Energy conditions in arbitrary dimensions

Energy conditions for matter fields are comprehensively investigated in arbitrary $n(\ge 3)$ dimensions without specifying future and past directions locally. We classify an energy-momentum tensor into $n$-dimensional counterparts of the Hawking-Ellis type I to IV, where type III is defined by a more useful form than those adopted by Hawking and Ellis and other authors to identify the type-III energy-momentum tensor in a given spacetime. We also provide necessary and sufficient conditions for types I and II as inequalities for the orthonormal components of the energy-momentum tensor in a canonical form and show that types III and IV violate all the standard energy conditions. Lastly, we study energy conditions for a set of physically motivated matter fields.


Introduction
In general relativity, energy conditions for an energy-momentum tensor T µν play a central role in proving powerful theorems independent of the concrete forms of matter fields, which in turn show a deep relation between geometry and matter configurations. The extensive work dedicated to this subject has not declined throughout the years even at the classical (and semi-classical) level, as revealed by very recent reviews [1,2] and articles [3][4][5]. Moreover, there are new developments at the quantum level in curved spaces such as those presented in Ref. [6]. Indeed, due to its importance, this topic has been discussed in widely used textbooks.
In the well-known book by Hawking and Ellis [7], a four-dimensional energy-momentum tensor is classified into four types (types I, II, III, and IV) based on the classification of real second-rank symmetric tensors 1 and necessary and sufficient conditions for the standard energy conditions are presented (see page 89 in Ref. [7], section 5 in Ref. [15], and Ref. [2]). Among them, types III and IV are unphysical because they do not satisfy the null energy condition. Hence, only types I and II are physically important and a variety of matter fields are included in these two types. However, unfortunately, the proofs for the necessary and sufficient conditions are absent in Ref. [7] and it is difficult to find them in the literature.
Remarkably, the classification of energy-momentum tensors into four types is also true in arbitrary n(≥ 3) dimensions [16][17][18]. Considering the Jordan canonical matrices, the classification of real second-rank symmetric tensors in five dimensions was done in Ref. [19] and then generalized in arbitrary dimensions by the same authors [16]. A different approach for the classification in five dimensions was used in Ref. [20], which can be extended by induction into n dimensions [18]. Theorem 2 in Ref. [17] claims that only types I and II also satisfy the dominant energy condition in n dimensions but again without a proof.
Indeed, it has been widely believed that all physically reasonable matter fields, such as a scalar field with potential in a certain form or a Maxwell field, respect standard energy conditions. In the literature, this has been proven individually for each matter field. For example, it has been shown in section 5.4 in Ref. [21] that a massless scalar field satisfies the dominant energy condition. Also, equivalent expressions of the standard energy conditions for a perfect fluid were derived in section 2.1 in Ref. [22]. Both of them were proven in four dimensions but generalizations in arbitrary dimensions are not difficult at all. In contrast, although it has been proven in the appendix in Ref. [23] that a Maxwell field satisfies the dominant energy condition in four dimensions, its higher-dimensional generalization is not so obvious. In Ref. [24], it was also proven in four dimensions that an SU(2) Skyrme field and its Born-Infeld generalization satisfy the dominant energy condition as well as the strong energy condition. Although there are many more physically motivated matter fields, it seems difficult to find proofs for them in arbitrary dimensions in the literature. Also, sometimes such proofs are presented under the assumption of time-orientability of spacetime.
Under these circumstances, in the present paper, we tidy up and present various known claims together with possible new ones related to the energy conditions with elementary proofs in arbitrary n(≥ 3) dimensions without assuming time-orientability of spacetime. After reviewing the standard energy conditions in the next section, we will first derive the most general canonical forms of the n(≥ 3)-dimensional counterparts of the Hawking-Ellis type-I-IV energy-momentum tensors in Sec. 3. Our expression of type III contains additional non-zero components to the one adopted by other authors [2], which are useful to identify the type-III energy-momentum tensor in a given spacetime. In the same section, we will provide, by means of a series of propositions, necessary and sufficient conditions for the energy conditions for type-I and II energy-momentum tensors and show that types III and IV violate the null energy condition. In Sec. 4, we will study the energy conditions for various physically motivated matter fields. Our results will be summarized in the final section. In Appendix A, we summarize the classification of real second-rank symmetric tensors in arbitrary dimensions with Lorentzian signature provided in Ref. [16].
Other types of indices will be specified in the main text.

