Abstract
All particles, whether massive or massless, experience the gravitational interaction due to their energy content.
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Notes
- 1.
- 2.
- 3.
- 4.
Here it is worth recalling that gravitational waves have been detected from the merger of two black holes 1.3 billion light-years from the Earth via the Laser Interferometer Gravitational Wave Observatory (LIGO). See B. P. Abbott et al.: Phys. Rev. Lett. 116, 061102 (2016), Astrophys. J. Lett. 818, L22 (2016).
- 5.
- 6.
Particle emission from a BH is formally explained through virtual pairs of particles created near the horizon with one particle falling into the BH while the other becoming free outside the horizon.
- 7.
Recall that entropy S represents a measure of the amount of disorder with information encoded in it.
- 8.
Bekenstein [14].
- 9.
- 10.
- 11.
Stelle [59].
- 12.
- 13.
- 14.
Recent recorded values are α = 3, and β = 41∕10π [15].
- 15.
- 16.
Arnowitt et al. [3].
- 17.
- 18.
- 19.
- 20.
See also Problem 1.2.
- 21.
Note that \(\partial _{\sigma }\,\varGamma _{\nu \,\mu }\:\!^{\mu } = \partial _{\sigma }\partial _{\nu }\,\mathrm{ln}\:\![\,\sqrt{-g}\,\,]\), see Problem 1.4, (ii).
- 22.
See Problem 1.9.
- 23.
We have used the familiar relation \(\partial _{j}\partial ^{j}\big(1/\vert \mathbf{X} -\mathbf{ X}'\vert \big) = -4\,\pi \,\delta ^{3}(\mathbf{X} -\mathbf{ X}')\).
- 24.
- 25.
Such a completeness relation is conveniently described by Schwinger [56], pp. 15–17. See also Appendix II at the end of this volume.
- 26.
See Problem 1.12.
- 27.
Gravitational waves have been detected from the merger of two black holes 1.3 billion light-years from the Earth via the Laser Interferometer Gravitational Wave Observatory (LIGO). See B. P. Abbott et al.: Phys. Rev. Lett. 116, 061102 (2016), Astrophys. J. Lett. 818, L22 (2016).
- 28.
- 29.
- 30.
See, e.g., [45, p. 989].
- 31.
This subsection is based on [38].
- 32.
The background field method was introduced by DeWitt [21].
- 33.
See Problem 1.13
- 34.
Schwinger [55].
- 35.
See the monumental work of DeWitt, e.g., in [21] and references therein.
- 36.
Such useful integrals are also considered in details in Appendix II of Vol. I [43].
- 37.
Note that unlike (1.5.15), in the problem in hand to be applied to the effective action in quantum general relativity, there is no exp [−i m 2 s] term in the present case.
- 38.
- 39.
- 40.
Note that K(x, x ; s) means setting x ′ = x in K(x, x ′; s) prior to considering limits on the s variable.
- 41.
Note that K(x, x ; s) means setting x ′ = x in K(x, x ′; s) prior to considering limits on the s variable.
- 42.
We note that due to dimensional reasons, we may introduce a mass parameter μ D , in the process of considering the limit ɛ → 0, when making the transition (dx) → (μ D )− ɛdD x and finally back to (dx). There is no need to write this explicitly here. See Appendix III of Vol. I [43], in particular, (III.8).
- 43.
Note, for example, referring to (1.5.15), we have the following expression for its right-hand side for x ′ = x :
$$\displaystyle{ \frac{\mathrm{i}} {(4\,\pi \,\mathrm{i})\,^{n/2}} \frac{1} {s\,^{n/2}}\,\mathrm{e}^{\,-\,\mathrm{i}\,s\,(m^{2}\,-\,\mathrm{i}\,\epsilon )}, }$$while K(x, x ′; s) → δ (n)(x − x ′) for s → 0, and the order in which limits are taken is important.
- 44.
- 45.
