brief courses on the theory of relativity for physicists, often carries a student away with speculations on the principles and postulates, instead of studying Riemannian geometry itself.
Following a brief introduction to Riemannian and pseudo-Riemannian geometry, we'll look at a bit of history ourselves before examining some claims made by authors who, taking a historical approach to relativity, got carried away.
This part 1 is rather technical, because it's mostly about mathematics. Part 2 is also somewhat technical, because it uses the mathematics of part 1 to explain how several different coordinate systems use different equations to provide different views of exactly the same spacetime manifold. Part 3, which is the least technical of the three parts, uses mistakes made by Crothers and others to illustrate points made in parts 1 and 2.
Some readers may prefer to read part 3 first, referring to parts 1 and 2 only as needed.
Antoni A Kosinski's Differential Manifolds begins with the following textbook definition. For the convenience of readers who may not be familiar with all of the prerequisite mathematics, I have added links to a few Wikipedia articles.
Antoni A Kosinski said:
1 Smooth Manifolds and Maps
A topological space is a manifold if it admits an open covering {Uα} where each set Uα is homeomorphic, via some homeomorphism hα, to an open subset of the Euclidean spaceRn. A near-sighted topologist transferred from Rn to a location in such a manifold would not notice a difference, at least not until he or she decided to do Calculus: A function which is differentiable when expressed in local coordinates relative to one Uα (i.e., its composition with hα) need not be differentiable relative to another set of local coordinates. For that to be true the covering must satisfy an additional condition.
(1.1) Definition Let M be a topological space. A chart in M consists of an open subset U ⊂ M and a homeomorphism h of U onto an open subset of Rm. A Cratlas on M is a collection {Uα, hα} of charts such that the Uα cover M and hβhα-1, the transition maps, are Cr maps on hα(Uα ∩ Uβ).
Some authors require the domain of each chart to be connected, which will become relevant when we discuss Hilbert's paper on the Schwarzschild chart. Other minor variations of the definition will be encountered as well.
[size=+1]Example: the 2-sphere.[/size]
The surface of the earth is approximated by a 2-sphere. The 2-sphere is not homeomorphic to any subset of the Euclidean plane, so the 2-sphere cannot be covered by a single chart. The 2-sphere is, however, covered by an atlas consisting of four charts that assign coordinates via:
an orthographic projection centered on the north pole and extending down to (but not including) the equator
an orthographic projection centered on the south pole and extending up to (but not including) the equator
an equirectangular projection showing the equatorial region between (but not including) the Tropic of Cancer and the Tropic of Capricorn and extending from (but not including) the prime meridian westward to (but not including) 30 degrees east longitude
an equirectangular projection showing the equatorial region between (but not including) the Tropic of Cancer and the Tropic of Capricorn and extending from (but not including) 5 degrees west longitude eastward to (but not including) 35 degress east longitude
The "(but not including)" phrases are needed because the domain of each chart must be an open subset of the 2-sphere.
If these four charts are collected within an atlas printed on flat pages, then every location on the 2-sphere will be represented within at least one chart. The coordinates of a location p within a chart can be defined as the pair of numbers <x,y> where x is the distance in millimeters from p to the left side of the page and y is the distance in millimeters from p to the bottom of the page.
Mexico City will appear on the first and third charts. Those two charts will assign completely different coordinates to Mexico City. That's okay. Coordinates are entirely chart-dependent and quite arbitrary, so long as they satisfy the mathematical restrictions stated within the definition of a differentiable manifold. What's important here is that the four charts are compatible with each other in the sense of having sufficiently differentiable transition maps.
If you take away any of the four charts, you will no longer have an atlas because some locations won't be covered by any of the three remaining charts. It is possible, however, to cover a 2-sphere with only two charts. For example, the first two charts listed cover all but the equator, so a slight extension of those two charts would obviate the need for the last two; the extended charts would no longer be orthographic projections, of course.
The maximal atlas compatible with the four listed charts contains infinitely many pairs of charts that cover the entire 2-sphere.
That maximal atlas also contains infinitely many charts that are so weirdly unintuitive that no one would ever want to use them when planning a trip or estimating the distance from one point to another. That doesn't matter: Even the weirdly unintuitive charts are compatible with the four charts listed above. If they weren't, they wouldn't be part of the maximal atlas that contains those four charts.
[size=+1]Example: a coordinate singularity.[/size]
R2 is the familiar 2-dimensional Euclidean space. We can define a chart that covers all of R2 and maps the point <x,y> to coordinates <x,y>. When expressed using the familiar coordinates defined by that chart, the Euclidean metric is ds2 = dx2 + dy2.
When the atlas containing that one chart is extended to the maximal atlas containing all charts compatible with that one, we get infinitely many charts. One of those charts, which is defined only on points <x,y> for which x > 0, maps <x,y> to <u,v> where u=x2 and v=y.
du/dx = 2x so dx = du/(2x). Similarly dv = dy. Rewriting the metric in terms of u and v, we get
Division by zero is undefined, so that particular formulation of the Euclidean metric is undefined at u = 0. Informally, we say this <u,v> chart has a coordinate singularity at u = 0.
Does that mean there's something wrong with the manifold at u = 0? Not at all. It means something goes wrong with the coordinates defined by this chart as the u-coordinate converges toward zero.
Technically, that means the chart can't be extended to cover any points of R2 whose u-coordinate would be zero. If we try to define the chart on all points of R2 whose familiar x-coordinate is non-negative, our definition won't be allowed because that isn't an open subset of R2. (Recall that charts have to be defined on an open subset of the manifold.) If we try to extend the chart to include an open neighborhood of any point whose familiar x-coordinate is zero, then that neighborhood will contain pairs of points <x,y> and <-x,y> that map to the same <u,v> coordinates, which means the mapping to <u,v> coordinates won't be one-to-one, which means it isn't a homeomorphism. (Recall that charts are required to be homeomorphisms.)
It's easy to see what's going on with this example because I started out by describing the underlying manifold, which is very familiar. If I had started by saying I'm going to define a manifold whose metric is given by
ds2 = du2/(4u) + dv2
where u > 0, you might have thought I was defining a manifold that can't be extended to include points beyond the coordinate singularity at u = 0.
That would be a mistake. That mistake was made during the early years of general relativity, and a few people are still making that mistake.
[size=+1]Tensor fields.[/size]
If you already understand tensors, I don't need to explain them here. If you don't already understand tensors, you aren't likely to learn enough about them by reading anything I could write here.
Tensor fields (often referred to simply as tensors) can be introduced in a rough and ready way by describing their local expressions in charts and then going on to explain how such expressions are transformed under a change of coordinates. With this approach one can gain proficiency with tensor calculations in short order, and this is usually the way physicists and engineers are introduced to tensors. However, since this approach hides much of the underlying algebraic and geometric structure, we will not pursue it here. Instead we present tensors in terms of multilinear maps.
