On Unitarity of Massive Gravity in Three Dimensions

Ichiro Oda; Masashi Nakasone

REVIEW

On Unitarity of Massive Gravity in Three Dimensions

Masashi Nakasone, Ichiro Oda

High Energy Physics - Theory · Submitted 20 Feb 2009 · 15 references

Pith's verdict: A correct, narrow no-go: the BHT three-dimensional massive gravity does not generalize to D≥5 because the wrong-sign massless graviton becomes dynamical there.

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arXiv ID0902.3531 (version 5)

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CitationIchiro Oda, Masashi Nakasone. "On Unitarity of Massive Gravity in Three Dimensions." arXiv:0902.3531 (2009). https://pith.science/paper/IRDPZJ6B

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The useful content here is the no-go, not the 3D confirmation. Nakasone and Oda take the Bergshoeff-Hohm-Townsend higher-derivative massive gravity, run the trace condition through D dimensions, and show the linearized action is still Pauli-Fierz at quadratic order with α, β, γ fixed as in (13). Then they compute the propagator residues: the massive spin-2 pole has positive residue in every D, but the massless pole residue vanishes only in D=3 and is negative otherwise. So the wrong-sign Einstein-Hilbert term, harmless in 3D because the massless graviton has no propagating degrees of freedom, becomes a ghost for D≥5. That is a clean, correct observation and it closes an obvious generalization direction. The 3D part is a re-derivation of BHT; the D-dimensional extension is the new content. The algebra I checked holds: the trace condition (5), the auxiliary-field integration to (12), and the residue signs (38)-(39) are standard and correct. Soft spots, in order of how much they matter. The 'D > 4' in Eq. (38) should be 'D > 3.' For D=4 the residue is -2(|t|² - ½|t_ii|²), which is non-positive for any symmetric 2×2 t, so the ghost is there too. D=4 is excluded anyway by the pole in (7), so the physics conclusion is unaffected, but the sentence is sloppy and a referee should ask for it to be fixed. The body says the theory 'is free from the ghost and describes an interacting unitary massive gravity theory,' which is stronger than what is shown. The abstract is careful ('in the tree-level'); the body is not. The residue test is a necessary condition on the quadratic propagator; it says nothing about whether the k–f interactions, which the authors themselves note appear beyond quadratic order, introduce negative-norm states. I would want the body to match the abstract's hedge. This is a scope issue, not an error. The stress-test worry that the quadratic analysis misses interaction-induced ghosts is fair in principle, but the paper does state tree-level and quadratic order explicitly, so I would call it an inconsistency between abstract and body rather than overreach. Who it is for: people working on 3D massive gravity, higher-derivative gravity, and anyone who has wondered whether BHT generalizes. It is a ten-page note; a focused read. Peer review: yes, send it out. It is a correct, modest result that closes a door, and the D>4 phrasing and the abstract/body mismatch are exactly what a referee should catch.

Referee report

Major objections (3)

[Below Eq. (29) and conclusion] Below Eq. (29) the paper acknowledges that at higher order there appear interaction terms between k_μν and f_μν, and that the theory is 'essentially an interacting theory of two symmetric tensor fields where one is a massless tensor field with the wrong sign and the other is a massive tensor one with the correct sign.' The conclusion then states that the model 'describes an interacting unitary massive gravity theory only in three space-time dimensions.' The residue computation (38)-(39) establishes unitarity only for the quadratic action; it does not establish that the interacting theory is unitary. The wrong-sign massless tensor could in principle reappear as a ghost once interactions are included. The conclusion should be scoped to 'unitary at the quadratic (tree) level,' and the paper should state explicitly that the unitarity of the interacting theory is not established.
[Eqs. (34) and (36)] Eq. (34) gives P^{-1}_{(2)} = (1/□)·2/(−1+ξ□), which upon □→−p² becomes +2/(p²(1+ξp²)). Eq. (36) writes the P(2) coefficient as 2/(p²+1/ξ) − 2/p² = −2/(p²(1+ξp²)). These differ by an overall sign. Please state the convention for the amplitude (e.g., the sign of the source coupling) or correct the sign, since the residue signs in (38)-(39) depend on it.
[Eqs. (5)-(7) and the D=3 case] The trace condition (5)-(7) is stated to be valid only for D>4, and the D=3 case is dispatched with 'it is easy to see that the present analysis naturally reduces to the work by Bergshoeff et al.' Since D=3 is the paper's central claim, the authors should either provide the explicit D=3 derivation (setting D=3 in (4)-(7) and checking the degeneracy of the R² terms) or justify why the analytic continuation of the D>4 formulas to D=3 is valid.