Standard energy conditions
Let us consider an n(≥ 3)-dimensional matter action written as which gives the energy-momentum tensor T µν for this matter field such that T µν := −2 ∂L m ∂g µν + g µν L m . (2. 2) The standard energy conditions for T µν are as follows: • Null energy condition (NEC): T µν k µ k ν ≥ 0 for any null vector k µ .
We follow the definitions adopted by Hawking and Ellis in Ref. [7], in which k µ and v µ are not assumed to be future-directed 2 . They allow us to use these conditions even in a globally non-time-orientable spacetime. Actually, even if a spacetime is not globally timeorientable, it is always locally time-orientable by defining "future" and "past" directions by two distinct light cones at a given point. An advantage of the above definitions is that they are free from the choice of these causal directions. In the present paper, we will present elementary proofs free from a local choice of future and past directions unless otherwise noted. We note that the classification of energy-momentum tensors presented in the next section is also irrelevant to time-orientability of spacetime because it is purely algebraic. Hence we adopt the definitions of the energy conditions in [7] as a suitable choice.
In the definitions of FEC and DEC, one often simply states that "J µ (= −T µ ν v ν ) is a causal vector" including the case where J µ is a zero vector implicitly [7,15]. However, a zero vector is actually not pointing in any direction, so we write it in the above statement explicitly. T µν = 0 is an example of such an energy-momentum tensor, which is realized not only for vacuum but also for a "stealth" configuration of matter fields 3 .
In the proofs presented in this paper, we use an orthonormal basis. A set of n vectors form an orthonormal basis in the local Lorentz frame in a given spacetime. Here η (a)(b) is the metric in the local Lorentz frame and the metric g µν in the spacetime is given by An orthonormal basis E µ (a) has a degree of freedom provided by the local Lorentz transfor- For a given vector field v µ , one can define the corresponding local Lorentz vector v (a) := v µ E (a) µ , which transforms as a vector under local Lorentz transformations but as a scalar under coordinate transformations. η (a)(b) and its inverse η (a)(b) are respectively used to lower and raise the indices (a) and v The orthonormal components of the energy-momentum tensor are given by For better physical interpretations of the energy conditions, we will use the following lemma.
Lemma 1 Let f (c, d) be a continuous scalar function of two causal vectors c µ and d µ .
Then, f (c, d) ≥ 0 for any set of timelike vectors c µ and d µ is equivalent to f (c, d) ≥ 0 for any set of causal vectors c µ and d µ .
Proof. We consider non-zero causal vectors c µ and d µ in the following general form: where ) 2 with d (0) = 0 are satisfied, with the equality holding in the case where c µ and d µ are null, respectively. Clearly f (c, d) ≥ 0 is satisfied for any set of timelike vectors c µ and d µ if f (c, d) ≥ 0 is satisfied for any set of causal vectors c µ and d µ . To show its inverse, suppose that f (c, d) ≥ 0 in the case where both c µ and d µ are timelike, namely for Then, by continuity, f (c, d) ≥ 0 keeps holding in the limit 2 from below and hence f (c, d) ≥ 0 holds for any set of causal vectors c µ and d µ .
While DEC clearly implies WEC, WEC implies NEC by Lemma 1. Therefore, if NEC is violated, then WEC and DEC are violated as well. Also by Lemma 1, there are the following equivalent descriptions of WEC, SEC, FEC, and DEC: • WEC2: T µν c µ c ν ≥ 0 for any causal vector c µ [26].
Thus, while NEC means non-negativity of the energy density of matter for any null observer, WEC means that the energy density of matter is non-negative for any causal observer.
SEC is related to the timelike convergence condition (TCC) R µν v µ v ν ≥ 0 in general relativity 4 . The scalar R µν v µ v ν appears in the Raychaudhuri equation for v µ and TCC implies that gravity is essentially an attractive force. In the absence of a cosmological constant, Einstein equations show that SEC and TCC are equivalent. On the other hand, since J µ := −T µ ν v ν is an energy current vector for an observer corresponding to v µ , FEC means that such an energy current is absent or does not propagate faster than the speed of light.
We note that DEC is equivalent to WEC with FEC. In a time-orientable region of spacetime, other equivalent descriptions of DEC are available: • DEC3: For any future-directed timelike vector v µ , J µ = −T µ ν v ν is a future-directed causal vector or a zero vector [22,31].
• DEC4: For any future-directed causal vector c µ ,J = −T µ ν c ν is a future-directed causal vector or a zero vector.
• DEC5: T µν u µ v ν ≥ 0 holds for any set of future-directed timelike vectors u µ and v µ .
Proof. DEC and DEC2 are equivalent by Lemma 1. DEC3 and DEC4 are shown to be equivalent in a similar manner to the proof of Lemma 1. DEC5 and DEC6 are equivalent by Lemma 1. So we complete the proof by showing that DEC, DEC3, and DEC5 are equivalent in a time-orientable region of spacetime. In the following, we write u µ , v µ , and J µ such that Since we now consider a time-orientable region of spacetime, we set E µ (0) being futuredirected without loss of generality.
We first prove that DEC and DEC3 are equivalent. Since v µ is timelike, we can set the frame such that v (i) = 0 for all i = 1, 2, · · · , n−1 by a local Lorentz transformation without loss of generality. Suppose that DEC is satisfied and then we have n−1 i=1 (j (i) ) 2 ≤ (j (0) ) 2 . Also in this frame, we have v (0) j (0) ≥ 0 from T µν v µ v ν ≥ 0. These two inequalities show that j (0) ≥ 0 is satisfied for v (0) > 0, where j (0) = 0 holds if and only if J µ is a zero vector. This implies that −T µ ν v ν is future-directed or a zero vector for any future-directed v µ and hence DEC3 is satisfied.
Inversely, we show that DEC3 implies DEC. We consider the frame with v (i) = 0 for all i = 1, 2, · · · , n − 1 without loss of generality. DEC3 implies that J µ is a future-directed causal vector or a zero vector for v (0) > 0 and hence j (0) ≥ 0 holds. Since we have Next we prove that DEC3 and DEC5 are equivalent. Since we have set E µ (0) as being future-directed, we have u (0) > 0 and v (0) > 0. First we show that DEC3 implies DEC5. For any given u µ , we can set the frame such that u (i) = 0 for all i = 1, 2, · · · , n − 1 without loss of generality. Since DEC3 implies j (0) ≥ 0 so that T µν u µ v ν = −u µ J µ = u (0) j (0) ≥ 0 holds in this frame, DEC3 implies DEC5.
In order to study energy conditions for concrete matter fields, the following lemma regarding the weighted sum of several distinct stress-energy tensors {T 1 µν , T 2 µν , · · · , T p µν } is sometimes useful; it will be used in Sec. 4.