In analogy to a one dimensional integral
$$\displaystyle{ \int _{-\infty }^{\infty }\!\mathrm{d}z\,\,\mathrm{e}^{\,-z\,^{2}}[1 + a_{n}(\kappa \,z)\,^{n}], }$$note that in addition to one in the square brackets, corresponding to one-loop, only even powers of n contribute to the above integral.
- 46.
In any case, one may modify the metric g μ ν to g μ ν plus a linear combination of R μ ν and g μ ν R, as indicated in (1.8.18), denoting the new expression, say, by g μ ν to write
$$\displaystyle{ \varGamma ^{[\,0\,]}[\:\!g_{\mu \nu }] +\varGamma ^{[\,1\,]}\Big\vert _{\mathrm{ div}}\![\:\!g_{\mu \nu }] =\varGamma ^{[\,0\,]}[\:\!\underline{g}\,_{\mu \nu }] + \cdots \,, }$$where Γ [ 0 ][ g μ ν ] is the Einstein-Hilbert action with metric g μ ν , and “⋯ ” is of the order of contributions of two loops.
- 47.
- 48.
- 49.
See also [34] for finiteness off the mass shell.
- 50.
- 51.
For the extension of the above alternative to higher order loops with a field redefinition, in the process of renormalization, such that the right behavior of a propagator for p 2 → ∞ is maintained thus avoiding non-positivity conditions, by expanding the metric around the Minkowski one: g μ ν = η μ ν +κ h μ ν , see [2]. See also [29].
- 52.
- 53.
- 54.
- 55.
Here one is considering a connected proper graph, where proper means that it cannot become disconnected by cutting a single line.
- 56.
See [59].
- 57.
A a quantum viewpoint treatment of a higher derivative modification of the linearized Einstein-Hilbert action in (1.2.1) obtained by replacing R in it by R + a R 2 with a > 0, shows that the theory involves, in addition to the graviton, a massive scalar particle which cannot be gauged away and no ghosts arise in the theory—see [42].
- 58.
These are referred to dimensionless couplings in the sense that if the mass dimension of a dimensionful coupling \(\tilde{g}\) is d, and \(\tilde{g}(k)\) denotes the coupling defined at a renormalization point with energy scale k, then one may define its dimensionless counterpart coupling by \(g(k) = (k)^{-d}\,\tilde{g}(k)\). The couplings are also restricted to those that do not change when a point-transformation of a field is carried since, in reference, to physical quantities the latter do not depend on how the fields are defined. Such inessential couplings g ′ give rise to partial derivatives of the Lagrangian density \(\partial \mathcal{L}/\partial g\,'\) which are at most total derivatives (or zero) when the Euler-Lagrange equations are used. See [67].
- 59.
We have encountered beta functions in abelian and non-abelian gauge theories in much details in Vol. I [43].
- 60.
One assumes at least one of the relevant couplings g i ∗ ≠ 0. The reason is that when one considers the application of the renormalization group to a physical quantity Q, such as a total cross section of a process, and D is the dimensionality of Q, and E is some energy characterizing, then the high energy behavior of the quantity (E)−D R is governed by g(E) defined in terms of the relevant couplings.
- 61.
This approach was introduced by Weinberg [67].
- 62.
- 63.
- 64.
- 65.
This is treated in Chap. 2.
- 66.
See, e.g., Sect. 2.10
- 67.
See, e.g., [19].
- 68.
That is, in particular, no corrections are assumed having the structures GNln(GN), GN 2ln(GN).
- 69.
We note that the integral may not be of a higher power in \(\mathrm{ln}\,(\mathbf{k}^{2})\).
- 70.
- 71.
- 72.
- 73.
- 74.
ADM refers to Arnowitt, Deser and Misner [3].
- 75.
This means, in particular, that N a = q ab N b. Note that q a b is rather a standard notation for this metric.
- 76.
For holonomies in quantum mechanics and applications, see Manoukian [40], Sect. 8.13, p. 524.
- 77.