Minor objections (5)

[Abstract and conclusion] The abstract says 'in higher dimensions' while the conclusion says 'more than four dimensions.' Since D=4 is excluded from the model entirely (the pole at D=4 in Eq. (7)), the abstract should say 'in D>4' or 'in more than four dimensions' to avoid implying D=4 is covered.
[Eq. (38)] The text says the residue 'becomes negative for D>4,' but the formula gives a negative value for D=4 as well. Since D=4 is excluded, this is not an error, but the text should note that D=4 is not a case of the model.
[Below Eq. (39)] The remark that 'the ghost does not show up in three dimensions even if there is a propagator like 1/(□(□−1/ξ))' would benefit from a sentence explaining that the 1/□ pole is the massless graviton, which has no propagating degrees of freedom in D=3.
[Eqs. (34) and (36)] The P(0,s) pole in (34) and (36) is at p²=0, i.e., massless, while the Pauli-Fierz massive graviton's scalar mode should be massive. Please clarify the physical interpretation of this massless scalar pole and whether it affects the unitarity conclusion.
[References] Reference [13] is arXiv:0901.1766; the paper should check the published version. The paper should also cite the standard result that the massless graviton has no propagating degrees of freedom in D=3.

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The paper

The paper claims

A higher-derivative extension of general relativity is shown to be a ghost-free massive gravity theory in three space-time dimensions and to fail in five or more, because a massless spin-2 mode with the wrong-sign kinetic term becomes a non

The reading

The paper asks whether a particular higher-derivative gravity theory — Einstein-Hilbert with the *wrong* sign plus curvature-squared terms — can serve as an interacting theory of a massive graviton, and whether the three-dimensional version recently proposed can be carried to other dimensions. The authors first impose a condition on the higher-derivative equations of motion: the trace of the extra tensor $K^{mn}$ must be proportional to the higher-derivative Lagrangian itself. That condition fixes the couplings ($lpha = -D/(4(D-1)) eta$) and makes the quadratic action reducible, after gauge fixing and integrating out an auxiliary field, to the Pauli-Fierz massive gravity action with the correct sign. The catch appears when the authors compute the propagator and read off the imaginary part of the residue at each pole with conserved transverse sources: at the massless pole the residue is $-2(|t_{ij}|^2 - |t_{ii}|^2/(D-2))$, which vanishes in $D=3$ but is negative for $D>4$, while the massive pole carries positive norm in every dimension. The conclusion is that the model is a unitary tree-level massive gravity theory only in three dimensions, and that the three-dimensional construction cannot be generalized to higher dimensions.

What carries the argument

The load-bearing device is the trace condition on the higher-derivative tensor $K_{mn}$: requiring $g^{mn}K_{mn}$ to be proportional to the original curvature-squared Lagrangian kills the $nabla^2 R$ term and forces $lpha = -D eta/(4(D-1))$, with the proportionality constant fixed to $c = 2/(4-D)$. This is what makes the scalar ($P^{(0,s)}$) ghost cancel between the $R_{mn}R^{mn}$ and $R^2$ terms, leaving only $xi R_{mn}R^{mn} + lambda R^2 = (xi/4) h^{mn}P^{(2)}_{mn,rs} Box^2 h^{rs}$. The second device is the decomposition of the quadratic Lagrangian into the complete set of spin projection operators $P^{(2)}, P^{(1)}, P^{(0,s)}, P^{(0,w)}, P^{(0,sw)}, P^{(0,ws)}$ in De Donder gauge, which makes the operator invertible so the propagator can be written down and its residues at $p^2=0$ and $p^2=-1/xi$ inspected.