Lemma 3
Let Π A (A = 1, 2, · · · , p) be a set of non-negative functions of the spacetime coordinates. If T A µν independently satisfies NEC, WEC, or SEC for each A = 1, 2, · · · , p, then T µν = p A=1 Π A T A µν satisfies the same energy condition.
µ be energycurrent vectors associated with T A µν . If T A µν independently satisfies FEC or DEC for each A and j A (0) j B (0) ≥ 0 holds for any A, B ∈ 1, 2, · · · , p, then T µν = p A=1 Π A T A µν satisfies the same energy condition.
Proof. The statement for NEC, WEC, and SEC is obvious. To prove for FEC, we use the following expression: Suppose that T A µν satisfies FEC for all A and then we have Then, the following expression shows J µ J µ ≤ 0 and hence FEC holds. The statement is true for DEC because DEC is a combination of WEC and FEC.
Note that Lemma 3 for DEC3 has been claimed in Ref. [24] under the assumption of time-orientability of spacetime. Actually, the condition j A (0) j B (0) ≥ 0 for DEC in Lemma 3 is not required in a time-orientable region of spacetime, as shown below.
Lemma 4 Let Π A (A = 1, 2, · · · , p) be a set of non-negative functions of the spacetime coordinates. If T A µν independently satisfies DEC for each A in a time-orientable region of spacetime, then T µν = p A=1 Π A T A µν satisfies DEC as well.
Proof. In a time-orientable region of spacetime, we can set E µ (0) as being future-directed without loss of generality. Suppose that T A µν satisfies DEC3 for all A and then j A (0) ≥ 0 holds for all A. This implies j A (0) j B (0) ≥ 0 for any set of A and B and therefore T µν = p A=1 Π A T A µν satisfies DEC by Lemma 3.

