Lecture Notes on General Relativity

Sean M. Carroll

REVIEW

Lecture Notes on General Relativity

Sean M. Carroll

General Relativity and Quantum Cosmology · Submitted 3 Dec 1997 · 7 references

Pith's verdict: Standard, correct, well-known GR lecture notes — useful to teach from, but the 'derivation-complete' label is a bit generous and there is nothing new here by the author's own admission.

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arXiv IDgr-qc/9712019 (version 1)

Pith IDL3VFYKLZ, Pith's permanent record of this paper and its review

CitationSean M. Carroll. "Lecture Notes on General Relativity." arXiv:gr-qc/9712019 (1997). https://pith.science/paper/L3VFYKLZ

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Editor's note

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You probably already know this document; it's Carroll's MIT 8.962 notes, posted in 1997 and cited mostly by people who learned GR from them. The preface says outright that none of the substance is new and that the only reason to read them is if you find the explanations clearer than Schutz, Wald, Weinberg, or MTW. That is an honest self-assessment and I'd take it at face value: this is a teaching document, not a research contribution, and it should be judged as one. On that measure it holds up well. The Christoffel connection is derived from metric compatibility plus torsion-freedom (3.21), the Riemann tensor from the commutator of covariant derivatives (3.67), the twice-contracted Bianchi identity gives the divergence-free Einstein tensor (3.96), geodesic deviation falls out cleanly (3.113), and the 20-component count in four dimensions (3.85) is done properly. I spot-checked the 2-sphere example (R = 2/a^2, 3.105) and the symmetry counting; both are right. The prose is unusually direct for a set of notes, and the asides (the redshift-as-grammatical-mistake passage in §3, the duality/monopole digression in §1) are the kind of thing students remember. Where I'd push back is on calling this 'derivation-complete.' The preface promises 'all necessary steps or leaving gaps that can readily be filled in by the reader,' which is weaker. Three places where a step is genuinely deferred: the cyclic identity (3.80) that the component count leans on is waved through with 'a little more work, which we leave to your imagination'; the sufficiency direction of Riemann-flat iff constant metric invokes Frobenius and says 'let's just take it for granted'; and the electromagnetic and scalar stress tensors (1.111, 1.112) are dropped in 'without any explanation at all.' None of these is a physics problem — they're routine or properly cited — but they mean the notes are a very good scaffold, not a self-contained proof. The reader's 'weakest assumption' point about the Christoffel connection being postulated as gravitational is a fact about GR, not a flaw in the notes; §4 presents it as the equivalence principle, which is the right move. Who should read it: a beginning grad student, or anyone who wants a fast, clean run at the standard formalism before hitting Wald. Would I cite it? Yes, as a pedagogical reference. Would I send it to referees? No — there's no research claim to referee, and the mathematics is textbook-standard. It's a good document that knows what it is.

Referee report

Significance

As a pedagogical document the notes are of real value, and the value lies in choices rather than in results. Three stand out. First, the text is unusually careful to separate statements that hold on any manifold from statements that hold only in flat spacetime in Cartesian coordinates: the discussion around Eq. (1.72) and Eq. (2.26) makes explicit why the partial derivative of a tensor is not a tensor, and the observation that the exterior derivative is a tensor regardless of connection (Eq. 3.29) is stated with the right emphasis. Second, the derivation of the Christoffel connection from metric compatibility plus torsion-freedom (Eqs. 3.19–3.21) is written out in full, as is the derivation of the Riemann tensor from the commutator (Eq. 3.65), the symmetry count reducing to n^2(n^2-1)/12 and hence 20 components in four dimensions (Eq. 3.85), and the geodesic deviation equation (Eq. 3.112). Third, the physical commentary is honest about what the formalism does and does not license, most notably the warning in §3 that the cosmological redshift does not define a well-defined relative velocity and that naive application of the Doppler formula produces an apparent superluminal recession

Minor objections (9)

