Wednesday, October 21, 2009
Physicists: Stop using the word “metric” to mean so many different things. A metric tensor is NOT the same object as a metric, it is NOT the same object as its matrix representation, and it is NOT the same object as its associated line element. You should not use those words interchangeably, they are not equivalent structures.
A metric tensor is a tensor field.
If local coordinates are known:
The matrix representation of a metric tensor is a matrix.
The line element is a function of a metric.
In mathematics, the word metric refers to a fairly general function which defines ‘distance’ between elements in a set (it takes in elements of a set, and produces a real number). Riemannian and pseudo-Riemannian metrics (there are many more kinds of classification of metric too) have different conditions on those functions, but that’s more detail than is required here.
A metric tensor is a function defined on a manifold (a vector space) that takes in two tangent vectors and produces a scalar quantity. Metric tensors are used to define the angle between and length of tangent vectors (somewhat analogous to the dot product of vectors in Euclidean space)
Defining a metric versus a metric tensor:
Consider a smooth manifold of dimension n. For every point x in our manifold, there is a vector space called a tangent space (a tangent space contains all of the tangent vectors to our manifold at the specific point x).
Now, a metric at our point x is a function gx(Xx,Yx), which takes in the two tangent vectors Xx and Yx (at x), and outputs a real number. The metric function must also be bilinear, symmetric, and nondegenerate, but we don’t need to go into further details.
Now we can define a metric tensor, g, on our manifold: The metric tensor assigns a metric, gx, to every point x in the manifold (such that it varies smoothly with x in the manifold). The metric tensor is then:
g(X,Y)(x) = gx(Xx,Yx)
For those familiar with tensors, it should be clear that the metric tensor is actually a tensor field (a tensor is assigned to each point of our mathematical space). A metric tensor is not the same as a metric (it’s more analogous to an ‘infinitesimal’ metric function), but it is usually understood in differential geometry and related areas in physics that when one says “metric”, they really mean “metric tensor”. Mathematically, they are not equivalent objects, but integration of a metric tensor does induce a metric function.
Most of the time when actually doing physics, we don’t want such a general object. If local coordinates are known, the metric tensor can be expressed in a variety of more useful forms.
If we are in a region of the manifold where we have defined a local coordinate system, ie. xμ (where μ runs from 0 to 3), we can re-write our metric tensor [field] as:
g = gμνdxμ⊗dxν
where, gμν are real-valued functions, and dxμ are one-forms.
If we have local coordinates defined, we can then represent the metric tensor in matrix form, where, for our four-dimensional spacetime, we will have a 4x4 matrix with elements gμν.
In our local coordinates, if we take dxμ to be an infinitesimal coordinate displacement, we can write out a line element: ds2 = gμνdxμdxν. The line element, we know, is incredibly useful, as it provides us with an invariant quantity and also imparts information about causal structure.
EDIT: A note from The Unapologetic Mathematician that I should add: "the metric tensor is a bilinear function of two vectors at a given point, while the line element is a quadratic function of a single vector. However, the polarization identities will allow you to recover the bilinear function from the quadratic one."
Why does this matter? Well, for starters, general relativity is really all about your frame of reference and choice of coordinates. Some structures are unchanged regardless of your choice of coordinates (ie. the metric function & metric tensor), and some structures change with change in coordinates (ie. the matrix representation of a metric and the associated line element).
Just a couple of (well cited) offenders:
C. Brans and R. H. Dicke, Mach's Principle and a Relativistic Theory of Gravitation . Phys. Rev. 124, 925 (1961), Cited 1,139 times.
“As in general relativity the metric tensor is written as
gij= ηij + hij …”
EDIT: If I included more of the quote, it would have been obvious that local coordinates had already been chosen and they weren't writing out a general metric tensor, but a coordinate specific object. The reference is cited for context. Abstract index notation for tensors uses indices to indicate the type of tensor, rather than to indicate components in a particular basis
As I said above, gij is not the metric tensor, or a tensor at all, but a set of real-valued function specified for a local coordinate system (gij are also the matrix elements in the matrix representation - in those coordinates - of the metric tensor). The same goes for ηij and hij as well.