Extended reading notes

Core claim

The central claim is that the higher-derivative action $S = -1/kappa^2 int d^Dx sqrt(-g)[R - R^{mn}R_{mn}/m^2 + (D/(4(D-1)))R^2/m^2]$ — equivalently the auxiliary-field action with a symmetric tensor $f_{mn}$ — describes a healthy massive graviton only in $D=3$. In three dimensions the would-be massless graviton has no propagating degrees of freedom, so the wrong-sign Einstein-Hilbert term costs nothing; the residue of the amplitude at $p^2=0$ vanishes identically. In $D>4$ that same wrong-sign term makes the massless spin-2 mode a genuine dynamical ghost, and the residue at $p^2=0$ is negative, so the theory is not unitary. The massive pole at $p^2=-1/xi$ has a positive residue, $2(|t_{ij}|^2 - |t_{ii}|^2/(D-1))$, in both cases, so the massive graviton itself is fine. The authors also note that the equivalence to Pauli-Fierz holds only at the linearized level: beyond quadratic order $k_{mn}$ and $f_{mn}$ interact, so the full theory is really two coupled symmetric tensor fields, one massless with the wrong sign and one massive with the correct sign. A four-dimensional version cannot be built at all, which the authors connect to the pole at $D=4$ in the coupling relation.

Load-bearing premise

The verdict rests on reading the sign of the residue of the tree-level propagator at each pole, with conserved transverse sources and only quadratic terms kept, as the test of unitarity; the paper itself notes that at higher orders the two tensor fields interact, and those interactions are not analyzed.

Editorial extensions

If this is right

The three-dimensional massive gravity model of Bergshoeff, Hohm and Townsend is confirmed to be ghost-free at tree level, since the massless spin-2 mode that would carry the wrong-sign kinetic term has no dynamics in three dimensions.
No dimension other than three works: for $D>4$ the massless graviton becomes a propagating ghost, so the higher-derivative route to an interacting massive graviton does not generalize.
A four-dimensional version of the construction is excluded outright, because the coupling relation required by the trace condition has a pole at $D=4$.
The massive graviton itself is healthy in every dimension, with positive residue $2(|t_{ij}|^2 - |t_{ii}|^2/(D-1))$ at $p^2=-1/xi$; the obstruction is entirely in the massless sector.
Because the equivalence to Pauli-Fierz holds only at quadratic order, the interacting theory is properly described as two coupled tensor fields, one massless with the wrong sign and one massive with the correct sign.

Reading between the lines

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

The residue test is a necessary condition, not a full unitarity proof: it inspects only the quadratic operator's poles with conserved transverse sources, so a negative-norm state generated by the acknowledged $k_{mn}$–$f_{mn}$ interactions at cubic and higher order would go undetected by this calculation.
The result suggests a general trade-off worth stating sharply: wrong-sign Einstein-Hilbert terms are harmless precisely when the massless graviton has no local degrees of freedom, which happens only in three dimensions, so any 'massive gravity without matter fields' built this way is dimensionally locked to $D=3$.
The same projection-operator bookkeeping could be applied to the three-dimensional model with a cosmological constant or a gravitational Chern-Simons term added, which is where the neighbouring topological massive gravity literature sits; whether the massless residue stays zero there is a concrete follow-up calculation.
If the model is really two coupled tensor fields beyond quadratic order, one could ask whether the wrong-sign massless field can be gauged or constrained away by a symmetry at the interacting level, which would be an alternative route to a higher-dimensional unitary theory than the one the authors rule out.

Original abstract

We examine a unitarity of a particular higher-derivative extension of general relativity in three space-time dimensions, which has been recently shown to be equivalent to the Pauli-Fierz massive gravity at the linearized approximation level, and explore a possibility of generalizing the model to higher space-time dimensions. We find that the model in three dimensions is indeed unitary in the tree-level, but the corresponding model in higher dimensions is not so due to the appearance of non-unitary massless spin-2 modes.