Hawking-Ellis classification of energy-momentum tensors
The Hawking-Ellis classification of energy-momentum tensors [7] is based on the classification of real second-rank symmetric tensors (such as the Ricci tensor) defined on a fourdimensional spacetime with Lorentzian signature [9][10][11][12][13][14] (see also section 5.1 in Ref. [15]). It is remarkable that such symmetric tensors and hence the energy-momentum tensor can also be classified into four types in arbitrary n(≥ 3) dimensions [16][17][18]. (The proof in [16] is summarized in Appendix A.) In this section, we analyze these four types of the energymomentum tensor, which can be specified using the Segre notation in Appendix A.
The energy-momentum tensor is classified by the extent to which its orthonormal components µ n µ . Although n (a) and n µ are certainly eigenvectors of T (a)(b) and T µν , respectively, n (a) is not a vector under coordinate transformations. For this reason, for distinction, we call n (a) and n µ a "local Lorentz eigenvector" and an "eigenvector", respectively, in the present section. The eigenvalue λ is determined by the following algebraic equation: As well known, two different local Lorentz eigenvectors n (a) 1 and n (a) 2 for different eigenvalues λ 1 and λ 2 are orthogonal, namely n 1(a) n (a) 2 (= n 1µ n µ 2 ) = 0, which is shown by constructing We will also study the energy conditions for all types of energy-momentum tensors. In the proofs presented below, we will write an arbitrary timelike vector v µ in the following normalized form: where a i (i = 1, 2, · · · , n−1) and γ( = 0) are arbitrary functions of the coordinates satisfying Also, we will write an arbitrary null vector k µ as whereā i (i = 0, 1, 2, · · · , n − 1) are arbitrary functions of the coordinates satisfyinḡ We will use T µν = T (a)(b) E (a)µ E (b)ν in the following proofs.

Type I
The n-dimensional counterpart of the Hawking-Ellis type-I energy-momentum tensor corresponds to the Segre type [1, 11 · · · 1], which admits one timelike eigenvector and (n − 1) spacelike eigenvectors of T µν . By a local Lorentz transformation, we can set the orthonormal bases E µ (a) identified by these eigenvectors n µ with normalization. Then, the orthonormal components of the type-I energy-momentum tensor are written as (3.7) The Lorentz-invariant eigenvalues of T (a)(b) are all non-degenerate and given by λ = {−ρ, p 1 , p 2 , · · · , p n−1 }. Their corresponding local Lorentz eigenvectors are n (a) = {t (a) , w 1(a) , w 2(a) , · · · , w n−1(a) }, respectively, where with which T (a)(b) can be written as Equivalent expressions of the standard energy conditions for the type-I energy-momentum tensor (3.7) are given by Here i = 1, 2, · · · , n − 1. The proofs in four dimensions are available in Section 2.1 of Ref. [22], but we will present more detailed ones below.

Proposition 3 SEC for type I is equivalent to
Proof. Using Eq. (3.3), we rewrite SEC as where we used Eq. (3.4). Since Eq. (3.14) is similar to Eq. (3.13), we can prove this proposition in the same way as Proposition 2.

Type II
The n-dimensional counterpart of the Hawking-Ellis type-II energy-momentum tensor corresponds to the Segre type [211 · · · 1], which admits one doubly degenerate 5 null eigenvector n µ =k µ and (n − 2) spacelike eigenvectors of T µν . In this case, we cannot let a coordinate axis point in the direction ofk µ . However, we can set coordinates such thatk µ lies in the plane spanned by E µ (0) and E µ (1) . Then,k µk µ =k (a)k (a) = 0 showsk (0) = ±k (1) ( = 0). Since we can reverse the direction of E µ (1) , we can setk (0) = −k (1) without loss of generality. Substituting this into Eq. (3.1) with a = 0 and 1, we obtain Thus, introducing new variables ν := T (0)(1) and ρ := −λ, we can write the orthonormal components of the type-II energy-momentum tensor in the following form: In the expression of the type-II energy-momentum tensor in Ref. [7] for n = 4, ν is chosen to be ν = ±1 but this is unhelpful, as pointed out in Ref. [2].
p n−1 } are non-degenerate and their corresponding local Lorentz eigenvectors are respectively given by n (a) = {w 2(a) , · · · , w n−1(a) } in Eq. (3.8), the eigenvalue λ = −ρ is doubly degenerate and its local Lorentz eigenvector is given by n (a) =k (a) , wherē In terms of these local Lorentz eigenvectors, T (a)(b) can be written as Equivalent expressions of the standard energy conditions for the type-II energy-momentum tensor (3.18) are given by • WEC: ν ≥ 0, ρ + p i ≥ 0, and ρ ≥ 0.
Proof. Using Eq. (3.3), we rewrite SEC as for any a i satisfying Eq. (3.4). Since Eq. (3.33) is similar to Eq. (3.27), this proposition can be proved as was done in Proposition 7.
To demonstrate the usefulness of the canonical expression (3.37), let us consider how to find orthonormal basis vectors in the following three-dimensional spacetime which is compatible with gyratons, namely, a matter field in the form of a null dust fluid (or equivalently a pure radiation) with an additional internal spin [33]. Since g rr is vanishing, one easily finds a null vector Then, one finds another null vector l µ satisfying k µ l µ = −1 and subsequently a unit spacelike vector m µ satisfying k µ m µ = l µ m µ = 0 such that Therefore, the simplest orthonormal basis vectors in the spacetime (3.38) are given by For gyratons in the spacetime (3.38), the non-zero components of T µν are T uu and T ux (= T xu ), which represent a classical null radiation and an inner gyratonic angular momentum, respectively. As shown in Ref. [33], the orthonormal components of T µν with the simplest basis vectors (3.41) are type III in the canonical form (3.37) such that where By contrast, it is not easy to find orthonormal basis vectors in the spacetime (3.38) leading to T (a)(b) with vanishing ν. For this reason, we adopt Eq. (3.37) as a canonical expression of the type-III energy-momentum tensor in the present paper 8 .
Proposition 11 NEC is violated for type III if ζ = 0.
Proposition 12 SEC is violated for type III if ζ = 0.
In the case of ρ = 0, Eq. (3.51) becomes which is not satisfied if ζ = 0. In the case of ρ = 0 and ζ = 0, Eq. (3.51) becomes where θ 0 is defined by tan θ 0 = (2νρ − ζ 2 )/(4ζρ). In the limit of α → 1 from below with θ = 0, in which the first term in the left-hand side is negligible, Eq. (3.53) gives where ε is a small positive constant. Since the limit α → 1 from below corresponds to ε → 0 + , there is always a finite range of θ for any given θ 0 , such that inequality (3.54) is violated.

