Is there a Cosmological Basis for E = mc^2?

Geraint F. Lewis

Is there a Cosmological Basis for E = mc^2?

T0 review · 0 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The claimed cosmological derivation of E = mc^2 misidentifies the first component of the 4-momentum as energy; projecting onto an observer's 4-velocity gives the rest energy mc^2 at every point.

desk verdict A correct, narrow comment that kills a bad derivation of E=mc^2; the rest-frame projection argument is standard and the conclusion survives the weak spot in the presentation. read the letter →

arxiv 1908.09267 v1 pith:E7IXECOS submitted 2019-08-25 gr-qc astro-ph.CO

Geraint F. Lewis This is my paper · ORCID

classification gr-qcastro-ph.CO PACS 04.20.-q98.80.-k

keywords E=mc^2generalrelativityFLRWmetric4-momentumobserverprojectiongravitationalhorizonHubbleradiusrelativisticenergy

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

Recent work claimed that $E=mc^2$ is the gravitational binding energy needed for a particle to escape from the origin of an expanding universe to its Hubble “gravitational horizon.” This paper argues that the claim comes from misidentifying the first component of the 4-momentum as the particle's energy. In general relativity, energy is the projection of a particle's 4-momentum onto an observer's 4-velocity; doing this projection for a comoving particle in the FLRW metric rewritten in proper-radius coordinates gives $E=mc^2$ everywhere along the trajectory, with no binding-to-kinetic transition. The conclusion matters because it keeps $E=mc^2$ in its standard role as rest-frame energy, and warns against importing coordinate-dependent components into physical interpretations.

What carries the argument

The load-bearing object is the 4-momentum of a comoving particle and the rule that measured energy is the projection $E=g_{\mu\nu}p^\mu u^\nu$ onto the observer's 4-velocity. Starting from the FLRW metric in proper-radius coordinates, the paper computes the comoving 4-velocity from the normalization $u^\mu u_\mu=c^2$, finds $u^\mu=(c,\dot R)$, and therefore $p^\mu=m(c,\dot R)$. Inserting this into the projection identity gives $E=mc^2$. The other key identity is the comoving relation $R_h=cR/\dot R$, which turns the messy metric coefficient into a simple constant 4-velocity.

What would settle it

Compute the projection $E=g_{\mu\nu}p^\mu u^\nu$ for a comoving particle in the observer-dependent FLRW metric at $R=0$ and $R=R_h$, using $p^\mu=m(c,\dot R)$ and $u^\mu=(c,\dot R)$. Both evaluations give $mc^2$; a result that varied with $R$, such as $mc^2\sqrt{1-(R/R_h)^2}$, would show that the cosmological-binding interpretation captures something real.

Watch

Extended reading notes

Core claim

The paper's central claim is that the cosmological derivation of $E=mc^2$ fails on both counts it relies on: the rewritten metric does not introduce a new gravitational horizon in the required sense, and the quantity called $E$ in the 4-momentum $p^\mu=(E/c,p^R)$ is not the energy. The energy measured by an observer is $E=g_{\mu\nu}p^\mu u^\nu$, where $u^\mu$ is the observer's 4-velocity. For a comoving particle the 4-velocity in the rewritten coordinates is $u^\mu=(c,\dot R)$, so the 4-momentum is $p^\mu=m(c,\dot R)$, and projecting onto its own 4-velocity yields $E=mc^2$. This holds at the origin, at $R_h$, and beyond; the spatial component $p^R$ grows without limit past $R_h$, but that growth is not an energy transition. The origin of $E=mc^2$ is therefore the invariant normalization $p^\mu p_\mu=(mc)^2$, not a binding energy against a cosmic horizon.

Load-bearing premise

The argument assumes that Equation 1 and Equation 2 are coordinate representations of one and the same spacetime, so equivalent observers in the two coordinate systems must measure identical quantities; if the coordinate transformation were not equivalent, the conclusion that the energy is always $mc^2$ would not follow.

Editorial extensions

If this is right

The Hubble radius $R_h$ does not act as a gravitational horizon for comoving particles: crossing it changes the spatial momentum component $p^R$ but leaves the measured energy fixed at $mc^2$.
The two-term form of the energy expression in the earlier derivation does not describe a transition from binding energy to kinetic energy; it is an artifact of taking a coordinate component as if it were energy.
For every comoving observer (constant $r$ in the FLRW metric), and for any expansion history $a(t)$, the same projection argument gives the particle energy $mc^2$ in its own rest frame.
The first component $E/c$ in the rewritten metric is coordinate-dependent and has no direct physical meaning until projected; treating it as the energy misreads 4-vector behavior in curved spacetime.