[Preface] The preface states that the notes include 'all necessary steps or leaving gaps that can readily be filled in by the reader.' This is an accurate description of the document and is the correct frame for the deferred steps noted below; the notes do not claim to be derivation-complete, and a reader who expects that should be redirected to the preface. A short appendix indexing the places where a step is deferred (with a one-line indication of what fills it) would make the notes considerably easier to use for self-study without changing their character.
[§3, Eq. (3.80)] The cyclic identity R_{ρσμν} + R_{ρμνσ} + R_{ρνσμ} = 0 is introduced with 'With a little more work, which we leave to your imagination,' and the 20-component count of Eq. (3.85) depends on it through Eq. (3.83). This is the one input to the counting argument that is not written out, and it is cheap to close: the same Riemann-normal-coordinate expansion already displayed in Eq. (3.77) yields it in two lines. I would encourage the author to add that sketch, since the counting argument is otherwise fully explicit and is one of the notes' nicer set pieces.
[§3, following Eq. (3.73)] The sufficiency direction of 'the Riemann tensor vanishes if and only if coordinates exist in which the metric components are constant' invokes Frobenius's theorem and says 'It's something of a mess to prove... Let's just take it for granted,' deferring to Schutz. The citation is appropriate and the result is standard, but the logical dependence should be flagged at the point of use, because the equivalence is relied on later when the notes identify vanishing curvature with flatness. A single sentence noting that the converse rests on Frobenius and pointing to the precise statement would suffice.
[§1, Eqs. (1.111) and (1.112)] The energy-momentum tensors for electromagnetism and for a scalar field are presented 'Without any explanation at all,' with only a check of the 00 component invited. This is defensible in lecture notes, but a one-line remark identifying these as the Hilbert (metric-variation) stress tensors, or as the Noether currents of translation invariance, would connect them to the variational machinery developed in §4 and would explain why they take the form they do.
[§1, Eq. (1.88)] The sign in the double Hodge star, **A = (-1)^{s+p(n-p)}A, depends on conventions for the Levi-Civita tensor and for the signature count s. A worked check in four Lorentzian dimensions for p = 1 and p = 2 would let the reader calibrate against the convention actually in use, which matters because the duality discussion around Eqs. (1.91)–(1.94) is otherwise easy to follow but hard to verify.
[§2, Eq. (2.34)] The canonical form is written with all -1 entries preceding all +1 entries, and the signature is then defined as s - t. Many references order the entries the other way and define the signature with the opposite sign, so a parenthetical noting the convention would prevent a recurring source of confusion for readers cross-referencing Schutz or Wald.
[§3, Eq. (3.72)] The symbol η_{µν}, reserved in §1 for the Minkowski metric, is reused here 'in a generalized sense' as a diagonal matrix of ±1 entries. The text flags the reuse, but a distinct symbol (or a subscript) would remove the momentary ambiguity, particularly since the same paragraph is making a point about the difference between coordinate and orthonormal bases.
[§8 (Cosmology)] The preface acknowledges that 'Time constraints during the actual semester prevented me from covering some topics in the depth which they deserved, an obvious example being the treatment of cosmology.' The table of contents confirms this: the cosmology chapter covers homogeneity and isotropy, the Robertson-Walker metric, Friedmann equations, parameters, the scale factor, redshift, and Hubble's law, but not perturbations, the microwave background, or structure formation. Since the abstract lists cosmology as one of three applications, a sentence in the introduction pointing to where the interested reader should go next would set expectations appropriately.
[Bibliography] The asterisk difficulty ratings, normalized to the notes themselves, are a genuinely useful feature. The D'Inverno entry, however, reads 'A book I haven't looked at very carefully, but it seems as if all the right topics are covered without noticeable ideological distortion.' The candor is disarming, but a bibliography entry that disclaims familiarity is of limited use as a recommendation; either a specific reason to consult it or omission would serve the reader better.

