Pith. sign in

REVIEW 1 major objections 5 minor 1 cited by

Gravitational energy in pp-wave spacetimes

T0 review · 1 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For pp-wave spacetimes with arbitrary polarization, the gravitational energy density is proportional to the determinant of the expansion tensor of a freely falling teleparallel frame, and the boxed four-momentum is proportional to its…

desk verdict Solid, modest TEGR extension; the unproved E(3)=0 assertion is a real gap, but the stress-test worry about it doesn't survive contact with the full Einstein equations. read the letter →

arxiv 2501.02290 v1 pith:HEYPE7RF submitted 2025-01-04 gr-qc math-phmath.MP

R. D. Costa , J. B. Formiga This is my paper · ORCID
classification gr-qcmath-phmath.MP MSC 83C4083C3583D05
keywords gravitationalenergypp-wavesteleparallelgravityTEGRenergy-momentumtensorexpansionwavesEinstein-Maxwellequations
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper works in the teleparallel version of general relativity, a reformulation in which the gravitational field is described by tetrads and the energy-momentum of gravity itself is a local tensor. The authors extend a previous analysis of plus-polarized pp-waves to arbitrary polarization. They show that, in a freely falling, non-rotating frame adapted to the pp-wave coordinates, the gravitational energy density is proportional to the determinant of the expansion tensor of the observers' worldlines, and the spacetime four-momentum inside a box is proportional to the trace of that same expansion tensor. The field equation in teleparallel form then displays the gravitational energy density as exactly the term needed to make the total derivative of the tetrad determinant appear. The paper's interest is that the local energy content of a gravitational wave becomes tied to the intrinsic geometry of the observer congruence, a step toward resolving the old problem of localizing gravitational energy.

What carries the argument

The key machinery is the orthonormal teleparallel frame (Eqs. (18)-(23)) adapted to the coordinates $(ct,x,y,z)$ of the pp-wave metric. Its tetrad determinant $e=f_1g_1-f_2g_2$ is assumed never to vanish, and the null vector $v^a=\hat t^a+\hat z^a$ defines the wave direction. The frame is freely falling ($\varphi_{ab}=0$) and vorticity-free, and its expansion tensor is $\theta_{(i)(j)}=-c\left(\begin{smallmatrix}\alpha&\beta\\\beta&\bar\alpha\end{smallmatrix}\right)$ with trace $\theta=d\ln e/du=-(c/e)\partial_z e$. The transport of the argument is the TEGR field equation (3), where the superpotential $\Sigma^{a\mu\nu}$ built from the Levi-Civita connection yields the quasi-local four-momentum as a surface integral (7). The central identity is that the gravitational stress tensor (45) equals $-\frac{4k}{ec^2}(\dot f_1\dot g_1-\dot f_2\dot g_2)v_av_b$, which the paper shows to be proportional to the determinant of $\theta_{(i)(j)}$, while the spacetime four-momentum over a box (41) is proportional to $\theta$ times the cross-sectional area $A=e\Delta x\Delta y$ at each face.

What would settle it

Construct a second allowed teleparallel frame for the same pp-wave spacetime by rotating the spatial triad $e_{(1)}, e_{(2)}$ by a fixed angle, keeping the time gauge and freeness, and recompute $t^b_a$ from Eq. (5). If the result is no longer proportional to the determinant of the expansion tensor, the central proportionality is frame-specific; if it survives every allowed rotation, the claim has a frame-independent core.