We recall an elementary aspect of group theory, showing that an SU(2) transformation induces a 3 dimensional rotation in Euclidean space with metric δ ij as follows. For (x 1, x 2, x 3) as components of a three-vector in Euclidean space, write X = σ i x i, where the σ i are the Pauli matrices. Then det X = −x i x i. For an SU(2) matrix M which induces a transformation on X via X → X′ = M X M −1, one has det X ′ = det X, i.e., x′ i x ′ i = x i x i.
- 78.
See Problem 1.19.
- 79.
Some authors define a holonomy as the trace of the expression in (1.9.12).
- 80.
- 81.
See also Sect. 6.12. of Vol. I [43].
- 82.
- 83.
In this section, the variables x, x′, … are understood to correspond to three components.
- 84.
- 85.
We will not need actual explicit representations of the so-called Wigner type matrices R j[A, h γ ]. For details of Wigner matrices D ( j ) in the theory of angular-momentum & spin, in general, see Manoukian [40].
- 86.
Clebsch-Gordan coefficients are proportional to so-called 3j symbols, see, e.g, Manoukian [40], p. 289.
- 87.
For the situation when some of the nodes and/or nodes and links lie on the surface, the analysis becomes quite involved, and we refer the reader for the underlying details to Rovelli [50], p. 254.
- 88.
- 89.
Note all the indices here, including α, β, are tensorial indices in curved spacetime with metric g μ ν , and not Lorentz ones.
- 90.
As mentioned in Appendix C, this is due to the fact that two of its indices must be equal implying the vanishing of a totally anti-symmetric tensor with five (or more) indices in four dimensional spacetime.
- 91.
- 92.
Bekenstein [14].
- 93.
- 94.
Particle emission from a BH is formally explained through virtual pairs of particles created near the horizon with one particle falling into the BH while the other becoming free outside the horizon.
- 95.
See, e.g., [39].
- 96.
For a pedestrian approach, this may be roughly inferred from Newton’s theory of gravitation that the escape speed of a particle in the gravitational field of a spherically symmetric massive body of mass M, at a distance r, is obtained from the inequality v 2∕2 − GN M∕r < 0, and by formally replacing v by the ultimate speed c gives for the critical radius R CRITICAL = 2 GN M∕c2 such that for r < R CRITICAL a particle cannot escape.
- 97.
A pedestrian approach in determining the temperature is the following. By comparing the expression of energy expressed in terms of the wavelength of radiation λ: E = h c∕λ, with the expression E = k B T, gives T = h c∕k B λ. On dimensional grounds λ ∼ 2 GN M∕c2, which gives T ∼ π ℏc3∕GN M k B. This leads the expression in (E-1.5) up to a proportionality constant.
Recommended Reading
DeWitt, B. (2014). The global approach to quantum field theory. Oxford: Oxford University Press.
Kiefer, C. (2012). Quantum gravity (3rd ed.) Oxford: Oxford University Press.
Manoukian, E. B. (2015). Quantum viewpoint of quadratic f(R) gravity, constraints, vacuum-to-vacuum transition amplitude and particle content. General Relativity and Gravitation, 47:78, 1–11.
Manoukian, E. B. (2016). Quantum field theory I: Foundations and abelian and non-abelian gauge theories. Dordrecht: Springer.
Manoukian, E. B. (1983). Renormalization. New York: Academic.
Oriti, D. (Ed.). (2009). Approaches to quantum gravity. Cambridge: Cambridge University Press.
Rovelli, C. (2008). Quantum gravity. Cambridge: Cambridge University Press.
Rovelli, C., & Vidotto, F. (2015). Covariant loop quantum gravity. Cambridge: Cambridge University Press.
Thiemann, T. (2007). Modern canonical quantum gravity. Cambridge: Cambridge University Press.
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Appendices
Appendix A: Variation of a Determinant
In varying \(\sqrt{-g}\) with respect to g μ ν where g(x) = det [ g μ ν (x) ], we need to know how to vary a determinant. This is explicitly used in Problem 1.3, and in obtaining the field equation in (1.2.2). This appendix shows you how to do that.