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The paper

The paper claims

Carroll builds general relativity as one continuous chain: a metric, the unique connection it forces, the curvature that connection produces, and the field equations that follow — then applies the whole machine to waves, black holes, and an

The reading

These are one semester of graduate lectures whose aim is to leave no gaps: every step from the definition of a manifold to the Friedmann equations is either derived in full or flagged as an exercise the reader can fill in. The spine of the argument is that a metric on a manifold singles out exactly one connection that is torsion-free and compatible with that metric, and that this connection in turn fixes the curvature. From there the notes derive the Riemann tensor as the commutator of covariant derivatives, extract the Bianchi identity, and show that its twice-contracted form makes the Einstein tensor $G^{munu}$ automatically divergence-free — which is what lets it be set equal to a conserved energy-momentum tensor. A sympathetic reader cares because this is the shortest honest route to seeing why Einstein's equations have the form they do, rather than simply being told the form and asked to check the Newtonian limit. The final third spends the machinery on three worked applications: weak-field gravitational radiation, the Schwarzschild solution and black holes, and homogeneous cosmology.

What carries the argument

The load-bearing object is the Christoffel (Levi-Civita) connection, Eq. 3.21, because everything downstream is built from it and nothing else. The notes prove it exists and is unique by writing metric compatibility for three cyclic permutations of the indices and solving the resulting linear system, then note that the difference of any two connections is a tensor while the connection itself is not. Curvature enters through the commutator of covariant derivatives, which the notes compute line by line for a vector field and for a general tensor; the same computation yields the geodesic deviation equation $a^mu = R^mu{}_{nurhosigma}T^nu T^rho S^sigma$, which turns curvature into a statement about the relative acceleration of neighbouring freely falling particles — that is, into tidal force. The twice-contracted Bianchi identity is the hinge that connects the geometry to matter: it is what makes $nabla_mu T^{munu} = 0$ and $nabla_mu G^{munu} = 0$ compatible rather than an extra constraint. A secondary but deliberate thread is the tetrad formalism, which rewrites the same connection and curvature in a locally orthonormal basis and makes the analogy with gauge theory explicit through local Lorentz transformations.

Extended reading notes

Core claim

The claim the notes set out to establish is a chain of uniqueness and consistency, not a single new result. First, demanding that a covariant derivative be torsion-free ($Gamma^lambda_{munu} = Gamma^lambda_{(munu)}$) and metric-compatible ($nabla_rho g_{munu} = 0$) leaves exactly one connection, and the notes derive its coefficients explicitly as $Gamma^sigma_{munu} = frac{1}{2} g^{sigmrho}(partial_mu g_{nurho} + partial_nu g_{rhomu} - partial_rho g_{munu})$ (Eq. 3.21). Second, the commutator of two covariant derivatives acting on a vector is not a differential operator but a multiplicative one, $[nabla_mu, nabla_nu]V^rho = R^rho{}_{sigmamunu}V^sigma$ minus a torsion term, which identifies the Riemann tensor as the local measure of curvature (Eq. 3.67). Third, the algebraic symmetries of that tensor reduce its independent components to $frac{1}{12}n^2(n^2-1)$ — twenty in four dimensions, exactly matching the twenty second derivatives of the metric that no coordinate choice can remove. Fourth, the Bianchi identity $nabla_{[lambda}R_{rhosigma]munu} = 0$, contracted twice, gives $nabla_mu G^{munu} = 0$ for $G^{munu} = R^{munu} - frac{1}{2}R g^{munu}$, so the geometric side of any field equation built from curvature is conserved identically. The notes then argue that identifying the geodesics of this connection with the worldlines of freely falling bodies reproduces Newtonian gravity in the weak-field limit and yields the three applications.

Load-bearing premise

The whole bridge from geometry to gravity rests on a physical postulate rather than a mathematical result: that the geodesics of the metric's unique torsion-free connection are the worldlines of freely falling bodies, which the notes assert in the discussion of geodesics and develop as the equivalence principle.