Tullio Regge and John A. Wheeler, Stability of a Schwarzschild Singularity . Phys. Rev. 108, 1063 (1957), Cited 476 times (two authors I respect immensely)
“Schwarzchild found long ago the solution of Einstein equations for the metric around a fixed spherically symmetrical center-of-mass:
ds2 = -(1-3m*/r)dT2 + (1 – 2m*/r)-1 dr2 + r2(dθ+sin2θdφ2) …”
This is the line element, not the metric.
Brandon Carter, Global Structure of the Kerr Family of Gravitational Fields . Phys. Rev. 174, 1559 (1968), Cited 383 times
“The covariant form of the metric tensor is expressed in terms of three parameters, m, e, and a by
ds2 = ρ2dθ2 – 2a sin2θdrdφ + 2drdu + …”
Again, this is a line element, not a metric tensor.
Marshall N. Rosenbluth, William M. MacDonald, and David L. Judd, Fokker-Planck Equation for an Inverse-Square Force. Phys. Rev. 107, 1 (1957), Cited 263 times.
“Let the expression for distance between two points whose coordinates differ by dq1, dq2, and dq3 be
Where aμν is a metric tensor…”
Again, aμν is not a metric tensor, but a coefficient, when working in local coordinates from this (local coordinate specific) representation of the metric tensor: aμνdxμ⊗dxν…
It isn’t that hard to say “line element”, or “matrix representation in local coordinates…”, or “matrix element in local coordinates…” instead of “metric tensor” or "metric" so why don't we?
Wednesday, October 14, 2009
Some informal background: a Riemannian manifold is a differentiable manifold (where the tangent space at each point has an inner product) with a positive-definite metric tensor, d(x,y) ≥ 0.
A familiar Riemannian manifold is a Euclidean manifold (where one has to add a smoothly varying inner product on the tangent space of the standard Euclidean space), with the familiar Euclidean (distance) metric (our 3-space, for example).
What is NOT a Riemannian manifold is the familiar Lorentzian manifold of general relativity (of which the Minkowskian manifold of special relativity is a special case). The Lorentzian manifold is a pseudo-Riemannian manifold, the generalization of the Riemannian manifold, such that the metric tensor need not be positive-definite. This apparently seems like a minor point to some, but pseudo-Riemannian and Riemannian manifolds are incredibly different because of this.
One of the underlying assumptions of general relativity is that spacetime can be represented by a Lorentzian manifold with signature (+,-,-,-) or (-,+,+,+) - where the signature of a metric tensor is just the number of positive and negative eigenvalues of the corresponding real symmetric matrix once it is diagonalised.
Unlike a Riemannian manifold, with a positive-definite metric, a Lorentzian manifold M, with non-positive-definite metric, g, allows tangent vectors, X, to be classified into timelike g(X,X) > 0, null g(X,X) = 0, or spacelike g(X,X) < 0.
The causal structure of relativity comes from this classification.
Interestingly, when you most often are reading a paper in a physics journal though, instead of seeing “pseudo-Riemannian” you will see the word “Riemannian”; doing a search in the Physical Review Letters this afternoon for “Riemannian Manifold” yields 526 results, while searching for “pseudo-Riemannian Manifold” only yields 51. While I am sure a few of those authors were actually are working with Riemannian manifolds (and the obvious overlap with the “pseudo-Riemannian” search), the vast majority are simply misusing the term.
Some sample offenders:
Stephen A. Fulling, “Nonuniqueness of Canonical Field Quantization in Riemannian Space-Time” (Phys. Rev. D 7, 2850 (1973), Cited 211 times) : Fulling technically means “pseudo-Riemannian space-time”, else he wouldn’t have any causal structure.