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We examine a unitarity of a particular higher-derivative extension of general relativity in three space-time dimensions, which has been recently shown to be equivalent to the Pauli-Fierz massive gravity at the linearized approximation level, and explore a possibility of generalizing the model to higher space-time dimensions. We find that the model in three dimensions is indeed unitary in the tree-level, but the corresponding model in higher dimensions is not so due to the appearance of non-unitary massless spin-2 modes.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We examine a unitarity of a particular higher-derivative extension of general relativity in three space-time dimensions, which has been recently shown to be equivalent to the Pauli-Fierz massive gravity at the linearized approximation level, and explore a possibility of generalizing the model to higher space-time dimensions.	contribution 0.84	0.52	0.05	asserted 0.47
s1	We find that the model in three dimensions is indeed unitary in the tree-level, but the corresponding model in higher dimensions is not so due to the appearance of non-unitary massless spin-2 modes.	result 1.00	0.96	0.91	asserted 1.00

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References

Works cited (15)

[1]

R. Percacci, ”The Higgs Phenomenon in Quantum Gravity”, Nucl. P hys.B353 (1991) 271, arXiv:0712.3545 [hep-th]

View on Pith Pith review arXiv 1991
[2]

Z. Kakushadze and P. Langfelder, ”Gravitational Higgs Mechan ism”, Mod. Phys. Lett.A15 (2000) 2265, arXiv:hep-th/0011245

View on Pith Pith review arXiv 2000
[3]

M. Porrati, ”Higgs Phenomenon for 4-D Gravity in Anti de Sitter S pace”, JHEP 0204 (2002) 058, arXiv:hep-th/0112166

View on Pith Pith review arXiv 2002
[4]

I. Kirsch, ”A Higgs Mechanism for Gravity”, Phys. Rev. D72 (2005) 024001, arXiv:hep- th/0503024

View on Pith 2005
[5]

G. ’t Hooft, ”Unitarity in the Brout-Englert-Higgs Mechanism for Gravity”, arXiv:0708.3184 [hep-th]

View on Pith Pith review arXiv
[6]

Z. Kakushadze, ”Gravitational Higgs Mechanism and Massive Gra vity”, Int. J. Mod. Phys.A23 (2008), arXiv:0709.1673 [hep-th]

View on Pith Pith review arXiv 2008
[7]

I. Oda, ”Gravitational Higgs Mechanism with a Topological Term”, Advanced Studies in Theoretical Physics 2 (2008) 261, arXiv:0709.2419 [hep-th]

View on Pith Pith review arXiv 2008
[8]

M. Maeno and I. Oda, ”Classical Solutions of Ghost Condensation Models”, Mod. Phys. Lett.B22 (2009) 3025, arXiv:0801.0827 [hep-th]

View on Pith Pith review arXiv 2009
[9]

M. Maeno and I. Oda, ”Massive Gravity in Curved Cosmological Bac kgrounds”, Int. J. Mod. Phys. A24 (2009) 81, arXiv:0808.1394 [hep-th]

View on Pith Pith review arXiv 2009
[10]

S. Deser, R. Jackiw and S. Templeton, ”Topologically Massive Gau ge Theories”, Ann. of Phys. 140 (1982) 372 [Erratum-ibid. 185 (1988) 406, 281 (2000) 409]; ”Three- dimensional Massive Gauge Theories”, Phys. Rev. Lett. 48 (1982) 975

View on Pith 1982
[11]

W. Li, W. Song and A. Strominger, ”Chiral Gravity in Three Dimens ions”, JHEP 0804 (2008) 082, arXiv:0801.4566 [hep-th]. 9

View on Pith Pith review arXiv 2008
[12]

M. Fierz and W. Pauli, ”On Relativistic Wave Equations for Particles of Arbitrary Spin in an Electromagnetic Field”, Proc. Roy. Soc. Lond. A173 (1939) 211

View on Pith 1939
[13]

E.A. Bergshoeﬀ, O. Hohm and P.A. Townsend, ”Massive Gravity in Three Dimensions”, arXiv:0901.1766 [hep-th]

View on Pith Pith review arXiv
[14]

K.S. Stelle, ”Renormalization of Higher-derivative Quantum Grav ity”, Phys. Rev. D16 (1977) 953