That's a polite way of saying the rough and ready approach gets you to the symbol-pushing level of proficiency faster, but you're less likely to make fundamental mistakes if you pay attention to the underlying mathematics.
Many of the mistakes we've been discussing in this thread are being made by people whose rough and ready approach to differential geometry has given them enough symbol-pushing proficiency to get into trouble.
[size=+1]The map is not the territory.[/size]
Curved surfaces (such as the surface of the earth) cannot be projected onto a flat Euclidean plane (such as a printed page) without distortion and/or tearing. The mathematical definition of a manifold forbids charts that involve tearing, and the definition of a differentiable manifold allows us to compensate for distortion using the tools of calculus, but anyone who expects all measurements made on a flat chart to coincide with measurements made on the charted territory is expecting the impossible.
We can describe the location of Mexico City using its latitude and longitude. Alternatively, we can describe its location using the number of millimeters from the edges of a chart in a printed atlas. Both coordinate systems are arbitrary. One coordinate system might be more convenient for some purposes (such as calculating the number of hours between sunrise and sunset at a solstice) while the other might be more convenient for other calculations (estimating the distance to Veracruz).
Arguing over which of those arbitrary coordinate systems is right and which is wrong would be silly. The location of Mexico City on earth is geometric and geographic fact, while its coordinates are arbitrary convention.
Einstein's theory of general relativity says you're allowed to calculate using any chart you think is convenient. In some cases, the answers you get may depend on the chart you use, which means those answers are relative to that particular chart. That's why we call it the theory of relativity.
Some things, however, are independent of the chart(s) we use.
When a number is independent of the chart(s) used to calculate it, we call it a scalar invariant.
Vectors, tensors, and tensor fields are geometric objects, and exist independently of coordinate charts. In particular, a vector is not just a sequence of numbers. Once we have made some arbitrary choice of vector basis, we can represent a vector by the sequence of numerical coefficients needed to express the vector as a linear combination of the basis vectors, but those numbers depend on the basis we have chosen.
More generally, we can represent vectors and tensors using numerical components derived from coordinates defined by some particular chart, and that's convenient when we need to calculate the distance from Mexico City to Veracruz, the precession of Mercury, or the bending of light within a gravitational field, but those numerical components depend on the chart we're using. The vectors, tensors, and tensor fields themselves do not depend upon the chart(s) used to represent their components.
That can be a hard fact to wrap your head around, especially if your knowledge of Riemannian geometry is of the rough and ready, symbol-pushing variety.
[size=+1]The tangent bundle of a manifold.[/size]
In calculus, the derivative of a differentiable function f can be defined as the unique function f' that makes g(x+∆x) = f(x) + ∆x f'(x) into the best linear approximation to f in the neighborhood of x. For functions of more than one argument, that definition generalizes to define the gradient of f, which is a vector field.
For Euclidean space, we can pretend all gradient vectors inhabit the same vector space V. When we generalize these notions to a manifold M, however, we need to know the point on the manifold at which each vector is rooted, and we also need to think of each vector as inhabiting a copy of V attached to that point. The tangent bundle of M consists of all those rooted copies of V.
A vector field on a manifold M is a sufficiently smooth map from M to the tangent bundle of M such that every point p maps to a vector rooted at p.
In Euclidean space, it's easy to translate a vector rooted at one point to any other point. On manifolds, however, the result of translating a vector depends on the path you take. Even then, the result isn't well-defined unless we define some connection between the copies of V that comprise the tangent bundle.
In general relativity, we define that connection by defining a pseudo-metric on the manifold.
[size=+1]Definition of pseudo-Riemannian manifolds.[/size]
A covariant tensor of rank 2 is a multilinear function that takes two vectors as input and produces a real number as output. That output is a scalar invariant.
Tensor fields are the obvious generalization of vector fields: A tensor field is a mathematical function that takes points of a manifold as input and produces a tensor as output. For each point p, the tensor at p operates on vectors (or dual vectors) that inhabit the vector space (or its dual) at p.
In the following definition, T2(M) is the space of all sufficiently smooth tensor fields that assign a covariant tensor of rank 2 to every point of a differentiable manifold M. ("Sufficiently smooth" means the following is true for every chart: the tensor's chart-dependent components are differentiable with respect to all of the chart's coordinates at every point of the chart, and those partial derivatives are themselves differentiable with respect to every coordinate at every point of the chart.)
The "sufficiently smooth" condition sounds like a restriction on tensor fields, but it can also be read as a restriction on charts.
I'm still trying to figure out whether that restriction on charts is implied by the definition of charts given above, or must be regarded as an additional restriction to be interpreted as a modification to the definition of charts given above. That's a purely mathematical question of no real interest to physicists.
Some folks may wonder why I even bother to read some of the papers I've discussed here. The main reason is they test my understanding.
I am not a physicist. Reading papers on relativity is just a hobby that lets me practice using mathematics I learned long ago.
With mainstream research papers, almost everything I read is either correct or goes above my head. I seldom spot an error, and most of the errors I spot are typos or easy to fix.
With crackpot papers, distinguishing correct from incorrect is a more interesting exercise. Most of the correct conclusions are within my ability to verify, while most of the errors I spot cannot be fixed. When I run into an anti-mainstream claim I can neither verify nor disprove, it usually means I don't understand something as well as I should.
One way to understand something better is to try to explain it to others. As I write my explanation, I encounter still more questions I can't answer. Most of those questions get left out of my exposition, because it's easy to explain where Crothers (for example) goes wrong without discussing anything subtle, but the questions that get left out of my exposition are often more important to me than any questions I answer.
Jeffrey M Lee said:
Definition 7.56. If g ∈ T2(M) is symmetric nondegenerate and has constant index on M, then we call g a semi-Riemannian metric and (M, g) a semi-Riemannian manifold or pseudo-Riemannian manifold.
Lee's semi-Riemannian metric g is often called a pseudo-metric, which is often abbreviated to "metric" even when g is not a true metric. (Warning: In other branches of mathematics, "pseudo-metric" and "semi-metric" can mean something different.)
A 4-dimensional Lorentz manifold is a pseudo-Riemannian manifold with signature (-,+,+,+). (Some authors, including Schwarzschild, use the opposite sign convention.)
[size=+1]Einstein's field equations.[/size]
A spacetime manifold is a 4-dimensional Lorentz manifold combined with a stress-energy tensor field that satisfies Einstein's field equations.
If a region of space contains no matter or energy for some interval of time, then the stress-energy tensor is zero at every point throughout the corresponding region of spacetime. Within that region of spacetime, and assuming the cosmological constant is zero, Einstein's field equations simplify to
Ric - ½Rg = 0
where g is the pseudo-Riemannian metric tensor field, Ric is the Ricci tensor field determined by g, and R is the Ricci scalar field determined by the Ricci tensor field.