C. N. Yang, “Integral Formalism for Gauge Fields” (Phys. Rev. Lett. 33, 445 (1974), Cited 208 times). Yang starts a paragraph off with “Introduction of a Riemannian metric”, when he then must actually be introducting a pseudo-Riemannian metric. Later, when Yang is defining “Pure Spaces”, he says, “A Riemannian manifold for which the parallel-displacement gauge field is sourceless will be called a pure space.” He then asserts, “A four-dimensional Einstein space, ie. For which Rαβ = 0, is a pure space.” From the definition, if he really mean a Riemannian metric, he could not conclude that “a four-dimensional Einstein space” was a pure space, because an Einstein space must have a different signature to be causal (even though with Rαβ = 0 he is specifying that the metric tensor is locally isometric to a Euclidean space).
Almost anytime you see the phrase “Riemannian space-time”, they are being sloppy. There is no such thing as a Riemannian space-time.
All of these highly respected papers incorrectly refer to the spacetimes they are working in as Riemannian:
Friedrich W. Hehl, Paul von der Heyde, G. David Kerlick, and James M. Nester, “General relativity with spin and torsion: Foundations and prospects” (Rev. Mod. Phys. 48, 393 (1976), Cited 612 times)
David G. Boulware, "Quantum field theory in Schwarzschild and Rindler spaces" (Phys. Rev. D 11, 1404 (1975), Cited 117 times)
Kenneth Nordtvedt, “Equivalence Principle for Massive Bodies. II. Theory” (Phys. Rev. 169, 1017 (1968), Cited 88 times)
Leonard Parker and S. A. Fulling, “Quantized Matter Fields and the Avoidance of Singularities in General Relativity” (Phys. Rev. D 7, 2357 (1973), Cited 87 times)
M. J. Rebouças and J. Tiomno, “Homogeneity of Riemannian space-times of Gödel type” (Phys. Rev. D 28, 1251 (1983), Cited 65 times)
J. S. Dowker and Raymond Critchley, “Stress-tensor conformal anomaly for scalar, spinor, and vector fields” (Phys. Rev. D 16, 3390 (1977), Cited 59 times)
M. A. Melvin, "Dynamics of Cylindrical Electromagnetic Universes" (Phys. Rev. 139, B225 (1965), Cited 43 times)
Leonard Parker, "Conformal Energy-Momentum Tensor in Riemannian Space-Time" (Phys. Rev. D 7, 976 (1973), Cited 36 times)
A. A. Coley, N. Pelavas, and R. M. Zalaletdinov, "Cosmological Solutions in Macroscopic Gravity" (Phys. Rev. Lett. 95, 151102 (2005), Cited 32 times)
F. W. Hehl, E. A. Lord, and Y. Ne'eman, "Hypermomentum in hadron dynamics and in gravitation" (Phys. Rev. D 17, 428 (1978), Cited 20 times)
The list goes on, and on, and on…
Physicists (& Journal Editors): if you’re working in a causal spacetime (and you know you should be), don’t say “Riemannian”. Say, “Lorentzian”, or “pseudo-Riemannian”, or “non-Riemannian”, don’t be lazy.
You wouldn't say "positive" when you mean "positive, zero, or negative", so why would you say "Riemannian" when you mean "pseudo-Riemannian"?
My problem with this paper starts right with the second sentence of the introduction with this statement: “Within the weak-field approximation of general relativity, exact solutions have been derived for the gravitational field of a mass moving with arbitrary velocity and acceleration (Felber, 2005a).”
There are several points that should stick out in the mind of the reader. First, “weak-field approximation” and “exact solution” should not go in the same sentence. Perhaps, within the approximation it is exact, but it is not an exact solution (else it wouldn’t be an approximation). Second, “a mass moving with arbitrary velocity” is a pretty dangerous statement, because it suggests possibly ignoring the speed of light constraint.