Energy conditions for canonical matter fields
In this section, we study energy conditions for a variety of physically motivated matter fields without assuming time-orientability of spacetime. In the following proofs, we will write timelike vectors u µ and v µ as

Perfect fluid and cosmological constant
A perfect fluid is phenomenologically defined by the following energy-momentum tensor: where ρ is the energy density, p is a pressure, and u µ is a normalized n-velocity of the fluid element such that u µ u µ = −1. A cosmological constant Λ corresponds to the case with ρ = Λ and p = −Λ.
Proposition 17 The standard energy conditions for a perfect fluid (4.2) are equivalent to • NEC: ρ + p ≥ 0.
Proof. Since u µ is timelike, we can set E µ (0) such that E µ (0) = u µ without loss of generality. Then, we have and hence This is type I with the same ρ and p i = p for any i = 1, 2, · · · , n − 1. Thus, the result follows from Propositions 1-5.

Proposition 18
For any value of a cosmological constant Λ, NEC and FEC are respected. While WEC and DEC are equivalent to Λ ≥ 0, SEC is equivalent to Λ ≤ 0.

Null dust fluid
A null dust fluid is phenomenologically defined by the following energy-momentum tensor: where µ is the energy density and k µ is a null vector; namely, k µ k µ = 0 holds.

Proposition 19
For a null dust fluid (4.5), FEC is respected and NEC, WEC, SEC, and DEC are all equivalent to µ ≥ 0.
Proof. We use a pseudo-orthonormal basis defined bȳ Since k µ is null, we can set E µ (0) such that k µ = ΩĒ µ (0) with a non-vanishing scalar function Ω without loss of generality.

Minimally coupled scalar field
The Lagrangian density for a minimally coupled scalar field φ with self-interacting potential V (φ) is given by where (∇φ) 2 := (∇ ρ φ)(∇ ρ φ) and the parameter ε is either 1 (for a real scalar field) or −1 (for a ghost scalar field). The equation of motion and the energy-momentum tensor for φ are respectively given by Proposition 20 For a minimally coupled real scalar field (4.11), NEC is respected if and only if ε = 1. Sufficient conditions for other energy conditions are as shown in the following table, where (R) and (V) stand for "Respected" and "Violated", respectively.
µ , where Φ (a) (a = 0, 1, · · · , n − 1) are functions, and then we have (4.12) For any given null vector k µ , we can set the frame such that k µ = ΩĒ µ (0) with a nonvanishing scalar function Ω without loss of generality by a local Lorentz transformation. In this frame, we have and hence (4.14) On the other hand, for any given timelike vector v µ , we can set the frame such that v (i) = 0 for all i = 1, 2, · · · , n − 1 by a local Lorentz transformation. In this frame, we have Using the following expression we compute We also obtain The proposition follows from Eqs.