Reading between the lines

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

A reader might push the argument one step further: any covariant version of the “cosmological binding energy” claim would need to produce a scalar that changes from the origin to $R_h$ under a legitimate observer projection; the present calculation indicates no such scalar exists for comoving particles.
The same projection rule applies in other coordinate systems, so the lesson generalizes: apparent energy expressions built from metric components in cosmology should be checked against observer-projection before being interpreted physically.
One could test the robustness of this conclusion by repeating the computation for non-comoving radial geodesics with nonzero initial peculiar velocity; the paper's method predicts that physical energy remains defined by projection, not by the first component of $p^\mu$.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Lewis has written a compact comment that does exactly what it should: it identifies a specific error in Melia's recent derivation of E=mc^2 from cosmology and corrects it with a standard GR calculation. The conclusion is right: Melia's p^0 is not the energy in those coordinates; the invariant projection onto the particle's own 4-velocity gives E=mc^2 in any spacetime, for any comoving observer, regardless of the expansion history. That is the real takeaway.

What's new is the specific rebuttal of the 4-vector normalization. Previous critiques of Rh=ct cosmology are set aside; Lewis focuses on the derivation itself. The central calculation is clean. The key identity p_mu u^mu = mc^2 is universal, so the headline conclusion does not depend on the contested metric in Eq. (2). That is a genuine strength.

Soft spots: the paper quotes the 'observer-dependent' metric from Melia rather than deriving it. If that line element were not actually the FLRW metric, or if it were trajectory-dependent rather than a spacetime metric, the explicit numerical refutation would need repair. But since the final argument does not rely on that metric, this is a presentation gap, not a load-bearing flaw. There are minor typos: Eq. (15) is garbled, and Section 3 says 'Using Equation 9' where it means the projection formula. None affect the argument.

Citation pattern is fine. Lewis cites his own earlier critiques but explicitly does not revisit them; the paper stands on the calculation. No fitting or free parameters; the derivation follows from textbook definitions.

Who this is for: anyone who cares about the Rh=ct debate, and instructors who want a clean example of why coordinate components of 4-vectors are not physical energies. It is a niche but legitimate comment. It deserves serious peer review as a comment/erratum-style paper; a good referee would check the algebra in Sec. 3 and the source of Eq. (2), then recommend publication with minor revisions.

Referee Report

0 major / 5 minor

Summary. The paper is a critical comment on Melia's recent claim that E = mc^2 has a cosmological origin as the gravitational binding energy needed for a particle to reach a cosmic horizon. The author argues that Melia's identification of the first component of the 4-momentum, E/c, with the physical energy is incorrect in general spacetimes. Using the standard relativistic definition of energy as the projection of the particle's 4-momentum onto an observer's 4-velocity, the paper shows that a comoving massive particle has E = g_{\mu\nu} p^\mu u^\nu = mc^2 in its own rest frame, independent of the scale factor, and that no transition to purely kinetic energy occurs at the Hubble radius. The conclusion is that E = mc^2 is the usual rest-energy identity rather than a cosmological binding energy.

Significance. If the cited work by Melia is faithfully represented, this is a decisive and useful correction: it identifies a concrete conceptual error, namely the treatment of coordinate components of 4-vectors as directly measurable energies. The derivation is clean, standard, and free of ad hoc parameters; it applies to any FLRW expansion history. The central invariant identity p^\mu u_\mu = mc^2 is robust and does not rely on the contested observer-dependent metric. The paper's contribution is primarily pedagogical and corrective rather than a new result, which is appropriate for a comment-style manuscript.

minor comments (5)

[Section 2, Eq. (6)] The invariant contraction of p^\mu is written as K^2 with K described as an 'undetermined constant', but since p^\mu = m u^\mu and u^\mu u_\mu = c^2, the invariant is fixed as (mc)^2 by Eq. (11). The role of K should be clarified so that the later derivation of K = mc does not appear to introduce a free parameter.
[Section 3, Eq. (15)] Equation (15) is typeset ambiguously: the square root should clearly extend over \Phi + (\dot R/c)^2 and the denominator should be enclosed in parentheses. The preprint version is easy to misread, although substitution of Eq. (7) does lead to u^t = c.
[Section 3, Eq. (17)] The sentence 'Using Equation 9, the particle's 4-momentum can be projected onto its own 4-velocity' appears to cite the wrong equation; Eq. (9) is Melia's energy expression, whereas the projection formula is Eq. (12). The cross-reference should be corrected.
[Section 3, final paragraph] The equivalence of the metrics in Eqs. (1) and (2) is asserted rather than demonstrated. Since the derivation of u^t = c for comoving observers passes through the metric in Eq. (2), the author should either provide the explicit coordinate transformation or cite the precise derivation in Ref. [2] so the manuscript is self-contained.
[Throughout] There are several typographical errors, including 'it has claimed' in Section 1, 'clams' in Section 1, 'in correct' in Section 3, and 'of of' in Section 3. These should be corrected in a final pass.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper derives E = mc^2 from the standard 4-velocity normalization and the projection formula, with no fitted inputs or load-bearing self-citation.