Confusingly, when one follows the reference to the paper he is citing, “Weak ‘antigravity’ fields in general relativity”, we get another version of our initial point: “We recently derived and analyzed exact time-dependent field solutions of Einstein’s gravitational field equation for a spherical mass moving with arbitrarily high constant velocity”, where the ‘recent derivation’ in 2005 takes you to a paper from 2008 called, “Exact ‘antigravity-field’ solutions of Einstein’s equation”.
But back to the initial paper we are considering, and onto the third sentence: “The solutions indicated that a mass having a constant velocity greater than 3-½ times the speed of light c gravitationally repels other masses at rest within a narrow cone.”
Totally ignoring the derivation of this result for the time being (which is not present in his paper or any of the initial citations), we will continue to analyze the language used here. The phrase “masses at rest” should stand out as odd to a relativist. Rest in terms of what, I wonder? Our arbitrarily fast, accelerating, mass? In what frame could the author possibly mean? “At rest” is a warning sign in any paper that claims to be written about relativity, because even basic students of special relativity should have the notion of ‘no absolute, well-defined state of rest’ drilled into them.
Fourth sentence: “At high Lorentz factors (γ >> 1), the force of repulsion in the forward direction is about -8γ5 times the Newtonian force.”
Again, simply looking at the language here, the phrase “Newtonian force” should jump out at you. What force are we talking about? In the Newtonian view of physics, we do refer to objects moving under the force of gravity, but in general relativity, we really should not. Gravity is simply a manifestation of the geometry of spacetime. An object moving along the curved spacetime manifold isn’t ‘moving under a force’, but rather, it is in inertial motion along a curved manifold. There is no force pushing objects out of straight paths, objects are still following the straightest path; gravity corresponds to the changes in the spacetime geometry along that path. Relativists should be careful not to ascribe a particle’s action to a ‘gravitational force’. While this is a pet peeve of mine, and a bad habit, good and respectable physicists do use the term “gravitational force”, partly out of habit, and partly because, in the Newtonian limit, it’s not so offensive.
Another quote to consider from the fourth sentence is, “in the forward direction”. Now our *Galilean* relativity should be telling us to be more precise with a statement like, but one can give the author the benefit of the doubt to assume he meant “forward” as along the path of our mass.
The second paragraph continually mentions this “exact-solution” to the Einstein equations, which is of course, just as dubious a claim as it was the first time the author made it. For those who aren’t familiar with the Einstein equations, they are non-linear PDEs that are quite difficult to solve exactly, which is why very few exact solutions exist (and they are all a big deal), and why most modern exact solutions are found numerically these days.
In the second paragraph, we have: “These exact ‘antigravity-field’ solutions were calculated from an exact metric first derived, but not analyzed, by (Hartle, Thorne and Price, 1986).”
Now, I am somewhat familiar with the reference he cites: “Black holes: The membrane paradigm”, edited by Thorne, Price, and MacDonald, but I am not familiar enough with the particular paper he citing, “Gravitational Interaction of a Black Hole with Distant Bodies” (by Hartle, Thorne, and Price) to know which “exact metric” he is referring to. Nevertheless, I do know that that particular paper was treating the “The long-term, secular evolution of a black hole weakly perturbed by gravitational forces of objects far from the event horizon is examined using the 3+1 formalism of the membrane paradigm”, which makes it fairly hard to guess what he would be referencing there.
It’s a little surprising that if Felber was actually working within the Membrane paradigm, that the word “membrane” doesn’t appear anywhere in the text of his paper, or “black hole”, or “event horizon”, for that matter. While often consequences derived from the study of event horizons are applicable in many other settings, it’s hard to see the connection the author is making in this case.
In Hartle, Thorne, and Price, an “exact analytical solution is found for the lapse, shift and spatial metric of a moving, nonrotating black hole” which leads Felber to claim, “The exact results confirm that a large mass moving faster than 3-½c could serve as a driver to accelerate a much smaller payload from rest to a good fraction of the speed of light.” While I know I said I was just going to address the language here, I must point out that this claim/inference Felber is making seems quite without merit. He also doesn’t bother to assert how he has come to such a conclusion.