Maxwell field
The Lagrangian density for the Maxwell field A µ is given by where α is a real constant, and the Faraday tensor F µν is F µν := ∂ µ A ν − ∂ ν A µ . The field equations and the energy-momentum tensor for a Maxwell field are respectively given by Proof. We write F µν in the orthonormal frame such as [µ E [µ E [µ E [µ E (4) ν] + · · · + 2f (2)(n−1) E [µ E [µ E (4) ν] + · · · + 2f (n−2)(n−1) E For any given timelike vector v µ , we set the frame such that v (i) = 0 for all i by a local Lorentz transformation without loss of generality. In this frame, we have This is a spacelike vector. We can still use a freedom of the Lorentz transformation in the spacelike section, namely spacelike rotation, such that v µ F µν is pointing the direction of E (1) ν , in which frame we have f (0)(i) = 0 for i = 2, 3, · · · , n − 1 and hence In this frame, we have and which shows that WEC (and hence also NEC) is respected for α > 0. On the other hand, we obtain Then, after a slightly tedious computation, we obtain (4.28) Equations (4.27) and (4.28) show that FEC is respected for α > 0. Since both WEC and FEC hold, DEC is respected.
Lastly, using the following expression: we obtain which shows that SEC is respected for α > 0.

Proca field
The Lagrangian density for the Proca field is given by where α and m are real constants. The field equations and the energy-momentum tensor for a Proca field are respectively given by Proof. Let us write the energy-momentum tensor (4.33) such that T µν =T µν + αm 2 τ µν , whereT T µν is the energy-momentum tensor for a Maxwell field and satisfies all the standard energy conditions for α > 0 by Proposition 21, so we focus on τ µν hereafter.
As in the proof of Proposition 21, we consider the frame such that v (i) = 0 for all i. In this frame, v µ and A µ are expressed as and hence we have The above equations show that τ µν satisfies WEC and SEC. Thus, by a combination of Proposition 21 and Lemma 3, the Proca field (4.33) with α > 0 also satisfies WEC and SEC.

Maxwell(Proca)-dilaton field
The Lagrangian density for a Proca field coupled with a dilaton φ with a potential V (φ) is given by where γ is a real coupling constant. The field equations and the energy-momentum tensor for this Proca-dilaton field are respectively given by We can write Eq. (4.45) as T µν = T φ µν + e −γφ T P µν , where T φ µν and T P µν are the energymomentum tensor for a minimally coupled scalar field (4.11) and that for a Proca field (4.33) with α = 1, respectively.
and hence j φ (0) j P (0) ≥ 0 holds for ε = 1 and V (φ) ≥ 0. Thus, the statement for FEC and DEC is shown by a combination of Propositions 20 and 22 and Lemma 3.

Yang-Mills field
Let us consider the Yang-Mills field with the non-Abelian symmetry group SU(N). The gauge field (or gauge potential) A is written as where τ a (a = 1, 2, · · · , N 2 − 1) are the generators of the su(N) Lie algebra satisfying Here f abc (= f [ab]c ) are structure constants of su(N). We note that the transition between contravariant and covariant components is trivial for indices a, b, and c; namely, τ a = τ a or f abc = f abc holds. The Yang-Mills field strength F µν is defined by where ζ is constant. Its matrix-valued components F a µν defined by F µν = F a µν τ a are given by The Lagrangian density for a Yang-Mills field is given by where α is a real constant 9 . The Yang-Mills equations and the energy momentum tensor for a Yang-Mills field are respectively given by For later use, we write Eq. (4.54) as T µν = α N 2 −1 a=1 T a µν , where T a µν is defined by without using the Einstein summation convention for a in the right-hand side. Hereafter, we will not use this convention for the index a.
In the following proof, we consider the frame such that v (i) = 0 for all i holds without loss of generality. Here we note that, as in the proof of Proposition 21, by using a remaining freedom of spacelike rotation of the orthonormal frame, we can still set one of the spacelike vectors v µ F a µν (a = 1, 2, · · · , N 2 − 1) to point in the direction of E (1) ν , which drastically simplifies the proof of Proposition 21; however, one cannot do this for all a simultaneously in the following proof. 9 The second equality is shown as .

Proposition 24
The Yang-Mills field (4.54) with α > 0 satisfies all the standard energy conditions.
Proof. Since the gauge field A a µ does not appear explicitly in its expression, T a µν for each a satisfies all the standard energy conditions as shown in the proof of Proposition 21 with α = 1. So, writing the energy-current vector associated with T a µν as J a µ : µ , we show j a (0) j b (0) ≥ 0 for any set of a and b.
We write the orthonormal components of F a µν as From the following expression and Eq. (4.57), we obtain which shows j a (0) j b (0) ≥ 0 for any set of a and b. Thus, by Lemma 3, all the standard energy conditions are satisfied for α > 0.