The equation shown above has geometric meaning independent of any charts we might use when calculating. When we want to calculate using chart-dependent components of the tensor fields, we can use the component form of that equation:
Rμν - ½Rgμν = 0
[size=+1]Example: the Schwarzschild manifold.[/size]
In 1915, Einstein was still tinkering with his field equations, trying to get them into a form that wouldn't restrict the choice of coordinate system (chart).
One of those charts used a radial coordinate named r (lower case), while the other used a radial coordinate named R (upper case).
Schwarzschild's equation (14) actually consists of two equations. The second equation implicitly defines the coordinate transformation (aka transition map) that relates Schwarzschild's two charts:
The first of those two equations is a line element, which is a chart-dependent statement of the pseudo-metric g that appears within the definition of a pseudo-Riemannian manifold.
Compare Schwarzschild's original line element (above) to the version most often attributed to him today:
There are only four differences between those line elements:
Part of Schwarzschild's original line element has been abbreviated as dΩ2 = dθ2 + sin2θ dφ2, which is the line element for a 2-sphere expressed in Schwarzschild's spherical coordinates θ and φ. That's just an abbreviation, so it makes no difference.
The signs are flipped because my version uses a sign convention opposite to Schwarzschild's. That's just a convention, so it makes no difference.
A tiny bit of high school algebra proves (1 - α/R) = (R - α)/R.
The upper case R in Schwarzschild's original line element has been renamed to a lower case r. Schwarzschild used upper case R because he had already been using lower case r for a related coordinate in a different chart. So long as we aren't already using lower case r to mean something, we can rename R to r and it makes no difference.
Schwarzschild's implicit assumption that his lower case r coordinate is non-negative, and
two-fold differentiability of the pseudo-metric tensor's chart-dependent components with respect to all coordinates of the chart, as required by the "sufficiently smooth" condition on g.
So Schwarzschild's equation (14) is entirely equivalent to the Schwarzschild metric as presented in modern textbooks, provided we regard that pseudo-metric as defined only for Schwarzschild's chart (with R renamed to r) and r > α.
[size=+1]Example: another coordinate singularity.[/size]
Those singularities correspond to division by zero in the terms of Schwarzschild's line element. To put it more technically, the partial derivative ∂s2/∂t2 does not exist at r = 0, and ∂s2/∂r2 does not exist at r = α. That means points with r = 0 or r = α cannot be part of the Schwarzschild chart. Those points can be part of the manifold, with their pseudo-metric behavior described using coordinates defined by other charts that overlap with the Schwarzschild chart, but those points can't be part of the Schwarzschild chart.
Hilbert defined only the one chart, so Hilbert failed to define a spacetime manifold that includes any of the points excluded from the Schwarzschild chart by the singularities at r = 0 and r = α.
It is unclear to me whether Hilbert intended for points with 0 < r < α to be regarded as part of the chart he defined. If he did, then Hilbert defined an extension of Schwarzschild's original chart that contains a separate region (with 0 < r < α) that isn't connected to Schwarzschild's original chart. As noted earlier, some authors use a definition of charts that requires them to be connected, which rules out extensions of the Schwarzschild chart that include points with 0 < r < α.
By modern standards, Hilbert's presentation was sloppy because he failed to distinguish between the manifold and a chart that assigns coordinates to the manifold's points. He also appears to have assumed his one chart covers the entire manifold. Both of those mistakes are still often made today, especially by authors who set out to criticize Hilbert for what they perceive as his mistakes.
Since then, however, the problem of the central symmetrical statical gravitational field has been completely solved by Schwarzschild and others; the derivation given by H. Weyl in his book, "Raum-Zeit-Materie," is particularly elegant.
So far as I know, Einstein's equation (96) in that paper was Einstein's first statement of his field equations in the form we see most often today.
By the way, equation (70) of that PDF contains a typographical error: The superscript alpha on the left hand side should be a sigma. I don't know whether that typo is present in the original typesetting, but this is a good example of the kind of error you'll occasionally find within mainstream research papers.
What Hilbert believed about points with r = 0 doesn't matter, because it is not possible to extend the Schwarzschild metric of equation (14) to points with r = 0 < α. Several coordinate-independent invariants of the pseudo-metric blow up (increase without bound) as r approaches 0.
Hilbert clearly understood that. Hilbert also understood that not all coordinate singularities correspond to genuine singularities of the manifold:
David Hilbert said:
Für α ≠ 0 erweisen sich r = 0 und bei positivem α auch r = α als solche Stellen, an denen die Maßbestimmung oder ein Gravitationsfeld gμν an einer Stelle regulär, wenn es möglich ist, durch umkehrbar eindeutige Transformation ein solches Koordinatensystem einzuführen, daß für dieses die entsprechenden Funktionen gμν' an jener Stelle regulär d.h. in ihr und in ihrer Umgebung stetig und beliebig oft differenzierbar sind und eine von Null verschiedene Determinante g' haben.
In a translation given by Salvatore Antoci:
For α ≠ 0, r = 0 and, with positive values of α, also r = α happen to be such points that in them the interval is not regular. I call an interval or a gravitation field gμνregular in a point if, through an invertible one-to-one transformation, it is possible to introduce a coordinate system such that for it the corresponding functions gμν' are regular in that point, i.e. in it and in its neighborhood they are continuous and differentiable at will, and have a determinant g' different from zero.
Hilbert made a mistake in that paragraph. His interval (by which he means the line element) is indeed not regular at points with r = 0, but part 2 of this 3-part series will describe a coordinate transformation that proves the line element is regular (by Hilbert's definition) at the points with r = α.
As several textbooks and many papers have shown, the manifold covered by the Schwarzschild chart can be embedded within a larger manifold whose atlas includes the Schwarzschild chart and also includes compatible charts whose spatial coordinates are identical to those of the Schwarzschild chart except they allow arbitrary r > 0.
A map is not the territory; a chart is not the manifold.
The Schwarzschild coordinate singularity at r = α means the Schwarzschild chart cannot be extended to include points with r = α.
On the other hand, the submanifold covered by the Schwarzschild chart is embedded within a larger spacetime manifold that does include points with 0 < r ≤ α.
The existence of that larger manifold is a mathematical theorem, not an empirical fact.
Schwarzschild's equation (14) describes the gravitational field of an idealized point mass in otherwise empty space. That abstraction provides an excellent, computationally tractable model for gravity in the vicinity of a planet, star, or black hole with negligible spin, but no one is saying the universe contains only one massive object. Furthermore, the applicability of Schwarzschild's metric in the near vicinity of planets and stars (because their exterior gravitational field is the same as for an equivalent point mass) does not imply true point masses exist.