View on Pith 1977
[15]

C.W. Misner, K.S. Thorne and J.A. Wheeler, ”Gravitation”, W H Fre eman and Co (Sd), 1973. 10

View on Pith 1973

Forward citations

Cited by 21 Pith papers

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How the literature uses it

49 arXiv papers cite this in 65 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 42 sentences without a confident label

Builds on it 2 papers

In order to investigate the changes that occur when torsion propagates, we start off by considering the same Lagrangian as the one analyzed by Nakasone and Oda in Ref. [ref] , with the exception that we consider here the right sign of the Einstein-Hilbert term.

A Discussion on Massive Gravitons and Propagating Torsion in Arbitrary Dimensions · Description of the Model · 0.82
which is nothing but the higher dimensional extension of NMG [ref] .

Massive Gravity Theories and limits of Ghost-free Bigravity models · An interesting four-derivative theory of gravity · 0.89

Uses its method or tool 2 papers

The spin projection operators in $D$ space-time dimensions, $P^(2), P^(1), P^(0, s), P^(0, w), P^(0, sw)$ and $P^(0, ws)$ form a complete set in the space of second rank symmetric tensors and are defined as [ref]

Massive Gravity with Mass Term in Three Dimensions · Propagator in a general higher-derivative gravity · 0.83
To this end, it is useful to take account of the spin projection operators in $3$ space-time dimensions [ref] .

Renormalizability of Topologically Massive Gravity · Graviton propagator · 0.95

Uses its data or values 1 paper

In the work of Ref. [ref] , their values are set to be

A Discussion on Massive Gravitons and Propagating Torsion in Arbitrary Dimensions · Description of the Model · 0.92

Compares with it 2 papers

These results highlight a remarkable difference with respect to [ref] , because we do not have spin-2 massive propagation for the vielbein; so, we do not expect spin-2 massive graviton in any dimension.

A Discussion on Massive Gravitons and Propagating Torsion in Arbitrary Dimensions · Description of the Model · 0.98
In that case there are no double poles and the residue at the ghost-like massless pole vanishes in a quite different way [ref] .

Unitarity of spin-2 theories with linearized Weyl symmetry in $D=2+1$ · Conclusion · 0.99

Cites it as support 1 paper

As shown in [ref] this is not the case.

A new spin-2 self-dual model in $D=2+1$ · The spin-2 case · 1.00

Cites it as background 13 papers

More recently, this new massive gravity model in three dimensions has been studied from various viewpoints such as the unitarity and the impossibility of generalization to higher dimensions [ref] , relation to the Pauli-Fierz mass term [ref] , the AdS black hole solutions [ref] , the properties of linearized gravitational excitations in asymptotically AdS space-time [ref] and AdS waves [ref] .

Renormalizability of Massive Gravity in Three Dimensions · Introduction · 0.97
Unitarity and renormalizability is studied in [ref] , though the issue of renormalizability is not on the firm foundation [ref] .

A Complete Classification of Higher Derivative Gravity in 3D and Criticality in 4D · Introduction · 0.80
Unitarity is proven for Minkowski space in [ref] , whereas it is discussed in [ref] for maximally symmetric spaces.

Unitarity versus Renormalizability of Higher Derivative Gravity in 3D · Introduction · 0.88

What it cites · 10 sentences without a confident label

Builds on 1 paper

The main aim of this paper is not olny explore a possibility of generalizing this three-dimensional model to higher space-time dimensions but also to examine the unitarity of this particular higher-derivative extension of general relativity in three space-time dimensions by Bergshoeff et al [ref] .

Massive Gravity in Three Dimensions · Introduction · 0.93

Cites as background 10 papers

The other interest of massive gravity is relevant to the problem of counting the microscopic physical degrees of freedom existing in black holes through a holographic two-dimensional dual theory where the well-known topological massive gravity with the Chern-Simons term [ref] plays an important role [ref] .

Chiral Gravity in Three Dimensions · Introduction · 0.88
For instance, some people conjecture that the massless graviton might acquire mass via spontaneous symmetry breakdown of general coordinate reparametrization invariance, whose dynamical mechanism is sometimes called "gravitational Higgs mechanism" [ref] .