Given two matrices N, M, integrate the following from λ = 0 to λ = 1:
to obtain
or for N = M + δM
We multiply the latter equation from the right by e−M, and take the trace to obtain
where the second equality follows from the fact that
For a given matrix A, let M = ln [ A ]. (A-1.5) then gives
On the other hand for a c-number a: δ ea = ea δ a, and for a = Tr (ln[ A ]), eTr ln[ A ] = det[ A ]. This c-number equation then gives
which upon comparison with (A-1.7) leads to
Appendix B: Parametric Integral Representation of the Logarithm of a Matrix
For a given matrix M into consideration, one may conveniently introduce another matrix N commuting with M, such as the unit matrix, and express
An integration of the right-hand side over λ gives the following parametric integral representation
where we note that the second term, within the square brackets, is independent of the matrix M in consideration. Of particular interest is the trace of lnM given by
where, again the second term on the right-hand side is independent of the matrix M. In a path integral representation, through exponentiation of the expression in (B-1.3), such a second term just gives rise to an overall multiplicative constant factor to the generating functional 〈 0+ | 0−〉 independent of the matrix M in consideration.
Appendix C: Content of the Euler-Poincaré Characteristic
The purpose of this Appendix is to establish the following fundamental result in four dimensional spacetime that
is a total derivative. Footnote 88 This is the content of the so-called Euler-Poincaré characteristic which holds true in four dimensional spacetime. It is also known as the Gauss-Bonnet Theorem in four dimensions. According to Problem 1.16, this is established by considering equivalently the far simpler expression
To establish that this is a total derivative, we make use of the following results: Footnote 89
established, respectively, in Problems 1.3, (i), Problem 1.15, Problem 1.7, (ii), Problem 1.17, Problem 1.18.
From (C-1.2)–(C-1.4), the variation in question is given by
The first two expressions on the right-hand side of the above equation may be combined and give
From (C-1.5), (C-1.6) and the fact that ∇ σ g μ ρ = 0, the last term in (C-1.7) may be equivalently rewritten as
which from Problem 1.5 (i) is a total derivative.
Therefore as an intermediary step we have
We rewrite the second term on the right-hand side of the above equation as
To investigate the nature of this second term on the right-hand side of (C-1.8), we proceed as follows.
A totally anti-symmetric tensor of rank 5, i.e., having five indices, in four dimensional spacetime, is necessarily zero due to the simple fact that at least two of its indices must be equal. Now consider the part
in this expression, concentrating on the following five indices (μ, ρ ′, ν, α, β). Suppressing the other indices in the just given part, we may, in turn, consider the expression:
By interchanging in turn the index ρ ′ with the other indices within the round brackets, we may generate the expression
where due to the anti-symmetry property of the multiplicative factor η μ ν α β, the above must be equal to zero as indicated, since an antisymmetric tensor of rank 5 is zero as mentioned above. By re-inserting the suppressed fixed indices, the above equation leads, upon multiplying by g ρ ρ ′ η σ λγ κ, to
In the second line simply write R ν ρ σ λ = −R ρ ν σ λ . In the third line exchange (σ λ) ↔ (γ κ), and α ↔ β. In the fourth line first write R α ρ γ κ = − R ρ α γ κ then exchange (σ λ) ↔ (γ κ), and α ↔ ν, and use, in the process, the anti-symmetry of η σ λ γ κ and, in particular, the anti-symmetry of R β α σ λ in its first two indices. These allow us to rewrite (C-1.12) as
We may now solve for δg μ ρ R ρ ν σ λ R α β γ κ η μ ν α β η σ λ γ κ, to obtain
This equation allows us to rewrite the term displayed in (C-1.9) as
Thus the first and second term on the right-hand side of the intermediary step in (C-1.8) cancel out. All told, we finally obtain
which from Problem 1.5 (i), may be rewritten in the form \(4\partial _{\sigma }(\sqrt{-g}\,\xi ^{\sigma })\), and is a total derivative.