For that matter, the mathematical existence of spacetime manifolds that contain black holes does not by itself imply the physical existence of black holes. It does, however, imply black holes are compatible with Einstein's general theory of relativity.
Part 2 will use several different charts to describe a larger spacetime manifold that includes the Schwarzschild chart as a submanifold. In part 3, we will give examples of how a historical approach to general relativity, combined with rough and ready symbol pushing without fully understanding the underlying mathematics, has contributed to a series of mistakes by Crothers and others.
black holes: history versus mathematics, part 2 of 3
Part 1 of this 3-part series defined the mathematical concept of a spacetime manifold and discussed Schwarzschild's construction of a chart that describes the idealized geometry of spacetime surrounding isolated masses such as planets, stars, and black holes. Part 1 concluded with these claims:
The Schwarzschild coordinate singularity at r = α means the Schwarzschild chart cannot be extended to include points with r = α.
On the other hand, the submanifold covered by the Schwarzschild chart is embedded within a larger spacetime manifold that does include points with 0 < r ≤ α.
The first of those claims was established within part 1. The second claim is the subject of this part 2.
The easiest and most direct way to establish the second claim is to define a spacetime manifold whose atlas includes a chart that covers the spacetime manifold of a black hole all the way down to the central singularity, and then to prove the Schwarzschild chart covers only the submanifold that lies outside the black hole's event horizon.
Many textbooks and research papers have done that, but none of the relativity textbooks in my possession describe or cite the Painlevé-Gullstrand chart, whose relationship to the Schwarzschild chart is almost as straightforward as the relationship between the familiar <x,y> chart for 2-dimensional Euclidean space and the <u,v> chart I described in part 1's first example of a coordinate singularity. Three years ago, in this forum, I used the Painlevé-Gullstrand chart to refute Stephen J Crothers's claim that the manifold covered by the Schwarzschild chart cannot be extended. That refutation relied on links to other posts whose LaTeX is no longer being rendered by this forum's software, so I will again summarize the Painlevé-Gullstrand chart and its relationship to the Schwarzschild chart.
[size=+1]Painlevé-Gullstrand coordinates.[/size]
The Painlevé-Gullstrand chart is convenient because its spatial coordinates are exactly the same as the spatial coordinates of the Schwarzschild chart.
The Schwarzschild coordinate singularity is caused by Schwarzschild's selection of a time coordinate t that corresponds to the proper time experienced by a particle or observer subjected to a radial acceleration that counteracts the gravitational attraction of the central mass, keeping him/her/it at a constant value of the Schwarzschild radial coordinate r. (Equivalently, the Schwarzschild time coordinate corresponds to the proper time of an observer "at infinity", where the gravitational attraction of the central mass is diminished to zero.)
When an observer whose proper time is measured by the Schwarzschild time coordinate t observes an object falling into a black hole, the time dilation observed for that object increases without bound as the in-falling object approaches the event horizon. That means there is no value of t for which the in-falling object is seen to reach the event horizon.
That mathematical fact is explained by the physical meaning of the event horizon. Any photons that might reveal the in-falling object's passage through the event horizon would have to be emitted at or inside the event horizon. Photons emitted at or inside a black hole's event horizon cannot escape beyond the event horizon, so they won't ever reach the distant observer.
To understand what happens to objects falling through a black hole's event horizon, we have to use a different time coordinate.
The most natural time coordinates correspond to the in-falling object's own proper time. That still leaves us with a choice of time coordinates, because in-falling objects experience time at different rates depending on where they started to fall (equivalently, on how fast they're falling when they pass some fixed reference point). The most natural reference points are the event horizon and "infinity". It is impossible to hold a massive object at the event horizon before releasing it, so the Painlevé-Gullstrand chart uses a time coordinate τ that corresponds to the proper time of an in-falling object that was released "at infinity". This notion of time is mathematically well-defined without assuming points at infinity because we can use calculus to extrapolate backwards in time to solve the boundary value problem that says the in-falling object's velocity converges to zero as its distance from the black hole increases without bound.
To express the relationship between the Schwarzschild time coordinate t and the Painlevé-Gullstrand time coordinate τ, define the parameters α and β by
where m is the mass of the black hole. Then
τ = t - ∫r∞ (β / (1-β2)) dr
for all r > α, so the relationship between τ and t is well-defined for all points of the spacetime manifold covered by the Schwarzschild chart. (The integral would involve a division by zero when β = 1, but that would mean r = α, and the Schwarzschild chart does not cover any points with r = α.)
Differentiating with respect to r on both sides of the equation that relates τ to t, we get
dτ/dr = dt/dr - (β / (1-β2))
so
dτ = dt - (β / (1-β2)) dr
and
dt = dτ + (β / (1-β2)) dr
Since the Schwarzschild and Painlevé-Gullstrand charts define exactly the same spatial coordinates, we can use that equation to eliminate dt from Schwarzschild's equation for the pseudo-metric, thereby expressing the Schwarzschild pseudo-metric using Painlevé-Gullstrand coordinates.
As explained in part 1, Schwarzschild's equation (14) is equivalent to
Substituting for dt and simplifying using high school algebra, we get the Schwarzschild pseudo-metric in Painlevé-Gullstrand coordinates:
By construction, those two equations for ds2 are just two different formulations of exactly the same tensor field.
The first of those, from Schwarzschild's equation (14), is restricted to r > α, so it is limited to spacetime outside the event horizon. As explained in part 1, the Schwarzschild chart cannot be extended to include the event horizon.
The second equation is defined for all r > 0. If we artificially restrict that second equation to r > α, we get a chart that covers the same submanifold that's covered by Schwarzschild's original chart. If we allow all positive values of r, we get the Painlevé-Gullstrand chart, which covers a spacetime manifold that extends the Schwarzschild submanifold all the way down to the central singularity of a black hole.
By construction, the Painlevé-Gullstrand chart is compatible with the Schwarzschild chart. Its spatial coordinates are exactly the same as the spatial coordinates of Schwarzschild's equation (14), and we have stated a differentiable and reversible relationship between the t and τ coordinates that holds throughout the submanifold covered by the Schwarzschild chart.
Schwarzschild's equation (14) describes a pseudo-metric tensor field that satisfies Einstein's field equations for empty space, and the Painlevé-Gullstrand line element describes exactly the same tensor field throughout the submanifold on which the Schwarzschild chart is defined, so the Painlevé-Gullstrand line element also satisfies Einstein's equations for all r > α. To prove it satisfies Einstein's equations for all r > 0, we can use its chart-dependent components
to calculate the 40 independent Christoffel symbols, the 20 independent components of the Riemann curvature tensor, and the 10 independent components of the Ricci tensor.
That's straightforward but exceedingly tedious. There are easier ways.