full rationale

The central derivation is self-contained against textbook relativity. The paper takes the invariant normalization of the 4-velocity, u_mu u^mu = c^2 (Eq. 10), and the 4-momentum, p_mu p^mu = (mc)^2 (Eq. 11), then defines the energy measured by an observer as the projection E = g_mu_nu p^mu u^nu (Eq. 12). For a particle projected onto its own 4-velocity, this gives E = m u_mu u^mu = mc^2. No constant is fitted, no subset of data is used to predict a related quantity, and the result is not assumed in advance; it is the standard rest-frame interpretation of the invariant mass. The simplification u^mu = (c, dot R) in Eq. (16) follows directly from the normalization in Eq. (14) together with the comoving condition Eq. (7), and the final projection is an identity that holds for any spacetime and any scale factor. The paper does cite the author's own earlier work [7, 9, 12], but only in a side remark questioning the Rh = ct interpretation; that citation is not load-bearing for the E = mc^2 argument. The borrowed 'observer-dependent' metric, Eq. (2), is taken from the target paper rather than rederived, which is a presentational dependency and a possible correctness risk, but not a circular reduction: the invariant conclusion p_mu u^mu = mc^2 does not depend on Eq. (2). Overall, the derivation chain has no step that reduces by construction to its own inputs.

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

The paper relies only on standard textbook definitions in general relativity and the assumption that the given coordinate transformations describe the same space-time. No free parameters are fitted, and no new entities are introduced.

assumptions (4)

standard math The magnitude of any 4-vector is invariant between coordinate systems: u^mu u_mu = c^2 and p^mu p_mu = (mc)^2.
Used in Section 3 to solve for the time component of the 4-velocity and to check the 4-momentum normalization.
domain assumption The energy of a particle measured by an observer is the projection of the particle's 4-momentum onto the observer's 4-velocity, E = g_{mu nu} p^mu u_obs^nu.
This is a standard definition in GR and is the central criterion used to determine the physical energy in Section 3.
domain assumption For a comoving particle, the proper radius and the Hubble radius satisfy R_h = c R / \dot{R}.
Used in Section 3 (Equation 7) to simplify the 4-velocity; it holds because \dot{R} = H R for comoving observers.
domain assumption The two metric forms (Equations 1 and 2) describe the same space-time, so equivalent observers must measure identical physical quantities.
Invoked in the final paragraph of Section 3 to reject the idea that the two coordinate choices yield different physics.

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Cite this review

Pith. "Pith review of Is there a Cosmological Basis for E = mc^2?." pith.science (2026). https://pith.science/paper/E7IXECOS

@misc{pith2026190809267,
  author       = {Pith},
  title        = {Pith review of: Is there a Cosmological Basis for E = mc^2?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E7IXECOS}},
  note         = {Machine review of arXiv:1908.09267}
}

read the original abstract

It has recently been claimed that relativity's most famous equation, E = mc^2, has a cosmological basis, representing the gravitational binding energy for a particle to escape from the origin to a gravitational horizon of the universe. In this paper, I examine these claims in detail, concluding that they result from a misinterpretation of motion of particles in the cosmological space-time, and an incorrect application of 4-vectors. Finally, I demonstrate that the origin of E = mc^2 comes from its usual relativistic interpretation, namely that it is the energy of a particle as seen in its own rest-frame.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 9 canonical work pages

[1]

Melia, International Journal of Modern Physics A 34(10), 1950055 (2019)

F. Melia, International Journal of Modern Physics A 34(10), 1950055 (2019). DOI 10.1142/S0217751X19500556

work page doi:10.1142/s0217751x19500556 2019
[2]

Melia, M

F. Melia, M. Abdelqader, International Journal of Modern Phys ics D 18(12), 1889 (2009). DOI 10.1142/S0218271809015746

doi:10.1142/s0218271809015746 2009
[3]

Melia, Astronomy & Astrophysics 553, A76 (2013)

F. Melia, Astronomy & Astrophysics 553, A76 (2013). DOI 10.1051/ 0004-6361/201220447

work page 2013
[4]

Melia, J.J

F. Melia, J.J. Wei, R.S. Maier, X.F. Wu, EPL (Europhysics Letters) 123(5), 59002 (2018). DOI 10.1209/0295-5075/123/59002 6 Geraint F. Lewis

work page doi:10.1209/0295-5075/123/59002 2018
[5]