Appendix D: Invariant Products of Three Riemann Tensors
In this appendix we show that the invariant products of three Riemans tensors on the mass shell, i.e., when R μ ν = 0 and hence also R = 0, are necessarily proportional to the invariant
At the very outset, we note the following:
-
(1)
The condition R μ ν = 0 (R = 0), and the anti-symmetry conditions of R μ ν σ λ in the interchange μ ↔ ν or of σ ↔ λ, imply that in order to generate a non-trivial, i.e., non-vanishing, invariant out a product of three Riemann, no two indices within each of the Riemann tensors may be set equal.
-
(2)
Exactly two indices must be common between any two of Riemann tensors in the product. If four indices are common between two of the Riemann tensors are common, no non-trivial invariant may be formed out of this scalar and the third Riemann fourth rank tensor. If three indices are common between two Riemann tensors, no non-trivial invariant may be constructed out of this second rank tensor and the third Riemann fourth rank tensor. Similarly if only one index is common between two of the Riemann tensors, no non-trivial invariant may be constructed of this sixth rank tensor and the third Riemann fourth rank tensor. Again we use the mass shell condition R μ ν = 0 (R = 0).
A direct application of the cyclic relation of the permutation of the indices of the Riemann tensor in the first equation in (1.1.41), which holds irrespective of the mass shell condition, gives
from which
Upon relabeling the indices on the right-hand side of the above equation, and using the elementary properties in (1.1.42), the above equation leads to
Upon multiplying the latter by R κ ρ μ ν gives the identity
where we have used the relation R λ σ κ ρ = −R σ λ κ ρ on the right-hand side of the above equation. With relabeled indices, we may apply the identity in (D-1.3) to R κ ρ μ ν R μ ν σ λ in the above equation to obtain
More appropriately, by relabeling indices, this may be rewritten as
We denote the left-hand side of the above equation by
Due to the identity in (D-1.4), and the fact that R μ ν σ λ is odd in the interchange μ ↔ ν, or of σ ↔ λ, and even in the interchange (μ, ν) ↔ (σ, λ), it is sufficient to establish that the following two invariants
are, in turn, also proportional to the invariant in (D-1.1). [ Note that in the products of the Riemann tensors corresponding to the ones in (D-1.9) and the one in (D-1.8) any two of the Riemann tensors, in a given product, have exactly two indices in common.]
To the above end, consider the invariant R μ ν σ λ R σ λ κ ρ R κ ρ μ ν , and concentrate on the indices in R μ ν ▪ ▪ R σ λ ▪ ▪ R κ ▪ ▪ ▪ , suppressing the other indices. Using the fact that a totally anti-symmetric tensor with five indices must be zero in four dimensional spacetime, Footnote 90 we can construct the following totally anti-symmetric tensor in four dimensions:
Inserting the remaining fixed indices, for example, in R μ ▪ ▪ ▪ , R ν ▪ ▪ ▪ , we obtain from the original expression R μ ρ μ ν , R ν ρ μ ν , which vanish on the mass shell. This means that the second and third lines in the above equation do not contribute on the mass shell. Similarly, within the first pair of round brackets only the first term gives a non-zero contribution. Within the fourth pair of round brackets, only the second and fourth terms are non-vanishing, and in the fifth pair of round brackets, only the third and the fifth terms give non-zero contributions. All told, (D-1.10) implies that
which upon relabeling the indices on the right-hand side of the equation, and using the elementary properties of the Riemann curvature tensor in (1.1.42), we obtain the following identity
Or more appropriately,
establishing that the first invariant in (D-1.9) is proportional to the invariant in question on the right-hand of the above equation.
On the other hand, we may rewrite the basic identity in (D-1.13) as
where in writing the fourth equality, we have relabeled the indices of the first term, and in writing the fifth equality we have used the identity in (D-1.7) for the term just mentioned. The above equation gives for the very last on its extreme right-hand side the identity
establishing that the second invariant in (D-1.9) is proportional to the invariant in question as well.