Watch

Extended reading notes

Core claim

The central claim is that for pp-wave spacetimes of the form $ds^2=-c^2dt^2+l_x^2 dx^2+l_y^2 dy^2+2l_{xy}dxdy+dz^2$ with arbitrary polarization functions satisfying constraint (A.8), and in the teleparallel frame defined by Eqs. (18)-(23), the gravitational energy-momentum tensor is $t^b_a = -\frac{4k}{e c^2}(\dot f_1\dot g_1-\dot f_2\dot g_2) v_a v^b$, where $e=f_1g_1-f_2g_2$ is the tetrad determinant and $v^a=\hat t^a+\hat z^a$ is a null vector along the wave. The same tensor is proportional to the determinant of the expansion tensor $\theta_{(i)(j)}=-c\left(\begin{smallmatrix}\alpha&\beta\\\beta&\bar\alpha\end{smallmatrix}\right)$ of the geodesic congruence, while the spacetime four-momentum (41) over a rectangular box is proportional to the trace $\theta=\dot e/e$ evaluated on the two faces. The authors also obtain the explicit Einstein-Maxwell equation (52), in which the gravitational energy density is exactly the term that converts the left-hand side into a total derivative $\ddot e$; for a circularly polarized electromagnetic wave with $f_2=g_2=0$, $f_1=g_1=f$, this reduces to the harmonic-oscillator equation $\ddot f=-\omega_0^2 f$ with $\omega_0^2=4\pi\epsilon_0 G E_0^2/c^2$. The vorticity and acceleration of the congruence vanish, so the frame is freely falling and Fermi-Walker transported; the weak-field limit of (45) reproduces standard linearized gravitational-wave energy results.

Load-bearing premise

The whole result is derived in one particular freely falling frame, and the paper acknowledges the spatial triad choice is ambiguous, so the neat proportionality to the expansion tensor could be an artifact of that choice.

Editorial extensions

If this is right

  • If Eqs. (41) and (45) are correct, the gravitational energy density and the boxed four-momentum of a pp-wave are completely determined by the expansion and determinant of the observers' congruence, so local gravitational energy can be read off the kinematics of a freely falling frame.
  • The earlier discreteness analysis for the cross-sectional area $A=e\Delta x\Delta y$ carries over unchanged, since Eq. (41) is written in terms of $A\theta$; hence the conclusion that $A$ must be discrete in the presence of an electromagnetic wave holds for arbitrary polarization.
  • The TEGR field equation (52) differs from the original Einstein equation (53) by exactly the gravitational energy-density term, so the teleparallel formalism gives a direct energy interpretation of the total derivative $\ddot e$ that the standard form lacks.
  • For circularly polarized electromagnetic waves and $f_2=g_2=0$, the system reduces to a harmonic oscillator with frequency $\omega_0^2=4\pi\epsilon_0 G E_0^2/c^2$, generalizing the fixed-polarization result.
  • In the weak-field limit, Eq. (45) agrees with known linearized gravitational-wave energy-momentum expressions, so the exact teleparallel result connects smoothly to the standard perturbative treatment.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • This is an editorial extension: the proportionality between energy and congruence expansion might hold for spacetimes beyond pp-waves, which would make gravitational energy a property of the observer congruence rather than a hidden substance; the authors themselves caution that the simple relations may not be general.
  • Because the frame's spatial triad is ambiguous, a natural check is to rotate $e_{(1)}, e_{(2)}$ and recompute $t^b_a$; if the determinant/trace relations survive only in the chosen gauge, they are frame-specific but still a coherent energy assignment.
  • The null character of the boxed four-momentum suggests that a region of pp-wave spacetime carries energy like a null pulse; a physical consistency test would compare the teleparallel energy flux with the standard high-frequency gravitational-wave energy-momentum in the short-wavelength limit.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 5 minor

Summary. The paper studies the TEGR description of gravitational energy for pp-wave spacetimes with arbitrary polarization. Starting from a Rosen-type line element with metric functions f1, f2, g1, g2 satisfying the constraint (A.8), the authors construct an orthonormal teleparallel frame, compute the torsion, Levi-Civita connection, superpotential, and gravitational energy-momentum tensor. They obtain that the spacetime four-momentum through a box is proportional to the difference of eθ on the two z-faces, Eq. (41); that the gravitational energy-momentum tensor has the null form (45), proportional to the determinant of the expansion tensor; and that the Einstein-Maxwell field equation can be written as (52), where the gravitational energy density is exactly the total-derivative term that converts Eq. (53) into a relation involving e. A circular-polarization example and a weak-field check are also given.