That proves the Schwarzschild manifold can be extended, even though the Schwarzschild chart cannot be extended. The chart is not the manifold. The map is not the territory.
[size=+1]Lemaître coordinates.[/size]
The Lemaître chart defines the same τ, θ, and φ coordinates as the Painlevé-Gullstrand chart but replaces the Schwarzschild radial coordinate r by a new ρ coordinate that's related to the Schwarzschild t and r coordinates by an equation similar to the equation that relates τ to t and r. In Lemaître coordinates, the line element for the pseudo-metric tensor field becomes
ds2 = - dτ2 + βdρ2 + r2dΩ2
As before, it's easy to prove the tensor field defined by the Lemaître line element is exactly the same as the tensor field defined by the Painlevé-Gullstrand line element. The restriction of that tensor field to the submanifold covered by the Schwarzschild chart is therefore exactly the same as the tensor field defined on that submanifold by the Schwarzschild line element; that fact can also be proved directly as was done above for Painlevé-Gullstrand coordinates. These proofs can be verified by anyone who's proficient with multivariate calculus and high school algebra.
A maximal atlas for the black hole manifold therefore contains all three of the Schwarzschild, Painlevé-Gullstrand, and Lemaître charts. The Schwarzschild chart is restricted to spacetime outside the event horizon, while the Painlevé-Gullstrand and Lemaître charts go all the way down to (but not including) the central singularity.
The Painlevé-Gullstrand and Lemaître charts describe almost all of the black hole manifold, omitting only points on the poles and directly opposite the prime meridian. Those points are omitted only because the mathematical definition of charts forces the spherical spatial coordinates θ and φ to be restricted to an open set in which no point can be described by two or more distinct values of θ or φ. A simple spatial rotation of the Painlevé-Gullstrand or Lemaître chart yields a second chart that covers those omitted points.
The same is true for the Schwarzschild chart, so it doesn't quite cover all of the spacetime manifold outside of the event horizon.
That's a well-known problem with spherical coordinates, and has nothing to do with relativity. Even so, the need for a second chart just to deal with the spherical coordinate singularity emphasizes the fact that coordinate singularities are common and don't imply any weirdness in the manifold described by a chart.
Because the Painlevé-Gullstrand and Lemaître charts define exactly the same pseudo-metric tensor field, any proof that one of those charts satisfies the Einstein field equations immediately implies the other does also. In 1933, when Lemaître published his chart and discussed its relationship to Schwarzschild and Painlevé-Gullstrand coordinates, he proved that all of those charts satisfy the Einstein equations.
Relativity textbooks often use Eddington-Finkelstein coordinates to motivate the Kruskal-Szekeres coordinates discussed below. There are two versions of Eddington-Finkelstein coordinates, which retain the Schwarzschild spatial coordinates but replace the Schwarzschild time coordinate t by a new coordinate U or V defined by
U = t - r*
V = t + r*
where
r* = r + α ln |r/α - 1|
is the "tortoise" coordinate, which diverges toward negative infinity as r approaches the event horizon at α.
A radially outgoing photon follows a world line on which U is constant, while a radially ingoing photon follows a world line on which V is constant. U and V are therefore referred to as outgoing and ingoing coordinates.
When expressed in Eddington-Finkelstein coordinates, the manifold's pseudo-metric tensor field is described by
ds2 = - (1 - β2) dU2 - 2 dU dr + r2dΩ2
or
ds2 = - (1 - β2) dV2 + 2 dV dr + r2dΩ2
Although the Schwarzschild coordinate singularity at β = 1 (where r = α) is no longer present, the outgoing coordinates (with U) cannot cover any time-like world line for an ingoing photon or particle that crosses the event horizon at r = α. The ingoing coordinates have a related pathology for outgoing world lines.
That means the outgoing and ingoing Eddington-Finkelstein charts each cover only part of the manifold covered by the Painlevé-Gullstrand and Lemaître charts. All of those charts are fully compatible on open sets where they overlap, so all will be present in the maximal atlas for that manifold.
[size=+1]Kruskal-Szekeres coordinates.[/size]
All of the charts discussed above describe the spacetime manifold of an isolated, non-spinning, uncharged black hole. The Painlevé-Gullstrand and Lemaître charts go all the way down to (but not including) the central singularity. The Schwarzschild chart only goes down to (without including) the event horizon at r = α. The Eddington-Finkelstein charts are partial as well, in ways that are harder to explain.
As we have seen, the spacetime manifold covered by the Scharzschild chart can be extended to a larger manifold that includes the event horizon and the interior of the event horizon, excluding only the central singularity.
It is natural to ask whether that larger manifold is the largest manifold whose atlas contains the Schwarzschild chart. Somewhat surprisingly, it is not.
The largest analytic extension of the manifold covered by the Schwarzschild chart contains the black hole manifold covered by the Painlevé-Gullstrand and Lemaître charts together with a "white hole" manifold that mirrors the black hole manifold, but with time running in the opposite direction. For example, a particle can fall into the black hole but cannot fall out of it, so a particle can fall out of the white hole but cannot fall into it.
The black hole and white hole submanifolds are connected by an Einstein-Rosen bridge, which is the fancy name for wormhole.
The presence of white holes and wormholes in a solution of Einstein's field equations tells us white holes and wormholes are consistent with Einstein's general theory or relativity, but does not imply white holes or wormholes actually exist as physical reality. Spacetime manifolds are mathematical objects. Although these mathematical spacetime manifolds are intended to model physical reality, how well they model physical reality is an empirical question.
[size=+1]Physics is strange.[/size]
Black holes are already pretty strange, but white holes and wormholes would be stranger still. That strangeness has led some people to hope the presence of black holes within solutions of the Einstein equations was just a mistake made by physicists and mathematicians. A select few, thinking their historical approach to relativity and differential geometry has uncovered that mistake, now deny the compatibility of black holes with Einstein's theory of relativity.
Those deniers are mistaken. Regardless of whether black holes actually exist in our physical universe, they are compatible with relativity. That's a mathematical fact.
Part 3 will identify concrete mistakes within several arguments made by Stephen J Crothers and others who deny the compatibility of black holes with Einstein's general theory of relativity.
J B S Haldane said:
Now my own suspicion is that the Universe is not only queerer than we suppose, but queerer than we can suppose.
black holes: history versus mathematics, part 3 of 3
Part 1 of this 3-part series defined the mathematical concept of a spacetime manifold and discussed Schwarzschild's construction of a chart that describes the idealized geometry of spacetime surrounding isolated masses such as planets, stars, and black holes.
Schwarzschild's chart is restricted to a region of spacetime in which the so-called radial coordinate r is greater than α = 2m, where m is the mass of the central object. That is not an issue for planets or ordinary stars, because r = α corresponds to a location inside the planet or star, and the chart we're discussing is only intended to model spacetime outside of the planet or star.