Summary
In the present paper, we have investigated energy conditions for matter fields in arbitrary n(≥ 3) dimensions. We have first tidied up and presented various known and possibly new claims related to the energy conditions. Then we have derived the most general canonical forms of the n-dimensional counterparts of the Hawking-Ellis type-I-IV energy-momentum tensors. Among them, our expression of type III contains additional non-zero components to the one adopted by other authors [2]. Although those components can be set to zero by local Lorentz transformations, our expression is useful to identify the type-III energymomentum tensor in a given spacetime. We have demonstrated this in a three-dimensional spacetime with a gyratonic matter.
We have also provided necessary and sufficient conditions for the standard energy conditions for the type-I and II energy-momentum tensors. These conditions have been presented as inequalities for the orthonormal components of the energy-momentum tensor in a canonical form. We have also shown that type-III and IV energy-momentum tensors violate the null energy condition. In all the proofs, we have not assumed time-orientability of spacetime.
Lastly, we have studied the energy conditions for a set of physically motivated matter fields. Among others, we have shown that the Maxwell field satisfies all the standard energy conditions in arbitrary dimensions. This result has been extended to a Proca field coupled with a dilaton field and also to a Yang-Mills field. Our result shows that powerful theorems in general relativity based on the energy conditions can be adopted with these matter fields. Nevertheless, many other canonical matter fields and also various non-canonical matter fields have been introduced in the modern research. The study of energy conditions for such matter fields is left for future investigations. classification as the Segre classification for simplicity in spite of the fact that the original classification by Segre [8] was performed with Euclidean signature.

A.1 Set up the problem
Let us consider an eigenvalue problem T µ ν v ν = λδ µ ν v ν at a spacetime point P for a real second-rank symmetric tensor T µν in n(≥ 3) dimensions, which is not necessarily an energymomentum tensor. Because of T µ ν = T ν µ in general, the eigenvalue equations T µ ν v ν = λδ µ ν v ν can be considered as a single matrix equation, where T µ ν is identified as an n × n matrix that is not symmetric in general, and the eigenvector v ν and δ µ ν are identified as an n × 1 matrix and an n × n identity matrix, respectively.
In a different coordinate systemx µ (=x µ (x)), the eigenvalue equations are written as T µ νv ν = λδ µ νv ν , whereT µ ν is given bȳ Around the spacetime point P , the transformationsx µ =x µ (x) and their inverse become linear, and then ∂x σ /∂x ν | P and ∂x µ /∂x ρ | P can be considered as a constant matrix S σ ν ≡ ∂x σ /∂x ν | P and its inverse matrix (S −1 ) µ ρ ≡ ∂x µ /∂x ρ | P , respectively. In terms of these matrices, Eq. (A.1) is written asT With a notation for simplicity such that T ≡ T µ ν , v ≡ v µ , I ≡ δ µ ν , and S ≡ S σ ν , where I is an identity matrix, the eigenvalue equation and Eq. (A.2) at a given spacetime point are described as T v = λIv andT = S −1 T S, respectively. Now the problem is, for a given T , how simple the matrixT can be by coordinate transformations defined by S. Since T (≡ T µ ν ) is a real but non-symmetric matrix, it cannot be diagonalized by choosing S as a real orthogonal matrix. Instead, one can adopt the theory of the Jordan canonical form for the eigenvalue equation T v = λIv.

A.2 Jordan canonical form
For later use, we present definitions of a Jordan block and Jordan matrix [34].
The scalar λ appears k times on the main diagonal; if k > 1, there are (k − 1) entries "+1" in the superdiagonal; all other entries are zero. An n × n Jordan matrix J is a direct sum of Jordan blocks J = J n 1 ⊕ J n 2 · · · ⊕ J nq , namely, where q k=1 n k = n.
In the Segre classification of real second-rank symmetric tensors, the following Jordan canonical form theorem is used (see Theorem 3.1.11 in Ref. [34]).
Theorem 1 Let an n × n matrix A be given. Then, there is a non-singular n × n matrix S, positive integers q and n 1 , · · · , n q with q k=1 n k = n, and scalars λ 1 , · · · λ q ∈ C such that A = SJ A S −1 , where the Jordan matrix J A is J A is uniquely determined by A up to the permutation of its direct summands. If A is real and has only real eigenvalues, then S can be chosen to be real.
Since the characteristic polynomial of J A , which is proportional to that of A, is given by the eigenvalue λ k is degenerate if n k ≥ 2 and then its algebraic multiplicity is n k .
By the Jordan canonical form theorem, there exists a non-singular matrix S, called a similarity matrix, such thatT is in the Jordan canonical formT = J T . Such a similarity matrix S can be constructed in the following manner. For each eigenvalue λ = λ k , we define n k contravariant vectors {v k1 , v k2 , · · · , v kn k } satisfying where q k=1 n k = n holds. By Eq. (A.7), v k1 is an eigenvector corresponding to the eigenvalue λ k . In terms of these contravariant vectors, an n × n k matrix V k is defined for each k by with which the similarity matrix S is constructed as S = (V 1 , · · · , V q ). (A.12) By Eqs. (A.7)-(A.10) for each k, all the vectors v k1 , · · · , v kn k (k = 1, · · · , q) are shown to be linearly independent, so that S is a non-singular matrix and hence has its inverse S −1 .