Significance. If the results hold, they provide explicit, checkable evidence that TEGR admits a localized gravitational energy for pp-wave spacetimes that are neither static nor asymptotically flat, with a simple geometric interpretation in terms of the expansion tensor. The strengths of the paper are the explicit algebra in Secs. III.A-C and the extension of Ref. [1] from plus polarization to arbitrary polarization; I have spot-checked Eqs. (35), (41), (45), (46), and (52) and found no algebraic errors. The weak-field limit (45) matching standard linearized results is a good consistency check. The significance is moderated by the acknowledged frame dependence of the TEGR energy-momentum tensor and by the missing derivation of the E(3)=0 assertion in Sec. III.D, discussed below.

major comments (1)
  1. [Sec. III.D] The statement that "the field equations force the component E(3) to vanish" is asserted without proof. This is load-bearing: it is the step that reduces the 4-potential to the transverse ansatz (47), and it leads directly to the two-term Maxwell tensor (51) and the explicit field equation (52). The assertion is not a consequence of Eq. (A.16), which only relates c(u) to the metric functions and does not constrain E(3). Nor does it follow from the source-free Maxwell equations alone: in the (u,v,x,y) form of the metric (11), a longitudinal component F_uv proportional to E(3) is allowed by Maxwell, with F_uv = C/e(u) for a constant C. A nonzero longitudinal component would add additional components to T^{μν} and would make Eq. (51) incomplete. To make Eq. (52) complete, the authors should derive E(3)=0 from the full Einstein-Maxwell system, for example by showing that the pp-wave Ricci tensor has R_vv=R_uv=0 while a longitudinal electric component contributes non-null stress-energy components, forcing the latter to vanish. Without this derivation, the central energy-density identification in Eq. (52) is incomplete.
minor comments (5)
  1. [Sec. III.D] The component E(3) should be defined explicitly (e.g., E(3) = -F_03) before the assertion that it vanishes; currently the reader has to infer this from the gauge condition dot A(0) = -dot A(3).
  2. [Eqs. (20)-(22) and (27)] The sign convention for hat t^mu appears inconsistent with Eq. (27): with hat t_mu = delta^0_mu and hat z^mu = delta^mu_3, one obtains v^mu = -delta^mu_0 + delta^mu_3, whereas partial^mu u = -(delta^mu_0 + delta^mu_3)/c; Eq. (27) requires v^mu = delta^mu_0 + delta^mu_3. Please check the time orientation and clarify the convention.
  3. [Eq. (24)] State explicitly that e = sqrt(-g) > 0 for a nondegenerate Lorentzian metric, so that the condition e != 0, which is used in denominators throughout, is the nondegeneracy of the metric rather than an additional physical assumption.
  4. [Abstract and Introduction] There are several grammar slips: "gravitation energy" should be "gravitational energy", and "long standing" should be hyphenated when used as a compound modifier.
  5. [Sec. IV] In the concluding remarks, "Eqs. (14) and (19)-(21)" is confusing; the frame components are given in Eqs. (18)-(23), so the reference should be adjusted accordingly.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the arbitrary-polarization energy calculation is a direct algebraic derivation from standard TEGR definitions, with self-citations supplying framework and consistency checks rather than the result itself.

full rationale

The derivation is self-contained in the relevant sense. The central results Eq. (41), Eq. (45), and Eq. (52) are obtained by direct substitution of the explicitly given orthonormal tetrad (18)-(23) into the standard TEGR definitions (3)-(5); no parameter is fitted and no target energy expression is assumed at the start. Eq. (45) reduces to the plus-polarization result of Ref. [1] only in the special case f2 = g2 = 0, and the arbitrary-polarization calculation is new. Eq. (41) follows algebraically from Eq. (35) and θ = d ln e/du, and Eq. (52) is the rearrangement τ = t + T of the same field equation, so the statement that the gravitational energy density is exactly the term making e a total derivative is an algebraic identity within the TEGR formalism, not a fitted input. The weak-field limit is checked against standard literature. Self-citations to Refs. [1,11,16,21,43] supply definitions, conventions, and consistency checks; they are not the load-bearing justification for the new results. Two limitations should be flagged, but they are not circularity: the assertion in Sec. III.D that the field equations force E(3) to vanish is stated without derivation and is a verification gap (if wrong, Eq. (51) and Eq. (52) would be incomplete), and footnote 4 acknowledges the ambiguity and frame dependence of the chosen teleparallel frame, which is a property of TEGR energy, not a self-referential reduction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on accepting the TEGR formalism and its identification of t^mu_a as gravitational energy-momentum, on the chosen pp-wave metric ansatz, and on the existence of a global adapted frame. No new physical entities are introduced and no free parameters are fitted; the only unproved step in the derivation is the asserted vanishing of E(3).