Schwarzschild's chart has passed many experimental tests of its accuracy, so there is no serious doubt about the physical reality of the portion of spacetime it depicts.
For black holes, however, r = α defines the event horizon surrounding the black hole. If we want to learn what Einstein's general theory of relativity says about spacetime at or inside the event horizon, we have to examine charts that cover more of the spacetime manifold than is covered by the Schwarzschild chart.
A map is not the territory; a chart is not the manifold.
The Schwarzschild coordinate singularity at r = α means the Schwarzschild chart cannot be extended to include points with r = α.
On the other hand, the submanifold covered by the Schwarzschild chart is embedded within a larger spacetime manifold that does include points with 0 < r ≤ α.
Part 2 began by discussing the Painlevé-Gullstrand chart, which is compatible with the Schwarzschild chart and uses the same spatial coordinates, but uses a different time coordinate τ that allows the range of the Schwarzschild radial coordinate to cover all r > 0.
That time coordinate τ corresponds to the proper time observed by a particle or observer in radial free fall starting at rest "from infinity" (with the mathematical meaning given in part 2) or from a finite value of r starting at the mathematically well-defined velocity such an observer would have at that value of r.
The Schwarzschild time coordinate t is different from τ. The Schwarzschild time t corresponds to the proper time observed by a particle or observer subjected to a radial acceleration that keeps him/her/it at a constant value of r. That's appropriate when we're observing distant planets or stars from earth, or earth from orbit, which is one of several reasons why the Schwarzschild chart is so important. Lower values of r are closer to the massive body whose gravitational field is being modelled, so lower values of r imply stronger gravity, hence greater radial acceleration needed to keep r constant.
The Schwarzschild coordinate singularity at r = α arises because gravity is so strong at r = α that no finite acceleration can keep a massive particle or observer at constant r. Only massless particles such as photons can remain at constant r = α. For 0 < r < α, the gravitational field is so strong that photons emitted in an outward direction will fall inward. Photons emitted in other directions will fall inward even faster.
Part 2 went on to list several more charts that are compatible with both the Schwarzschild and Painlevé-Gullstrand charts but use different coordinates. As explained in part 1, a spacetime manifold's maximal atlas contains infinitely many charts. Most of those charts use really weird coordinates, so we generally focus on a few charts that define intuitive or convenient coordinate systems.
If we focus too much on a single chart or family of charts, we may forget that the chart is not the manifold.
That's what Stephen J Crothers has been doing for the past ten years. In the following quotation from his most recent paper, equation (18) is a generalization of the Schwarzschild chart to a parameterized family of charts.
Stephen J Crothers said:
It is apparent that (18) is not ‘extendible’ to produce a black hole universe. Since (18) generates all the possible equivalent solutions in Schwarzschild form, if any one of them is extendible then all of them must be extendible. In other words, if any one from (18) cannot be extended then none can be extended.
...snip...
"It is evident from (18) that this is impossible."
...snip...
Therefore, owing to equivalence, no solution generated by (18) can be extended. Consequently, the supposed extension of Droste’s solution to values 0 ≤ r by means of the Kruskal-Szekeres ‘coordinates’, the Eddington-Finkelstein ‘coordinates’ and also the Lemaître ‘coordinates’, are all fallacious.
...snip...
No equivalent solution generated by (18) is extendible to produce a black hole universe (Crothers, (2014b) for a detailed analysis)....A further consequence of this is that the ‘singularity theorems’ of Hawking and Penrose are invalid (Crothers, 2013a).
When people say "It is apparent that..." or "It is evident that...", they usually mean they haven't actually proved it. In this case, however, the parts I snipped contain alleged proofs.
The problem here is simple: Crothers's proofs don't justify his conclusions. What Crothers actually proves is that his parameterized family of Schwarzschild-like charts cannot be extended to obtain a larger chart using the same (parameterized family of) spatial coordinates while keeping the same time coordinate.
Crothers does not even consider the possibility of using a different time coordinate while keeping the same spatial coordinates, nor does he consider the possibility of using different coordinates for both time and space.
[size=+1]Historical note: the Painlevé-Gullstrand chart.[/size]
When Painlevé and Gullstrand published their charts in 1921 and 1922, they thought their charts were incompatible with Schwarzschild's. In a letter replying to Painlevé, Einstein tried to explain the arbitrariness of coordinate systems (charts), but Einstein himself didn't accept the Painlevé-Gullstrand chart until several years after Lemaître had proved its restriction to r > α was equivalent to Schwarzschild's chart.
Lemaître therefore deserves credit for proving the manifold covered by Schwarzschild's chart is just part of a larger spacetime manifold.
[size=+1]Historical note: the Schwarzschild chart.[/size]
Salvatore Antoci and Dierck-Ekkehard Liebscher said:
The content of this review would be hardly understandable without a proviso of historical character: Schwarzschild’s original solution, as undisputably testified by Schwarzschild’s “Massenpunkt” paper [1], describes a manifold that is different from the one defined by the solution that goes under the name of Schwarzschild in practically all the books and the research articles written by the relativists in nearly nine decades. That solution must be instead credited to Hilbert [2].
The manifold described by Schwarzschild's original solution is a proper submanifold of the manifold described by modern textbooks and research papers. On that submanifold, Schwarzschild's original equation (14) is equivalent to the modern version of his equation.
The quoted paragraph continues:
Salvatore Antoci and Dierck-Ekkehard Liebscher said:
The readers should not be induced by this assertion into believing that it is our intention to deprive Schwarzschild of the merit of his discovery, and to attribute it to the later work by Hilbert. It is not so: an accurate reading of Schwarzschild’s paper and of the momentous Communication by Hilbert [3] shows in fact that, while Schwarzschild’s derivation of the original solution is mathematically flawless, Hilbert’s rederivation contains an error. Due to this overlooked flaw, Hilbert’s manifold happened to include Schwarzschild’s manifold, but by chance it resulted not to be in one-to-one correspondence with it.
The larger manifold described by modern textbooks is not diffeomorphic to Schwarzschild's smaller submanifold, basically because it's larger.
As explained in part 1 and below, Hilbert's big mistake was to believe the Schwarzschild coordinate singularity cannot be removed by going to another chart. Citing Abrams, who made that same mistake while accusing Hilbert of a different mistake, Antoci and Leibscher repeat Hilbert's big mistake:
Salvatore Antoci and Dierck-Ekkehard Liebscher said:
This fact was deemed rather irrelevant by Hilbert, but soon developed into a conundrum that puzzled theoreticians like Marcel Brillouin [4], and had to become of crucial importance more than forty years later. In fact the birth of the black hole idea, as noted for the first time by Abrams [2], can be considered just a legacy of Hilbert’s magnanimity.