A.3 Segre classification
In the Segre classification, only the dimension of the Jordan blocks and the degeneracies of the eigenvalues are relevant. The Segre type is a list [n 1 n 2 · · · n q ] of the dimensions of the Jordan blocks in the Jordan matrix (A.5). If two eigenvalues are complex conjugates, the symbols Z andZ are used in the list instead of a digit to denote the dimension of a block with a complex eigenvalue. The digits in the list are arranged in such an order that those corresponding to spacelike eigenvectors appear last and the digit corresponding to a timelike eigenvector is separated from the others by a comma. Equal eigenvalues in distinct blocks are indicated by enclosing the corresponding digits inside round brackets.
Lemma 5 Suppose that all the eigenvalues of T are real. Let v r1 be an eigenvector corresponding to an n r -dimensional Jordan block J nr (λ r ) in the Jordan canonical form J T of T . Then, if n r ≥ 2, v r1 is a null vector.
Using Lemma 5, we can show the following proposition [16].

Proposition 25
Suppose that all the eigenvalues of T are real. Then, the Jordan canonical form of T cannot contain (i) more than one Jordan block with dimensions larger than one, and (ii) Jordan blocks with dimensions larger than three.
Proof. First we prove (i) by contradiction. Suppose that the Jordan canonical form of T contains more than one Jordan block with dimensions larger than one. Then, there exist different integers r and s (1 ≤ r, s ≤ q) such that there exist the following sets of equations; and Acting (v s1 ) µ and (v r1 ) µ on Eqs. (A. 19) and (A.21), respectively, and using the symmetry of T µν , we obtain Thus, (v r1 ) µ (v s1 ) µ = 0 holds in the case of λ r = λ s . In the case of λ r = λ s , acting (v s2 ) µ and (v r1 ) µ on Eqs. (A. 19) and (A.22), we also obtain (v r1 ) µ (v s1 ) µ = 0. Since both (v r1 ) µ and (v s1 ) µ are null by Lemma 5, (v r1 ) µ (v s1 ) µ = 0 implies that (v r1 ) µ and (v s1 ) µ are collinear, which contradicts the assumption that the similarity matrix S of T is non-singular.
Next let us also prove (ii) by contradiction. Suppose that the Jordan canonical form of T contains Jordan blocks with dimensions larger than three. Then, there exists an integer r (1 ≤ r ≤ q) such that the following set of equations holds: (A.27) By Lemma 5, (v r1 ) µ is null. In a similar manner to the proof of (i), we can show (v r1 ) µ (v r2 ) µ = 0 and (v r1 ) µ (v r3 ) µ = 0 from Eqs. and hence (v r2 ) µ is null. Since both (v r1 ) µ and (v r2 ) µ are null, (v r1 ) µ (v r2 ) µ = 0 implies that (v r1 ) µ and (v r2 ) µ are collinear, which contradicts the assumption that the similarity matrix S of T is non-singular.
By Propositions 25 and 26, possible Segre types of a real second-rank symmetric tensor T µν defined in an n(≥ 3)-dimensional Lorentzian spacetime are [311 · · · 1], [211 · · · 1], [1, 11 · · · 1], and [ZZ1 · · · 1]. A different proof is also available by showing that T admits at least one real and non-null eigenvector [18]. If the eigenvector is timelike, by a spacelike rotation in the spacelike (n − 1)-dimensional subspace M n−1 orthogonal to this eigenvector, T µ ν can be diagonalized and hence the corresponding Segre type is [1, 11 · · · 1]. If the eigenvector is spacelike, one can reduce the n-dimensional classification to the one in (n − 1) dimensions. Then, by induction, the problem reduces to the well-known four-dimensional classification, which is given for instance in section 5.1 of Ref. [15].