assumptions (4)
  • domain assumption The TEGR framework, including the superpotential definition (4), gravitational energy-momentum tensor (5), and quasilocal momentum (7).
    The paper assumes teleparallel equivalence as the framework for defining gravitational energy; these definitions come from Refs. [11,16,21] and are not derived here.
  • domain assumption The pp-wave metric ansatz (11) with the constraint (A.8) on f1, f2, g1, g2.
    The line element and the relation among the metric functions are taken from Ref. [41] as the class of spacetimes under study.
  • domain assumption The pp-wave spacetime is parallelizable and admits a global teleparallel frame.
    Footnote 2 acknowledges that parallelizability is a strong global topological restriction; the calculations require the global frame (13)-(14).
  • domain assumption The longitudinal electric component E(3) vanishes in the Einstein-Maxwell pp-wave system.
    Sec. III.D states that field equations force this, but no derivation is given. This underpins the choice A(0)=A(3)=0 and the simplified Maxwell tensor (51).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Gravitational energy in pp-wave spacetimes." pith.science (2026). https://pith.science/paper/HEYPE7RF

@misc{pith2026250102290,
  author       = {Pith},
  title        = {Pith review of: Gravitational energy in pp-wave spacetimes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HEYPE7RF}},
  note         = {Machine review of arXiv:2501.02290}
}
abstract

The description of the gravitation energy is a long standing problem. Although some success has been achieved, there is no satisfactory solution to this problem yet. Probably the most promising approach to this problem is given by teleparallelism. Many consistent and interesting results have been obtained in the context of the so-called Teleparallel Equivalent of General Relativity, including results obtained with spacetimes that are neither static nor asymptotically flat. One example is the analysis of the plus-polarized $pp$-waves that has been made recently [Phys. Rev. D 108, 044043 (2023)]. In this paper, this analysis is extended to arbitrary polarization and some new results that shows the consistency of the teleparallel approach is obtained.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Elliptically Polarized Plane Gravitational Waves

    gr-qc 2025-01 accept novelty 6.0 of 10

    Exact elliptically polarized plane gravitational wave solutions exhibit a cosmic jet effect: generic timelike and null geodesics asymptotically align with the propagation direction at speed c.

Reference graph

Works this paper leans on

44 extracted references · 35 canonical work pages · cited by 1 Pith paper

  1. [1]

    J. B. Formiga and J. A. C. Duarte, Gravitational energy pr oblem and the energy of photons, Phys. Rev. D 108, 044043 (2023). 7 The exact form may vary from author to author. 10

    work page 2023

  2. [2]

    Rovelli, Statistical mechanics of gravity and the thermodynamical origin of time, Class

    C. Rovelli, Statistical mechanics of gravity and the thermodynamical origin of time, Class. Quantum Grav. 10, 1549 (1993)

    work page 1993

  3. [3]

    C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W.H. Freeman and Company, New York, 1973)

    1973

  4. [4]

    Einstein, On the general theory of relativity, in Volume 6: The Berlin Years: Writings, 1914–1917 , edited by M

    A. Einstein, On the general theory of relativity, in Volume 6: The Berlin Years: Writings, 1914–1917 , edited by M. J. Klein, A. J. Kox, and R. Schulman (Princeton University Pres s, Princeton, 1996) pp. 98–110

    work page 1914

  5. [5]

    Papapetrou, Einstein’s theory of gravitation and flat space, Proc

    A. Papapetrou, Einstein’s theory of gravitation and flat space, Proc. R. Irish Acad. (Sect. A) 52, 11 (1948)

    work page 1948

  6. [6]

    P. G. Bergmann and R. Thomson, Spin and angular momentum i n general relativity, Phys. Rev. 89, 400 (1953)

    work page 1953

  7. [7]

    Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (John Wiley and Sons, New York, 1972)

    S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (John Wiley and Sons, New York, 1972)

    1972

  8. [8]