As we saw in part 1, Schwarzschild's equation (14) is entirely equivalent to this familiar modern formulation of the Schwarzschild pseudo-metric:
Hilbert derived the first of those two equations, but Hilbert also considered a wider range for the so-called radial coordinate: 0 ≤ r. By considering that wider range, Hilbert was able to discuss the singularities at r = 0 and r = α. That was a major contribution to the research literature.
Hilbert's biggest mistake was to say the Schwarzschild coordinate singularity at r = α could not be removed by transforming to a different coordinate system. As discussed above and in parts 1 and 2, the Schwarzschild coordinate singularity at r = α can be removed by a simple coordinate transformation; more precisely, you can use other charts (such as the Painlevé-Gullstrand chart) that use the same spatial coordinates as the Schwarzschild chart but are defined for all r > 0.
As I said in part 1, it is unclear to me whether Hilbert actually intended for his chart to be defined for all r ≥ 0. Did he consider the range 0 ≤ r ≤ α just so he could talk about the singularities, or did he think he had defined a chart that covered that entire range?
Abrams thinks Hilbert thought he had defined a chart that covered 0 < r but not r = 0. Crothers, who is less careful than Abrams, says Hilbert allowed any non-negative value of r. Antoci and Liebscher, citing Abrams but not Crothers, nevertheless agree with Crothers on this instead of Abrams.
It seems clear to me, however, that Hilbert realized r = 0 and r = α were both excluded from his chart because of the singularities.
It also seems clear to me that Hilbert made the mistake of failing to distinguish between his chart and the underlying manifold. As noted in part 1, Hilbert was writing at a time when the number of people in the world who understood the importance of that distinction could be counted on your fingers.
The historical question that remains is whether Hilbert intended for his chart to be defined on 0 < r < α as well as r > α. As was mentioned in part 1, some authors require charts to cover a connected region of the manifold, which would rule out 0 < r < α.
Even if 0 < r < α is not ruled out on grounds of connectedness, it would be ruled out by a requirement that the pseudo-metric tensor have constant index and signature throughout the chart.
Crothers, Antoci, and Liebscher use those arguments to say Hilbert made a mistake by allowing 0 < r < α.
Whether Hilbert actually made the mistake of allowing 0 < r < α in his chart is a purely historical question of no scientific importance.
[size=+1]Substantive note: the Schwarzschild chart.[/size]
Abrams, Crothers, Antoci, and Liebscher made a scientific mistake of considerable importance when they used Hilbert's alleged mistakes to conclude "black holes have no scientific basis" or "can be considered just a legacy of Hilbert's magnanimity."
The chart is not the manifold. As was shown in part 2, it's fairly easy to define a spacetime manifold whose atlas includes both the Schwarzschild and Painlevé-Gullstrand charts. That proves the manifold covered by the Schwarzschild chart can be extended to a larger spacetime with r > 0, even though the Schwarzschild chart itself cannot be so extended.
The maximal atlas for that larger manifold also includes the Lemaître and Eddington-Finkelstein charts, and the black hole half of the Kruskal-Szekeres chart.
Not all spacetime manifolds have a consistent arrow of time. The Gödel universe, for example, is not time-orientable.
Antoci and Liebscher argue that Hilbert's chart is inadmissible because (they say) it has no consistent arrow of time, but time-orientability is a global property of manifolds, not a property of charts. Their argument against an arrow of time in Hilbert's chart involves extrapolation across the coordinate singularity at r = α, which is inadmissible. No matter what Hilbert may have intended, the Schwarzschild pseudo-metric's chart-dependent components are not definable at r = α in a Schwarzschild or Hilbert chart. To see whether the "arrow of time" changes across the event horizon, you have to use a chart that's defined on an open set that includes the event horizon. Examination of the Painlevé-Gullstrand and other charts shows the arrow of time maintains its orientation all the way down to the genuine black hole singularity at r = 0.
I think Antoci and Liebscher got confused by the fact that, if you really insist upon extending the Schwarzschild chart by allowing 0 < r < α, you have to treat the r coordinate as a measure of time and the t coordinate as a measure of space throughout that region. Yes, that's really confusing. You can avoid that confusion by requiring charts to be connected or by restricting the Schwarzschild spacetime to r > α, as Schwarzschild himself did, and by using other charts (compatible with but not the same as Schwarzschild's) to understand black hole spacetime at and inside the event horizon.
That's what modern textbooks and research papers do.
Fantastic! Your explanations of charts, tensors and manifolds in elementary terms are beautiful and quite hrlpful even though I am already familiar with the concepts and the mathematical fundamentals
But boy, differential geometry is tough! Still, I am working my way through Spivak's tome.
When it comes to tensors, it might help to reference elementary tensors like the angle between two vectors. This particular example is useful in this context since local angle invariance (that may not be the technical term) distinguishes shape-preserving maps from shape-distorting ones. And differential geometry, especially with a basic understanding of determinants, shows why the former was often preferred in nautical contexts despite looking so grossly distorted; you can easily if not intuitively correct for area distortion (e.g. you can still sail in straight lines parallel to the equator) whereas if you distort shape, you (may?) no longer have a proper chart with easily translated metrics, you can't do calculus, etc. Angles are in such a sense more fundamental than area.
Wow! Beyond excellent; superlative! Hands down, the best post of the year.
Clinger didn’t just put the final nail in the coffin; he hoisted the coffin atop the Empire State Building, doused it in gasoline, set it afire and shoved it over the side.
It’s going to be very interesting to see how Crothers reacts to this -- we know he’s closely watching this thread.
My guess is that he reacts the same way he's reacted to every differential geometry lesson he's ever been exposed to in his entire non-career. He'll read three lines into it, decide that it's written by a gormless malicious hater, and all further processing of the contents will shut down. Then he'll repeat one of his standard hobbyhorses ("clinger has fallen for some gibberish excuse for ignore the real singularity at r=a") as though this were not addressed in the post.
You can safely bet the farm that Crothers is watching this thread; and a few other threads too.
First off, Crothers has listed Clinger as one of his “malicious” critics.
Second, there have been discussions of having a debate between Crothers and Clinger held at this forum – see the previous comments by Strawman, who is apparently an official spokesperson for Crothers.
And finally, the e-mail correspondence between Strawman and Crothers has been posted at the Myron Evans website:
Stephen Crothers (July 6 said:
It’s interesting and very telling that Clinger continues to make ad hominem references to me and other demonstrably erroneous remarks about me and my mathematics after he backed out of the public debate with me that you invited him to. It seems he prefers a one-sided ‘debate'; of course!
That’s how Crothers tells it. But in reality, he can start a debate with Clinger (or anyone else at this forum) by simply posting a comment. So Crothers is the one who has “backed out”, as he puts it.