Editorial extensions

If this is right

Linearizing the field equations about flat space gives a wave equation whose solutions have two transverse-traceless polarizations travelling at the speed of light, and the quadrupole formula fixes the energy loss rate of a radiating source.
The Schwarzschild solution follows from spherical symmetry alone (Birkhoff's theorem), and its maximal extension through Kruskal coordinates exhibits an event horizon, a genuine curvature singularity, and a Penrose diagram structure with conformal infinity.
Because the field equations are built from a conserved geometric tensor, the energy-momentum tensor of a perfect fluid, $T^{munu} = (rho+p)U^mu U^nu + p g^{munu}$, can be coupled to them without adding new constraints.
Assuming homogeneity and isotropy gives the Robertson-Walker metric and the Friedmann equations, from which the redshift of distant sources and Hubble's law follow as consequences of the expansion of the metric rather than of motion through space.
The Newtonian limit is recovered from the $00$ component of the weak-field equations, so the same connection that defines geodesics also reproduces the inverse-square law in the appropriate limit.

Reading between the lines

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

The notes deliberately keep the connection logically independent of the metric and only later single out the metric-compatible torsion-free one, which leaves room for theories with torsion or non-metricity; the same commutator derivation would then carry extra terms, so the formalism itself marks where such alternatives would have to differ.
The observation that the twenty independent curvature components match the twenty unremovable second derivatives of the metric is presented as a consistency check, but it also explains why curvature rather than the connection must be the coordinate-invariant carrier of gravity: the connection can always be made to vanish at a point, and only its derivatives cannot.
The path-ordered exponential that solves parallel transport around a loop, and the remark that knowing the holonomy of every loop is equivalent to knowing the metric, point toward treating holonomies rather than the metric as fundamental — which the notes mention only in passing as a route to quantization.
The claim that the exterior derivative requires no connection at all, while the covariant derivative does, suggests a clean split between the parts of the formalism that survive any choice of connection and the parts that do not; the notes use this to argue that torsion can never enter the exterior derivative of a form.

Original abstract

These notes represent approximately one semester's worth of lectures on introductory general relativity for beginning graduate students in physics. Topics include manifolds, Riemannian geometry, Einstein's equations, and three applications: gravitational radiation, black holes, and cosmology.

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The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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What the abstract itself states, as probabilities: claims first 0.02 · states limitation 0.12 · human participants 0.03 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.13 · compares with prior results 0.02 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

These notes represent approximately one semester's worth of lectures on introductory general relativity for beginning graduate students in physics. Topics include manifolds, Riemannian geometry, Einstein's equations, and three applications: gravitational radiation, black holes, and cosmology.

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	These notes represent approximately one semester's worth of lectures on introductory general relativity for beginning graduate students in physics.	contribution 0.49	0.45	0.03	asserted 0.92
s1	Topics include manifolds, Riemannian geometry, Einstein's equations, and three applications: gravitational radiation, black holes, and cosmology.	contribution 0.88	0.14	0.04	asserted 0.99

What kind of paper the classifier reports this is: review 0.93 · evidence from proof or derivation 0.54.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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References

Works cited (7)

[1]

The union of the Uα is equal to M; that is, the Uα coverM

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[2]

The charts are smoothly sewn together. More precisely, if two charts overlap,Uα∩Uβ ⁄= ∅, then the map ( φα ◦φ−1β ) takes points in φβ(Uα ∩Uβ) ⊂ Rn ontoφα(Uα ∩Uβ) ⊂ Rn, and all of these maps must be C ∞ where they are deﬁned. This should be clearer in pictures: Uα ϕ ( ) ϕ ( ) ϕ ϕ ϕ ϕ ϕ ϕ β α α β α β Uβ Uα α β Uβ -1 -1 these maps are only defined on the sha...

View on Pith 1997
[3]

linearity: ∇(T +S) = ∇T + ∇S

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[4]

Leibniz (product) rule: ∇(T ⊗S) = ( ∇T ) ⊗S +T ⊗ (∇S) . If ∇ is going to obey the Leibniz rule, it can always be written as t he partial derivative plus some linear transformation. That is, to take the covari ant derivative we ﬁrst take the partial derivative, and then apply a correction to make the r esult covariant. (We aren’t going to prove this reason...