Melia, Monthly Notice of the Royal Astronomical Society 481(4), 4855 (2018)

F. Melia, Monthly Notice of the Royal Astronomical Society 481(4), 4855 (2018). DOI 10.1093/mnras/sty2596

work page doi:10.1093/mnras/sty2596 2018
[6]

van Oirschot, J

P. van Oirschot, J. Kwan, G.F. Lewis, Monthly Notice of the Royal As- tronomical Society 404(4), 1633 (2010). DOI 10.1111/j.1365-2966.2010. 16398.x

arXiv 2010
[7]

Lewis, P

G.F. Lewis, P. van Oirschot, Monthly Notice of the Royal Astrono mical Society 423(1), L26 (2012). DOI 10.1111/j.1745-3933.2012.01249.x

arXiv 2012
[8]

Bilicki, M

M. Bilicki, M. Seikel, Monthly Notice of the Royal Astronomical Socie ty 425(3), 1664 (2012). DOI 10.1111/j.1365-2966.2012.21575.x

arXiv 2012

Show all 14 references

[9]

Lewis, Monthly Notice of the Royal Astronomical Society 432(3), 2324 (2013)

G.F. Lewis, Monthly Notice of the Royal Astronomical Society 432(3), 2324 (2013). DOI 10.1093/mnras/stt592

2013 doi
[10]

Mitra, Monthly Notice of the Royal Astronomical Society 442(1), 382 (2014)

A. Mitra, Monthly Notice of the Royal Astronomical Society 442(1), 382 (2014). DOI 10.1093/mnras/stu859

2014 doi
[11]

Kim, A.N

D.Y. Kim, A.N. Lasenby, M.P. Hobson, Monthly Notice of the Royal As- tronomical Society 460(1), L119 (2016). DOI 10.1093/mnrasl/slw079

2016 doi
[12]

Lewis, L.A

G.F. Lewis, L.A. Barnes, R. Kaushik, Monthly Notice of the Royal Astro- nomical Society 460(1), 291 (2016). DOI 10.1093/mnras/stw1003

2016 doi
[13]

Narlikar, American Journal of Physics 62, 903 (1994)

J.V. Narlikar, American Journal of Physics 62, 903 (1994). DOI 10.1119/ 1.17679

1994
[14]

Hartle, Gravity: An Introduction to Einstein’s General Relativity , il- lustrate edn

J.B. Hartle, Gravity: An Introduction to Einstein’s General Relativity , il- lustrate edn. (Benjamin Cummings, 2003)

2003

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Melia, International Journal of Modern Physics A 34(10), 1950055 (2019)

F. Melia, International Journal of Modern Physics A 34(10), 1950055 (2019). DOI 10.1142/S0217751X19500556

work page doi:10.1142/s0217751x19500556 2019

[2] [2]

Melia, M

F. Melia, M. Abdelqader, International Journal of Modern Phys ics D 18(12), 1889 (2009). DOI 10.1142/S0218271809015746

doi:10.1142/s0218271809015746 2009

[3] [3]

Melia, Astronomy & Astrophysics 553, A76 (2013)

F. Melia, Astronomy & Astrophysics 553, A76 (2013). DOI 10.1051/ 0004-6361/201220447

work page 2013

[4] [4]

Melia, J.J

F. Melia, J.J. Wei, R.S. Maier, X.F. Wu, EPL (Europhysics Letters) 123(5), 59002 (2018). DOI 10.1209/0295-5075/123/59002 6 Geraint F. Lewis

work page doi:10.1209/0295-5075/123/59002 2018

[5] [5]

Melia, Monthly Notice of the Royal Astronomical Society 481(4), 4855 (2018)

F. Melia, Monthly Notice of the Royal Astronomical Society 481(4), 4855 (2018). DOI 10.1093/mnras/sty2596

work page doi:10.1093/mnras/sty2596 2018

[6] [6]

van Oirschot, J

P. van Oirschot, J. Kwan, G.F. Lewis, Monthly Notice of the Royal As- tronomical Society 404(4), 1633 (2010). DOI 10.1111/j.1365-2966.2010. 16398.x

arXiv 2010

[7] [7]

Lewis, P

G.F. Lewis, P. van Oirschot, Monthly Notice of the Royal Astrono mical Society 423(1), L26 (2012). DOI 10.1111/j.1745-3933.2012.01249.x

arXiv 2012

[8] [8]

Bilicki, M

M. Bilicki, M. Seikel, Monthly Notice of the Royal Astronomical Socie ty 425(3), 1664 (2012). DOI 10.1111/j.1365-2966.2012.21575.x

arXiv 2012