The remarkable property of invariant products of three Riemann curvature tensors, on the mass shell, being proportional to the invariant product in (D-1.1) is a key identity in investigating the divergent part of quantum general relativity in the two-loop contribution as discussed above within the text. Footnote 91
Appendix E: Bekenstein-Hawking Entropy Formula of a Black Hole
The Bekenstein-Hawking Entropy formula Footnote 92 was briefly discussed in the introduction to this chapter. It relates the entropy of a Black Hole (BH) to the surface area of the event horizon. It arises as a consequence of investigations by HawkingFootnote 93 that a BH is a thermodynamic object, it radiates and has a temperature associated with it. Footnote 94 To find its temperature of a BH, note that the so-called thermodynamic partition function e−β H, in performing averages (by taking a trace), where H is the Hamiltonian of a system,Footnote 95 may be obtained from the time development unitary operator e − i t H∕ℏ, by the substitution: t → −iℏ β. The trace operation gives rise to a periodicity condition on β.
The infinitesimal distance squared in Minkowski space, i.e., in flat spacetime is given by ds 2 = −η μ ν dx μdx ν, with metric η μ ν = diag[ −1, 1, 1, 1]. For a spherically symmetric (uncharged, non-rotating) body of mass M, for example, the gravitational effect of the body causes a curvature of spacetime around it, which amounts in replacing the line element squared by ds 2 = −g μ ν dx μdx ν, where g μ ν is the corresponding metric which takes this distortion into account. The line element squared outside the body now takes the form
where the point r = R BH = 2 GN M∕c2 at which the coefficient of (dr)2 blows out defines the radius of the BH, specifying the location of the horizon.Footnote 96
The above metric is referred to as the Schwarzschild metric, and R BH is also referred to as the Schwarzschild radius.
We note that near the horizon \(r\gtrsim 2\,\mathrm{G_{N}}M/\mathrm{c}^{2}\)
This suggests that near the horizon, we may define the variable
carry out the transformation t → −iℏ β, as discussed earlier, to obtain
The first two terms represent the line element squared in a two dimensional space (a disc), with [ ℏc3∕4GN M ]β representing an angle. For [ ℏc3∕4GN M ] β = 2π and β = 1∕kB T, where kB is the Boltzmann constant, the temperature of the black hole is thus given byFootnote 97
where note that a very massive black hole is cold.
Recall that entropy S represents a measure of the amount of disorder with associate encoded information, and invoking the thermodynamic interpretation of a BH, we may write
which upon integration with boundary condition that for M → 0, S → 0, gives the celebrated result
referred to as the Bekenstein-Hawking Entropy formula of a BH, where A is the (surface) area of the horizon. Using the fact that GN ℏ∕c3 = ℓ P 2 denotes the Plank length squared, we may appropriately rewrite (E-1.7) as
As mentioned earlier, the proportionality of entropy to the area rather than to the volume of a BH horizon should be noted.
Problems
1.1. Establish the anti-symmetry property:
e β μ (∂ ν e α μ +Γ ν σ μ e α σ) = − e α μ (∂ ν e β μ +Γ ν σ μ e β σ),
obtained by simply interchanging the Lorentz indices α, β.
1.2. Show that:
1.3. Show that:
1.4. Establish the equalities:
1.5.
-
(i)
For a vector field ξ μ, show that \(\sqrt{-g}\,\nabla _{\mu }\xi ^{\mu }\, =\, \partial _{\mu }(\sqrt{-g}\,\xi ^{\mu })\).
-
(ii)
For a scalar field ϕ, show that \(\sqrt{-g}\,g^{\mu \nu }\partial _{\mu }\phi \partial _{\nu }\phi \, = -\sqrt{-g}\,\phi (g^{\mu \nu }\nabla _{\mu }\!\nabla _{\nu }\phi )\), up to a total derivative.