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[5]

commutes with contractions: ∇µ(T λλρ) = ( ∇T )µλλρ ,

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[6]

reduces to the partial derivative on scalars: ∇µφ =∂µφ . There is no way to “derive” these properties; we are simply de manding that they be true as part of the deﬁnition of a covariant derivative. Let’s see what these new properties imply. Given some one-fo rm ﬁeld ωµ and vector ﬁeld V µ, we can take the covariant derivative of the scalar deﬁned by ωλV λ...

View on Pith 1997
[7]

Of course, if we have a proposed solution to a set of diﬀeren tial equations such as this, it would suﬃce to plug in the proposed solution in order to ver ify it; we would like to do better, however. In fact, we will sketch a proof of Birkhoﬀ’s theorem, which states that the Schwarzschild solution is the unique spherically symmetric solution to Einstein’s...

View on Pith 1963

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How later papers used it

73 later papers cite it

Evidence: substantial. 73 of the citing papers share no author with it, and 8 of those carry a confident label from Pith's citation classifier.

Pith read 25 of the 119 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

5 later papers by other authors are listed below: 5 used its method or tool.

Uses its method or tool 5 of 6 papers with this label

Proceeded to write the Ricci tensor in second-order perturbations following this paper.

Weak field and slow motion limits in energy-momentum powered gravity (2022) · 0.84 · paraphrased: the licence does not allow quoting
We will follow the derivation in [ref] for as long as we can.

A variational derivation of the field equations of an action-dependent Einstein-Hilbert Lagrangian (2022) · Variation of the metric · 0.99
Considered the procedure followed in this paper.

Penrose process in charged axion-dilaton coupled black hole (2014) · 0.94 · paraphrased: the licence does not allow quoting
Consulted this paper for the definition of the Lie derivative for a rank-2 tensor-valued background field.

Cosmology in the presence of diffeomorphism-violating, nondynamical background fields (2022) · 0.95 · paraphrased: the licence does not allow quoting
Benefited from the Christoffel symbol expressions of this paper in the next steps of its calculations.

Some Aspects of Morris-Thorne Wormhole in Scalar Tensor Theory (2017) · 0.92 · paraphrased: the licence does not allow quoting

How the literature uses it

76 arXiv papers cite this in 123 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 · 74 sentences without a confident label

Uses its method or tool 6 papers

Accordingly, let us proceed with writing the Ricci tensor in the second order $h_μν$'s as [ref] [equation] Including the second order perturbations in Lorenz gauge implies $h^μν(∂_μ h_αν+∂_ν h_αμ-∂_α h_μν)=0$, and hence, the second order Ricci tensor becomes [equation] Then, the $tt$ component of Eq. [ref] up to second order perturbations reads [equation] and yields the solution [equation] Therefore, in the EMPG model, the components of the post-Newtonian metric take the following form: [equation]

Weak field and slow motion limits in energy-momentum powered gravity · Post-Newtonian limit · 0.84
We will follow the derivation in [ref] for as long as we can.

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Here we consider the procedure followed in [ref] .

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Compares with it 2 papers

The approximate equation of orbit ([ref]) predicts a reduced radius of circular orbit, [equation] This prediction is in agreement [ref] to first order with that obtained by minimizing the Schwarzschild effective potential ([ref]), [equation] (A distinction is made here between the radius of circular orbit as determined from the Schwarzschild effective potential $R_c$ and that determined from the equation of orbit $r̃_c$.) For $≪1/12$, the radius of circular orbit is predicted to be reduced, [equation] The Schwarzschild effective potential ([ref]) is compared to that derived from Newtonian mechanics ([ref]) in Fig. [ref].

