Hamilton's equations of motion from the SU(1,1)/U(1) symmetry of Schwarzschild radial dynamics

S. H. Castles

REVIEW

Hamilton's equations of motion from the SU(1,1)/U(1) symmetry of Schwarzschild radial dynamics

S. H. Castles

General Relativity and Quantum Cosmology · Submitted 21 Feb 2006 · 6 references

Pith's verdict: A clean classical reformulation of radial Schwarzschild dynamics undercuts itself by calling non-canonical coordinates canonical and promising quantization that isn't there.

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arXiv IDgr-qc/0602080 (version 2)

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CitationS. H. Castles. "Hamilton's equations of motion from the SU(1,1)/U(1) symmetry of Schwarzschild radial dynamics." arXiv:gr-qc/0602080 (2006). https://pith.science/paper/LFM2KJM4

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Dear colleague,

The one thing worth keeping from this paper is the explicit change of variables Q=dr/dτ, P=√(1−2m/r). On a radial geodesic, these satisfy Q²+P²=E², so the motion lies on a circle, and the angle ε defined by dε = (P dQ − Q dP)/E² makes the equations dQ/dε=P, dP/dε=−Q. That is a clean classical reformulation, and the Poincaré-disk pictures give a real geometric intuition for the energy shells. The author also credits Witten for the SU(1,1)/U(1) symmetry and Gautreau–Hoffmann for the global time T, so the historical record is handled honestly.

The soft spot is the word 'canonical.' On the physical phase space with symplectic form Ω=dp_r∧dr, the variables (Q,P) are not Darboux. A direct computation gives {Q,P}=mP/r², not 1. So H=½(Q²+P²) is not the physical Hamiltonian in canonical coordinates; the oscillator equations hold because of the energy identity, not because the Hamilton–Jacobi structure was carried over. This is not a minor gap: the advertised quantization rests on regarding (Q,P) as canonical operators with [Q,P]=iħ. The paper never defines a transformation that would justify that step, and the stress-test's computation shows the plain reading fails.

The global-time claim is in better shape than the canonicity claim. ε=arctan(Q/P) is a well-defined phase angle for each particle, so something global may well be true. But the conformal-angle argument in Section III is only suggestive, and the paper does not pin down the constant across different energy shells. A referee should ask for a direct statement: for two particles with different E, under what initial-condition convention is ε the same?

The paper is a modest reformulation of known radial dynamics, not a new result. The geometric presentation is careful and the appendices are fine. I would not cite it in my own work, but I would send it to a referee: the classical core is correct as a reparameterization, and if the author revises the canonical/quantization claims to be a claim about an auxiliary phase space, the paper could be a useful pedagogical note.

Recommendation: peer review, with major revision on the symplectic and quantization points.

Referee report

Major objections (3)

[Section V, Eqs. (23)-(24)] The variables Q and P are not canonical on the physical phase space. With the reduced radial phase space (r, p_r), the physical momentum is p_r = (1 - 2m/r)^{-1} dr/dτ = Q/P², and the symplectic form is Ω = dp_r ∧ dr. Substituting r = 2m/(1 - P²) gives Ω = [4m/(P(1 - P²)²)] dQ ∧ dP, so the Poisson bracket is {Q, P} = P(1 - P²)²/(4m), not 1. Hence H = (1/2)(Q² + P²) with dQ/dε = P and dP/dε = -Q is not Hamilton's equation for the physical radial particle; the paper has silently replaced the physical symplectic structure by dQ ∧ dP. The quantization announced in the abstract, which requires [Q, P] = iℏ, therefore has no canonical basis. This is a load-bearing defect in the paper's central claim, not a matter of presentation.
[Section III, paragraphs following Eq. (16b)] The argument that ε is a global time shared by all particles on a radial ray is incomplete. Conformality of Möbius transformations and circle inversion preserves angles locally, but the paper does not specify an identification of initial-condition points on different energy shells (different E or ρ0) that would allow the same numerical value of ε to represent a common instant for all particles. If ε is simply the angular coordinate of the line through (0,1) in the upper half-plane representation, that explicit definition should be given and the conformal argument replaced by a direct coordinate check; without this, the globality claim is not established.
[Section V, Eqs. (23)-(24), and Introduction] The harmonic-oscillator form is obtained by construction rather than derived from the symmetry. Defining P = sqrt(1 - 2m/r), Q = dr/dτ, and ε through Eq. (3) automatically yields Q² + P² = E² and the oscillator equations; this part does not require the SU(1,1)/U(1) machinery. The genuinely group-theoretic content lies in the geometric correspondence of Sections II-III and in the claim that ε is a global time. The paper should separate these two ingredients and avoid the suggestion that the oscillator Hamilton equations are a consequence of the symmetry alone.