My guess is that he reacts the same way he's reacted to every differential geometry lesson he's ever been exposed to in his entire non-career. He'll read three lines into it, decide that it's written by a gormless malicious hater, and all further processing of the contents will shut down. Then he'll repeat one of his standard hobbyhorses ("clinger has fallen for some gibberish excuse for ignore the real singularity at r=a") as though this were not addressed in the post.
Gerard 't Hooft wrote the following comment with Myron Evans specifically in mind. But the same thing applies to Stephen Crothers; probably even more so:
General Relativity is a great example of a doctrine that is simple enough for self-taught "scientists" to put their noses into, and complicated enough for them to make numerous mistakes.
I think Antoci and Liebscher got confused by the fact that, if you really insist upon extending the Schwarzschild chart by allowing 0 < r < α, you have to treat the r coordinate as a measure of time and the t coordinate as a measure of space throughout that region. Yes, that's really confusing.
Does anyone have a link to a good article that explains that in detail? I see it mentioned in the literature all the time, but the brief explanations given are usually just too mind-boggling to comprehend.
Hi Crothers, how about registering here and address Clinger's critique directly? As you see, people are making all kind of assumptions about how your replies would be, but it would be absolutely splendid if you could prove them all wrong!
black holes: history versus mathematics, erratum 1
In part 2, I said the Painlevé-Gullstrand chart uses the same spatial coordinates as the Schwarzschild chart but a different time coordinate τ that's related to the Schwarzschild time coordinate t by this equation:
That's wrong because the integral diverges for all finite r > α. Please replace the equation above by
τ = t - f(r)
where
f(r) = 2α ( 1/β + ln ((√(1-β2))/(1+β)))
As shown in the spoiler, ∫ (β / (1-β2)) dr = f(r) plus a constant of integration that could be chosen to make an in-falling object pass through the event horizon at any convenient value of the Painlevé-Gullstrand time coordinate τ. In the repaired equation for τ, I've set that constant to zero.
I got my original equation from a research paper, but I am now pretty sure that equation is incorrect.
Recall that α = 2m and β = (α/r)½. I'm defining a transition map (coordinate transformation) on the intersection of the Schwarzschild chart with a Painlevé-Gullstrand chart, so r > α and β < 1.
We can use the identities for logarithms and a Taylor series to rewrite f(r) as follows:
That proves f(r) is the anti-derivative (indefinite integral) of (β / (1-β2)) to within a constant of integration.
As a falling object's r coordinate approaches α, f(r) increases without bound, and does so at a rate that transforms the infinite positive range of the Schwarzschild time coordinate t onto a finite range of the Painlevé-Gullstrand time coordinate τ.
To prove that, consider a photon that's emitted at Schwarzschild time t = 0 and aimed directly toward the center of the black hole. In Schwarzschild coordinates, t increases without bound as r approaches α.
In Painlevé-Gullstrand coordinates, that photon's coordinate velocity at r is dr/dτ = -1 - β (as can be calculated directly from the line element, expressed in Painlevé-Gullstrand coordinates). That implies dτ = - dr/(1+β). ∫rα - dr/(1+β) = ∫αr dr/(1+β) and ∫0r dr/(1+β) are both finite, so the photon will pass through the event barrier in finite time and will also go off the charts ("reach the singularity") in finite time.
If a photon is emitted directly away from the black hole, its coordinate velocity at r is dr/dτ = 1 - β. If r > α, then dr/dτ > 0, which means the photon is able to make outward progress. If r = α, then dr/dτ = 0, which means the photon is stuck at the event horizon. If r < α, then dr/dτ < 0, which means the photon is being sucked toward the singularity even though it is trying to travel in the opposite direction.
Does anyone have a link to a good article that explains that in detail? I see it mentioned in the literature all the time, but the brief explanations given are usually just too mind-boggling to comprehend.
I read Robert Wald’s explanation (Section 6.4) repeatedly, and the concept of treating “the r coordinate as a measure of time and the t coordinate as a measure of space”, still remained hazy.
So I kept searching for another source and found this excellent and very thorough discussion of the topic:
In post #96, you mentioned Crothers's claim that the derivation of Schwarzschild's solution from Tab=0 means there is no possible source of gravity. From 1:46:15 (a student asking "Can we go back, way back, to where the metric comes from?") to 1:52:26 ("Does that answer your question?"), Susskind explains the correspondence between Tab=0 outside the source and the Newtonian equation ∇2φ=0.
Leonard Susskind's Lecture 12 can also be used to refute Stephen Crothers' latest article:
Stephen Crothers (July 29 said:
Cosmologists claim that they have found black holes all over the Cosmos. The black hole is however entirely a product of mathematics. The simplest case is the ‘Schwarzschild’ black hole, from the solution to Einstein’s field equations in the absence of matter, for a static, uncharged, non-rotating mass. In the absence of "matter” involves linguistic legerdemain, but in any event all types of black holes reduce, mathematically speaking, to a very simple question: Can a squared real number take values less than zero? Symbolically this is restated as follows. Let r be any real number. Is r2 < 0 possible? No, it’s not possible. Thus, the black hole is not possible. Anybody who can square a real number is capable of understanding why the black hole is a fantasy of mathematical physicists and cosmologists, illustrating once again why it can be very dangerous to put trust in the word of an Authority.
A Few Things You Need to Know to Tell if a Mathematical Physicist is Talking Nonsense: the Black Hole - a Case Study: Here
At time 34:55 to 38:40 Susskind discusses how r2 can be less than zero: Here
But this subject has already been brought up previously by Christian Corda; he has discussed letting r take on negative values:
Christian Corda (March 28 said:
The r coordinate becomes negative (this is possible because the origin of the coordinate system is the surface of the Schwarzschild sphere). The r coordinate reaches a minimum r = −rg for η = pi. Thus, we understand that at this point the collapse terminates and the star is totally collapsed in a singularity at r = −rg. In other terms, in the internal geometry all time-like radial geodesics of the collapsing star terminate after a lapse of finite proper time in the termination point r = −rg and it is impossible to extend the internal space-time manifold beyond that termination point. Thus, the point r = −rg represents a singularity based on the rigorous definition by Schmidt.
Clearly, as all the particles of the collapsing star fall in the singularity at r = −rg values of r > −rg do not represent the internal geometry after the end of the collapse, but they will represent the external geometry. This implies that the external solution (30), i.e. “the original Schwarzschild solution” to Einstein field equations which has been derived for the first time by Karl Schwarzschild in [2] can be analytically continued for values of −rg < r ≤ 0 and it results physically equivalent to the solution (1) that is universally known like the ”Schwarzschild solution”.
A clarification on the debate on “the original Schwarzschild solution”: Here
I am beginning to suspect that Crothers doesn't actually read what his critics are writing; he just assumes that anyone who disagrees with him must be "malicious" and "gormless".
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