Gravitational Higgs Mechanism and Massive Gravity · Introduction · 0.90
For instance, some people conjecture that the massless graviton might acquire mass via spontaneous symmetry breakdown of general coordinate reparametrization invariance, whose dynamical mechanism is sometimes called "gravitational Higgs mechanism" [ref] .

Gravitational Higgs Mechanism with a Topological Term · Introduction · 0.90

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

R. Percacci, ”The Higgs Phenomenon in Quantum Gravity”, Nucl. P hys.B353 (1991) 271, arXiv:0712.3545 [hep-th]

View on Pith Pith review arXiv 1991

[2] [2]

Z. Kakushadze and P. Langfelder, ”Gravitational Higgs Mechan ism”, Mod. Phys. Lett.A15 (2000) 2265, arXiv:hep-th/0011245

View on Pith Pith review arXiv 2000

[3] [3]

M. Porrati, ”Higgs Phenomenon for 4-D Gravity in Anti de Sitter S pace”, JHEP 0204 (2002) 058, arXiv:hep-th/0112166

View on Pith Pith review arXiv 2002

[4] [4]

I. Kirsch, ”A Higgs Mechanism for Gravity”, Phys. Rev. D72 (2005) 024001, arXiv:hep- th/0503024

View on Pith 2005

[5] [5]

G. ’t Hooft, ”Unitarity in the Brout-Englert-Higgs Mechanism for Gravity”, arXiv:0708.3184 [hep-th]

View on Pith Pith review arXiv

[6] [6]

Z. Kakushadze, ”Gravitational Higgs Mechanism and Massive Gra vity”, Int. J. Mod. Phys.A23 (2008), arXiv:0709.1673 [hep-th]

View on Pith Pith review arXiv 2008

[7] [7]

I. Oda, ”Gravitational Higgs Mechanism with a Topological Term”, Advanced Studies in Theoretical Physics 2 (2008) 261, arXiv:0709.2419 [hep-th]

View on Pith Pith review arXiv 2008

[8] [8]

M. Maeno and I. Oda, ”Classical Solutions of Ghost Condensation Models”, Mod. Phys. Lett.B22 (2009) 3025, arXiv:0801.0827 [hep-th]

View on Pith Pith review arXiv 2009

[9] [9]

M. Maeno and I. Oda, ”Massive Gravity in Curved Cosmological Bac kgrounds”, Int. J. Mod. Phys. A24 (2009) 81, arXiv:0808.1394 [hep-th]

View on Pith Pith review arXiv 2009

[10] [10]

S. Deser, R. Jackiw and S. Templeton, ”Topologically Massive Gau ge Theories”, Ann. of Phys. 140 (1982) 372 [Erratum-ibid. 185 (1988) 406, 281 (2000) 409]; ”Three- dimensional Massive Gauge Theories”, Phys. Rev. Lett. 48 (1982) 975

View on Pith 1982

[11] [11]

W. Li, W. Song and A. Strominger, ”Chiral Gravity in Three Dimens ions”, JHEP 0804 (2008) 082, arXiv:0801.4566 [hep-th]. 9

View on Pith Pith review arXiv 2008

[12] [12]

M. Fierz and W. Pauli, ”On Relativistic Wave Equations for Particles of Arbitrary Spin in an Electromagnetic Field”, Proc. Roy. Soc. Lond. A173 (1939) 211

View on Pith 1939

[13] [13]

E.A. Bergshoeﬀ, O. Hohm and P.A. Townsend, ”Massive Gravity in Three Dimensions”, arXiv:0901.1766 [hep-th]

View on Pith Pith review arXiv

[14] [14]

K.S. Stelle, ”Renormalization of Higher-derivative Quantum Grav ity”, Phys. Rev. D16 (1977) 953

View on Pith 1977

[15] [15]

C.W. Misner, K.S. Thorne and J.A. Wheeler, ”Gravitation”, W H Fre eman and Co (Sd), 1973. 10

View on Pith 1973