-
(iii)
For a fixed real scalar field ϕ, show that \(\updelta [\sqrt{-g}\,(g^{\mu \nu }\partial _{\mu }\phi \,\partial _{\nu }\phi + m^{2}\phi ^{2})]\) is given by
$$\displaystyle{ \sqrt{-g}\,[\partial _{\mu }\phi \,\partial _{\nu }\phi -\frac{1} {2}\,g_{\mu \nu }(\partial ^{\rho }\phi \,\partial _{\rho }\phi + m^{2}\phi ^{2})]\updelta g^{\mu \nu }. }$$
1.6. Use (1.1.14) to infer that although Γ ρ ν γ is not a tensor, the variation δΓ ρ ν γ is a tensor.
1.7. Show that the infinitesimal variations of the Riemann tensor, the Ricci tensor and the connection may be written as:
1.8. Establish that \(\sqrt{-g}\,g^{\mu \nu }\,\updelta R_{\mu \nu }\, =\, 0\), up to a total derivative.
1.9. Verify that \(\updelta (\sqrt{-g}g^{\mu \nu }R_{\mu \nu })\, =\, \sqrt{-g}(R_{\mu \nu } -\frac{1} {2}\,g_{\mu \nu }R)\updelta g^{\mu \nu }\).
1.10. Derive the following fundamental equalities which hold true up to total derivatives. These are generalization of results to be used in partial integrations in curved spacetime.
for n rank and n + 1 rank tensors \(T^{\mu \,_{1}\ldots \mu \,_{k}\mu \,_{k+1}\ldots \mu \,_{n}}\), \(S_{\mu \,_{1}\ldots \mu \,_{k}}\,^{\mu }\,_{ \mu \,_{k+1}\ldots \mu \,_{n}}\), respectively,
for n + 1 rank and n rank tensors \(T^{\mu \,_{1}\ldots \mu \,_{k}\mu \mu \,_{k+1}\ldots \mu \,_{n}}\), \(S_{\mu \,_{1}\ldots \mu \,_{k}\mu \,_{k+1}\ldots \mu \,_{n}}\), respectively. In particular, for n = 0, i.e., for a scalar field ϕ, (i) reads
1.11. The action of a massive real scalar field is defined by:
-
(i)
Derive the field equation for the field ϕ.
-
(ii)
The energy-momentum tensor is defined in (1.2.22) by
$$\displaystyle{ T_{\mu \nu } = - \frac{2} {\sqrt{-g}}\, \frac{\updelta } {\updelta g^{\mu \nu }}W_{\mathrm{matter}}.\;\;\;\;\;\mathrm{Find}\;\;T_{\mu \nu }. }$$ -
(iii)
Show that ∇ μ T μ ν = 0, ∇ ν T μ ν = 0.
1.12. Show that the action in (1.3.36) of first order formulation of linearized gravity is invariant under the infinitesimal gauge transformations:
1.13. Show that the expressions in (1.4.3) and (1.4.4) are the same.
1.14. Find δΓ μ ν σ up to second order in h μ ν , with the latter describing a fluctuation about a background metric g μ ν .
1.15. Given the tensor \(\eta ^{\mu \nu \lambda \sigma } =\varepsilon ^{\mu \nu \lambda \sigma }/\sqrt{-g}\), with ɛ μ ν λ σ totally anti-symmetric, and ɛ 0123 = +1. Show that, under an infinitesimal variation δg ρ κ, δη μ ν λ σ = (1∕2)η μ ν λ σ g ρ κ δg ρ κ = − (1∕2)η μ ν λ σ g ρ κδg ρ κ .
1.16. Show that R μ ν ρ σ R λ κ γ ε η μ ν λ κ η ρ σ γ ε \(= 4\big(4R_{\mu \nu }R^{\mu \nu } - R_{\mu \nu \rho \sigma }R^{\mu \nu \rho \sigma } - R^{2}\big)\).
1.17. Show that η ρ σ α λ∇ λ R μ ν ρ σ = 0.
1.18. Show that ∇ λ η μ ν α β = 0.
1.19. Show that the solution of the equation in (1.9.10) is given by the one in (1.9.11), thus leading to the explicit expression for the holonomy in (1.9.12).
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Manoukian, E.B. (2016). Introduction to Quantum Gravity. In: Quantum Field Theory II. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-33852-1_1
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