Minor objections (5)

[Equations (14a), (14b), (16a), (17a), (21), (22)] Many display equations are typeset in a garbled form that makes it impossible to verify the algebra; the manuscript needs a careful revision of the equations throughout.
[Section IV, Eq. (19)] There are two equations labeled (19); the second should be renumbered as (20).
[Introduction, first sentence] The text refers to 'SU(1/1)/U(1)'; this should be SU(1,1)/U(1).
[Section II, paragraph before Eq. (4)] The phrase 'with and 1yˆxˆ 2 2= +' appears to be missing a variable and should be corrected.
[Section III, Eq. (14a)] The definition of h = +1 (-1) should be stated more carefully for bound trajectories where the sign of dr/dτ can change at turning points.

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The paper

The paper claims

Radial Schwarzschild geodesics are expressed as a harmonic oscillator in a time variable motivated by SU(1,1)/U(1) symmetry, but this is a reparameterization of earlier results.

The reading

When an object falls straight toward a black hole, its motion is described by Einstein's equations. For motion along a single radial line, the equations can be written as two first-order equations: one for how fast the radius changes with the object's own time, and one for how the clock of a distant observer runs. This paper rewrites those equations. It chooses new variables: one is the radial speed, the other is a factor depending on distance from the black hole. It then defines a new time variable, called epsilon, using the geometry of a mathematical shape called the hyperbolic disk, associated with a symmetry group named SU(1,1)/U(1). In terms of these new variables, the two equations become exactly the pair that describes a simple harmonic oscillator, like a mass on a spring: the rate of change of one variable equals the other, and vice versa with a minus sign.

The paper claims that this time variable is global, meaning all objects falling along the same radial line share it, up to a fixed offset. That would be convenient for comparing clocks and for quantizing the motion. However, the proof of globality rests on a geometric argument about preserving angles, and the paper itself notes that the new time variable becomes imaginary inside the black hole's horizon. The final section gestures at quantization by citing known work on SU(1,1) quantum mechanics, but it does not actually perform a quantization.

The end result is a neat reformulation of a known piece of general relativity. The harmonic oscillator form is a consequence of choosing the right variables, not a new physical prediction.

Extended reading notes

Core claim

With the time variable epsilon, Hamilton's equations for particles on a radial ray have the form of the harmonic oscillator. Section V defines Q = dr/dtau and P = sqrt(1 - 2m/r) and states that dQ/depsilon = P and dP/depsilon = -Q, with Hamiltonian H = (1/2)(Q^2 + P^2). This is claimed to follow from the SU(1,1)/U(1) dynamical symmetry of the radial Schwarzschild geometry.

Load-bearing premise

The assertion that epsilon is a global time shared by all particles on a radial ray, up to a constant set by initial conditions, relies on the conformality of Mobius transformations across different energy shells (Section III, paragraphs following Eq. (16b)). If this synchronization is not well defined, the claimed globality fails, while the per-particle harmonic oscillator form could still hold.

Original abstract

Particles moving on a radial ray with respect to a Schwarzschild mass are shown to have SU(1,1)/U(1) dynamical symmetry. This symmetry is used to identify a global time variable shared by all test particles moving on a radial ray. With this time variable, Hamilton's equations for particles on a radial ray have the form of the harmonic oscillator. The SU(1,1)/U(1) symmetry, in conjunction with the well defined time variable and observer, assists in determining the quantization of the motion of the test particles.

The compiled record

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.06 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.05 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

Particles moving on a radial ray with respect to a Schwarzschild mass are shown to have SU(1,1)/U(1) dynamical symmetry. This symmetry is used to identify a global time variable shared by all test particles moving on a radial ray. With this time variable, Hamilton's equations for particles on a radial ray have the form of the harmonic oscillator. The SU(1,1)/U(1) symmetry, in conjunction with the well defined time variable and observer, assists in determining the quantization of the motion of the test particles.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Particles moving on a radial ray with respect to a Schwarzschild mass are shown to have SU(1,1)/U(1) dynamical symmetry.	result 0.90	0.85	0.41	asserted 1.00
s1	This symmetry is used to identify a global time variable shared by all test particles moving on a radial ray.	result 0.52	0.84	0.14	asserted 1.00
s2	With this time variable, Hamilton's equations for particles on a radial ray have the form of the harmonic oscillator.	result 0.99	0.84	0.40	asserted 1.00
s3	The SU(1,1)/U(1) symmetry, in conjunction with the well defined time variable and observer, assists in determining the quantization of the motion of the test particles.	interpretation 0.58	0.71	0.21	asserted 0.80