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Cites it as background 37 papers

Note that this does not mean that torsion is ”physically” there, something both conjectured in some extensions of gravity [ref] and used to rewrite general relativity in a more ”gauge-like” form [ref] ; The fluid in a heavy ion collision [ref] lives in a space free of either Riemannian or torsional curvature.

Gaussian pseudogauge invariant hydrodynamics with spin · Introduction · 0.84
Diffeomorphism invariance translates as invariance under, h →h + L_ϵ(G) , ξ→ξ+ L_ϵ(Φ) , a_q→a_q + L_ϵ(A_q) where $L_ϵ(...)$ stands for the Lie derivative w.r.t. the vector field $ϵ= ϵ^M(x)∂_M = ϵ^A(x)e_A.$ More explicitly, it is not difficult to show that ([ref])-([ref]) are left invariant under the field transformations [This is a generalization of the usual perturbative treatment around flat space in the context of General Relativity, see for example [ref] , chapter 6. ] , ^ϵh_AB = h_AB + D_Aϵ_B + D_Bϵ_A^ϵξ = ξ+ ϵ^A D_AΦ^ϵa_q_A_1…A_q = a_q_A_1…A_q+ ϵ^B D_B A_q_A_1…A_q + D_A_1ϵ^B A_q_BA_2…A_q + …+D_A_qϵ^B A_q_A_1…A_q-1B As the system is linear, it follows that $(L_ϵ(G),L_ϵ(Φ),L_ϵ(A_q))$ is solution for any $ϵ$, the pure gauge, trivial, solution.

Gauge invariant perturbation theory and non-critical string models of Yang-Mills theories · The gauge-invariant perturbative setup · 0.87
The most direct approach would be to treat the metric deviations $h_μν(x)$ in equation [ref], away from the flat metric $η_μν$ (or from a more general classical solution [ref] equation 16, for which linearised gravity can also be adopted as noted in [ref] after equation 6.3), as the field to be quantised.

Quantum Gravity from the Composition of Spacetime Constructed through Generalised Proper Time · Quantum Theory with Gravity · 0.83

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]

The union of the Uα is equal to M; that is, the Uα coverM

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[2] [2]

The charts are smoothly sewn together. More precisely, if two charts overlap,Uα∩Uβ ⁄= ∅, then the map ( φα ◦φ−1β ) takes points in φβ(Uα ∩Uβ) ⊂ Rn ontoφα(Uα ∩Uβ) ⊂ Rn, and all of these maps must be C ∞ where they are deﬁned. This should be clearer in pictures: Uα ϕ ( ) ϕ ( ) ϕ ϕ ϕ ϕ ϕ ϕ β α α β α β Uβ Uα α β Uβ -1 -1 these maps are only defined on the sha...

View on Pith 1997

[3] [3]

linearity: ∇(T +S) = ∇T + ∇S

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[4] [4]

Leibniz (product) rule: ∇(T ⊗S) = ( ∇T ) ⊗S +T ⊗ (∇S) . If ∇ is going to obey the Leibniz rule, it can always be written as t he partial derivative plus some linear transformation. That is, to take the covari ant derivative we ﬁrst take the partial derivative, and then apply a correction to make the r esult covariant. (We aren’t going to prove this reason...

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[5] [5]

commutes with contractions: ∇µ(T λλρ) = ( ∇T )µλλρ ,

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[6] [6]

reduces to the partial derivative on scalars: ∇µφ =∂µφ . There is no way to “derive” these properties; we are simply de manding that they be true as part of the deﬁnition of a covariant derivative. Let’s see what these new properties imply. Given some one-fo rm ﬁeld ωµ and vector ﬁeld V µ, we can take the covariant derivative of the scalar deﬁned by ωλV λ...

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[7] [7]

Of course, if we have a proposed solution to a set of diﬀeren tial equations such as this, it would suﬃce to plug in the proposed solution in order to ver ify it; we would like to do better, however. In fact, we will sketch a proof of Birkhoﬀ’s theorem, which states that the Schwarzschild solution is the unique spherically symmetric solution to Einstein’s...

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