Onto the opening sentence of the third paragraph of the introduction: “The exact results are consistent with the repulsion of relativistic particles by a static Schwarzschild field, discovered
by (Hilbert, 1924).” Interestingly, his first attempt to back his claims up, outside of referencing himself or the strange appeal to Hartle, Thorne, and Price, comes in the form of a scan of section of Hilbert’s German version of his memoir, Die Grundlagen der Physik. Now, my German is pretty rusty, but Felber does cite a recent, English account of Hilbert’s curious result here: http://arxiv.org/PS_cache/arxiv/pdf/0904/0904.1578v1.pdf.
Instead of going over the whole debate on gravitation repulsion, I’ll direct any curious reader to the paper that I believe should be the current authority on the topic. It is “Gravitational repulsion in the Schwarzschild field”, by McGruder (Phys. Rev. D 25, 3191 - 3194 (1982)). Very nicely, he goes over the historical background of the initial results of gravitational repulsion and the great number of papers that followed them. A small point to mention, Hilbert’s results, found independently by Bauer, were for particles near the Schwarzschild radius. Later, McVittie and Jaffe and Shapiro showed that repulsion could occur anywhere in the Schwarzschild field, so long as the total particle velocity was greater than 2-½c, not 3-½c, like Felber is using.
Anyway, McGruder concludes an important result, which shouldn’t be a surprise these days to people who are familiar with similar solutions, that “gravitational repulsion can occur in the Schwarzschild field; but, it can only be detected by an observer whose meter sticks and clocks are not affected by gravity”. The important final line of his conclusion is, “that gravitational repulsion is not a function of the total particle velocity or energy; rather, its occurrence depends on the relationship between the transverse and radial velocity.” Unfortunately, it seems as if Felber is not familar with this work (ie. didn't do a google search of "repulsive gravity").
Now back to Felber: Nowhere near finished with the introduction, we have come to some fairly major issues. He is using a metric (although I see no evidence of him actually ‘using’ it anywhere), taken from the Membrane paradigm (not for a Schwarzschild field), using out of date results that only apply to near the Schwarzschild radius, and a very faulty interpretation of how these results can be interpreted/observed.
The ‘meat’ of the paper is his outline for an experiment to test his notion of gravitational repulsion at the LHC… so it can be assessed for the “potential of relativistic ‘antigravity’ for propulsion of payloads in the distant future.” Now, this claim seems so fanciful on it's own, that many readers wouldn't have bothered to give Felber a chance. Ruling something out, purely because it doesn't fit with conventional knowledge is bad science. However, sloppy mathematics, ignoring current research, poor foundations, and leavings things as undefined as possible is also bad science.
More often than not, the introduction to a scientific paper can tell you the weight of the results before you need bother get to the conclusion. Faulty reasoning has been leaking into theoretical physics for as long as theoretical physics has existed. Science fiction and pop-science lead many people into a false sense of familiarity with concepts that they do not truly understand. It is this, non-rigorous science though, that inspires and drives many new people into the field. Sometimes, it leads to new scientists and to new and important discoveries. More often than not though, it leads to crack-pot futurists filling the arXiv with nonsense, just because they can.
Spotting a crack-pot is not too hard, and it doesn’t require much time at all. Sure, words like “time-travel” and “anti-gravity” are often dead giveaways on their own, but, sometimes, for the sake of science, you have to give them the benefit of the doubt. The introduction to any paper can let you gauge the competence level of the author and help you decide, through the use of their language and inferences, whether they are doing science, or crack-pot science.
Posts will be separated into three categories:
1. Bad Papers - highlighting poor logic and bad physics in published work that, for whatever reason, is garnering attention.
2. Bad Language - often good physicists use sloppy terminology and incorrect definitions that end up taking away from the overall quality of the work (ie. pet peeves of mine).
3. Bad Physics - highlighting commonly bought into physical theories that are built on bad foundations.