    L. D. Landau and E. M. Lifschitz, The Classical Theory of Fields , 4th ed., Vol. 2 (Pergamon Press, Oxford, 1975)

    1975

Show all 44 references
  1. [9]

    ¨Ozkurt and S

    S. . ¨Ozkurt and S. Ayg¨ un, Energy distributions of Bianchi type- VIhuniverse in general relativity and teleparallel gravit y, Pramana 88, 66 (2017)

    2017

  2. [10]

    Norton, What was Einstein’s principle of equivalenc e?, Stud

    J. Norton, What was Einstein’s principle of equivalenc e?, Stud. Hist. Phil. Sci. 16, 203 (1985)

    1985

  3. [11]

    J. B. Formiga and V. R. Gon¸ calves, Generalization of th e ADM gravitational energy momentum, Phys. Rev. D 106, 044021 (2022)

    2022

  4. [12]

    H. C. Ohanian, What is the principle of equivalence?, Am . J. Phys. 45, 903 (1977)

    1977

  5. [13]

    Di Casola, S

    E. Di Casola, S. Liberati, and S. Sonego, Nonequivalenc e of equivalence principles, Am. J. Phys. 83, 39 (2015), https://doi.org/10.1119/1.4895342

    2015 doi

  6. [14]

    Penrose, Quasi-local mass and angular momentum in ge neral relativity, Pro

    R. Penrose, Quasi-local mass and angular momentum in ge neral relativity, Pro. R. Soc. Lond. A 381, 53 (1982)

    1982

  7. [15]

    J. D. Brown and J. W. York, Quasilocal energy and conserv ed charges derived from the gravitational action, Phys. Rev. D 47, 1407 (1993)

    1993

  8. [16]

    J. W. Maluf, The teleparallel equivalent of general rel ativity, Ann. Phys. (Berlin) 525, 339 (2013)

    2013

  9. [17]

    J. B. Formiga, On the teleparallel frame problem, Mod. P hys. Lett. A 37, 2250222 (2022)

    2022

  10. [18]

    Oshita and Y.-P

    N. Oshita and Y.-P. Wu, Role of spacetime boundaries in avierbein formulation of gravity, Phys. Rev. D 96, 044042 (2017)

    2017

  11. [19]

    L. P. Eisenhart, Non-Riemannian geometry, Vol. 8 (Dover Publications, Mineola, 1927)

    1927

  12. [20]

    J. M. Nester, Positive energy via the teleparallel Hami ltonian, Int. J. Mod. Phys. A 04, 1755 (1989)

    1989

  13. [21]

    J. B. Formiga, The meaning of torsion in teleparallel th eories, Int. J. Geom. Methods Mod. Phys. 21, 2450211 (2024), https://doi.org/10.1142/S0219887824502116

    2024 doi

  14. [22]

    Y. N. Obukhov and G. F. Rubilar, Covariance properties a nd regularization of conserved currents in tetrad gravity, Phys. Rev. D 73, 124017 (2006)

    2006

  15. [23]

    J. L. Cervantes-Cota, S. Galindo-Uribarri, and G. F. Sm oot, A brief history of gravitational waves, Universe 2, 10.3390/universe2030022 (2016)

    2016 doi

  16. [24]

    Sauer, Field equations in teleparallel space–time: Einstein’s fernparallelismus approach toward unified field theory, Hist

    T. Sauer, Field equations in teleparallel space–time: Einstein’s fernparallelismus approach toward unified field theory, Hist. Math. 33, 399 (2006), Special Issue on Geometry and its Uses in Physic s, 1900-1930

    2006

  17. [25]

    Møller, Further remarks on the localization of the en ergy in the general theory of relativity, Ann

    C. Møller, Further remarks on the localization of the en ergy in the general theory of relativity, Ann. Phy. (N.Y.) 12, 118 (1961)

    1961

  18. [26]

    Y. M. Cho, Einstein Lagrangian as the translational Yan g-Mills Lagrangian, Phys. Rev. D 14, 2521 (1976)

    1976

  19. [27]

    Hayashi and T

    K. Hayashi and T. Shirafuji, New general relativity, Ph ys. Rev. D 19, 3524 (1979)

    1979

  20. [28]

    J. W. Maluf, Hamiltonian formulation of the teleparall el description of general relativity, J. Math. Phys. (N.Y.) 35, 335 (1994)