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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sources read

arXiv listing abstract v2

not examined

full text
figures and tables
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relations to other papers

levels run

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S sections
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** The solid arc centered on the yaxis is the flow line of the geodesic transformation, Eq. (2a), acting on the point (x,y) = (0,tanh(0.4)) or, equivalently, an extremum transformation, Eq. (2b), of the x-axis, ׀x ׀ > 1. The solid arc centered on the x-axis is the geodesic transformation with χ1 = – 0.4 of the y-axis, ׀y ׀ > 1. It makes an angle ε relative to the yaxis at y = ±1. x = tanh(χ1). Similarly, the extremum transformation, acting on the points on the x-axis, − < x < 11 , transform the x-axis into points on the arc of a circle centered on the y-axis, this arc being the flow…

**Figure 2.** The arcs from Fig. 1 are repeated

Figures reproduced from arXiv:gr-qc/0602080 by the authors.

References

Works cited (6)

[1]

It makes an angle ε relative to the y- axis at y = ±1. x = tanh( χ1). Similarly, the extremum transformation, acting on the points on the x-axis, 1 1<<− x , transform the x-axis into points on the arc of a circle centered on the y-axis, this arc being the flow lines of a geodesic transformation. The points in the unit disk can be defined by the product of...

View on Pith
[2]

Perform an inversion in ω on the points z in the unit disk and label the new points by , namely ) 1 , 0 (− z′ i) z 4/( i z z−+−=′a . (9) The arcs of circles centered on the y-axis are transf ormed into concentric arcs of circles centered at (0,1). Specifically, th e arc that intercepts the y-axis at tanh( χ2) is transformed under the inversion in ω into a...

View on Pith
[3]

The existence of two families of curves that are everywhere orthogonal, with one family of curves given by the geodesics. This geometry dictates (18)

View on Pith
[4]

The first order equation of motion for dr/d τ, given by (14a) in Schwarzschild coordinates

View on Pith
[5]

The first order equation of motion for dt/d τ, given by (14b) in Schwarzschild coordinates

View on Pith
[6]

The two-dimensional Schwarzschild metric, (11). For example, if elements 1, 2 and 4 are given, the first order equation of motion for dt/dτ can be derived. This interdependence is not surprising. The time coordi nate, t, appearing in the metric and in the first order equation of motion for dt/d τ, describes the time measured by a pa rticular observer rela...

View on Pith Pith review arXiv 1991

Discussion

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[1] [1]

It makes an angle ε relative to the y- axis at y = ±1. x = tanh( χ1). Similarly, the extremum transformation, acting on the points on the x-axis, 1 1<<− x , transform the x-axis into points on the arc of a circle centered on the y-axis, this arc being the flow lines of a geodesic transformation. The points in the unit disk can be defined by the product of...

View on Pith

[2] [2]

Perform an inversion in ω on the points z in the unit disk and label the new points by , namely ) 1 , 0 (− z′ i) z 4/( i z z−+−=′a . (9) The arcs of circles centered on the y-axis are transf ormed into concentric arcs of circles centered at (0,1). Specifically, th e arc that intercepts the y-axis at tanh( χ2) is transformed under the inversion in ω into a...

View on Pith

[3] [3]

The existence of two families of curves that are everywhere orthogonal, with one family of curves given by the geodesics. This geometry dictates (18)

View on Pith

[4] [4]

The first order equation of motion for dr/d τ, given by (14a) in Schwarzschild coordinates

View on Pith

[5] [5]

The first order equation of motion for dt/d τ, given by (14b) in Schwarzschild coordinates

View on Pith

[6] [6]

The two-dimensional Schwarzschild metric, (11). For example, if elements 1, 2 and 4 are given, the first order equation of motion for dt/dτ can be derived. This interdependence is not surprising. The time coordi nate, t, appearing in the metric and in the first order equation of motion for dt/d τ, describes the time measured by a pa rticular observer rela...

View on Pith Pith review arXiv 1991