    1994

  21. [29]

    J. W. Maluf and J. F. Da Rocha-Neto, General relativity o n a null surface: Hamiltonian formulation in the teleparall el geometry, Gen. Relativ. Gravit. 31, 173 (1999)

    1999

  22. [30]

    J. W. Maluf and J. F. da Rocha-Neto, Hamiltonian formula tion of general relativity in the teleparallel geometry, Phys. Rev. D 64, 084014 (2001)

    2001

  23. [31]

    J. W. Maluf, J. F. da Rocha-Neto, T. M. L. Tor´ ıbio, and K. H. Castello-Branco, Energy and angular momentum of the gravitational field in the teleparallel geometry, Phys. Rev . D 65, 124001 (2002)

    2002

  24. [32]

    J. F. da Rocha Neto, J. W. Maluf, and S. C. Ulhoa, Hamilton ian formulation of unimodular gravity in the teleparallel geometry, Phys. Rev. D 82, 124035 (2010)

    2010

  25. [33]

    Oko/suppress l´ ow, ADM-like Hamiltonian formulation of grav ity in the teleparallel geometry, Gen

    A. Oko/suppress l´ ow, ADM-like Hamiltonian formulation of grav ity in the teleparallel geometry, Gen. Relativ. Gravit. 45, 2569 (2013)

    2013

  26. [34]

    Oko/suppress l´ ow, ADM-like Hamiltonian formulation of gravity in the teleparallel geometry: derivation of constraint algebra, Gen

    A. Oko/suppress l´ ow, ADM-like Hamiltonian formulation of gravity in the teleparallel geometry: derivation of constraint algebra, Gen. Relativ. Gravit. 46, 1636 (2014)

    2014

  27. [35]

    Ferraro and M

    R. Ferraro and M. J. Guzm´ an, Hamiltonian formulation of teleparallel gravity, Phys. Rev. D 94, 104045 (2016)

    2016

  28. [36]

    Blixt, M

    D. Blixt, M. Jos´ e Guzm´ an, M. Hohmann, and C. Pfeifer, R eview of the hamiltonian analysis in teleparallel gravity, Int. J. Geom. Methods Mod. Phys. 18, 2130005 (2021)

    2021

  29. [37]

    J. M. Nester and H.-J. Yo, Symmetric teleparallel gener al relativity, Chin. J. Phys. 37, 113 (1999)

    1999

  30. [38]

    J. B. Jim´ enez, L. Heisenberg, and T. Koivisto, Coincident general relativity, Phys. Rev. D 98, 044048 (2018)

    2018

  31. [39]

    F. W. Hehl, J. McCrea, E. W. Mielke, and Y. Ne’eman, Metri c-affine gauge theory of gravity: Field equations, Noether identities, world spinors, and breaking of dilation invari ance, Phys. Rep. 258, 1 (1995)

    1995

  32. [40]

    Blagojevi´ c, Gravitation and Gauge Symmetries , Series in High Energy Physics, Cosmology and Gravitation ( IoP, London, 2002)

    M. Blagojevi´ c, Gravitation and Gauge Symmetries , Series in High Energy Physics, Cosmology and Gravitation ( IoP, London, 2002)

    2002

  33. [41]

    J. B. Griffiths, Colliding Plane Waves in General Relativity , kindle ed. (Dover Publications, Mineola, New York, 2016)

    2016

  34. [42]

    Newman and R

    E. Newman and R. Penrose, An Approach to Gravitational R adiation by a Method of Spin Coefficients, J. Math. Phys. 3, 566 (1962), https://pubs.aip.org/aip/jmp/article-pdf /3/3/566/19147601/566 1 online.pdf. 11

    1962

  35. [43]

    J. B. Formiga, The energy–momentum tensor of gravitati onal waves, Wyman spacetime, and freely falling observers, Ann. Phys. (Berlin) 530, 1800320 (2018)

    2018

  36. [44]

    Maggiore, Gravitational Waves: Theory and Experiments , Vol

    M. Maggiore, Gravitational Waves: Theory and Experiments , Vol. 1 (Oxford University Press Inc, New York, 2008)

    2008

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.