Tensor density
In differential geometry, a tensor density (also known as a relative tensor) is a generalization of a tensor field that transforms under a change of coordinates according to the standard tensor transformation law multiplied by a factor of the determinant of the Jacobian matrix of the transformation raised to a power called the weight w.[1] The weight w is typically an integer (such as $0, \pm 1, \pm 2), though it can be any real number, and when w = 0, the object reduces to an ordinary tensor.[1] Specifically, for a tensor density of type (k, l) with weight w, the components in the new coordinate system \overline{x} transform as \overline{T}^{j_1 \dots j_k}_{i_1 \dots i_l} = T^{s_1 \dots s_k}_{r_1 \dots r_l} \cdot \det\left( \frac{\partial x}{\partial \overline{x}} \right)^w \cdot \prod_{m=1}^k \frac{\partial \overline{x}^{j_m}}{\partial x^{s_m}} \cdot \prod_{n=1}^l \frac{\partial x^{r_n}}{\partial \overline{x}^{i_n}}, where x are the old coordinates and the products account for the usual tensor index transformations.[1] Tensor densities arise naturally in contexts requiring coordinate-independent formulations of integrals, such as volume elements on manifolds.[2] For instance, the coordinate volume element d^n x = dx^1 \wedge \dots \wedge dx^n in n-dimensional space is an n-form that behaves as a tensor density of weight -[1](/page/−1), transforming as d^n \overline{x} = \det\left( \frac{\partial \overline{x}}{\partial x} \right) d^n x to ensure the integral \int f \, d^n x is invariant when f is a scalar.[1] In Riemannian or pseudo-Riemannian geometry, the scalar \sqrt{|\det g|}, where g is the metric tensor, serves as a scalar density of weight +[1](/page/1), enabling the construction of the invariant volume form \sqrt{|\det g|} \, d^n x.[3][4] These objects are particularly prominent in physics, especially in general relativity and field theories, where they facilitate the densitization of tensors for Hamiltonian formulations or covariant integration.[2] A classic example is the Levi-Civita symbol \varepsilon_{i_1 \dots i_n}, which is a tensor density of weight -1 rather than a true tensor, as its transformation includes an extra \det\left( \frac{\partial x}{\partial \overline{x}} \right)^{-1} factor; multiplying by \sqrt{|\det g|} yields the Levi-Civita tensor, which is a proper tensor.[3] Tensor densities also appear in the Hodge star operator on oriented manifolds, where they adjust for metric-induced volume scalings in exterior algebra.[5] In non-orientable manifolds, modified versions like ψ-densities incorporate orientation bundles to allow integration.[1]Motivation and Fundamentals
Role in Coordinate Transformations
In differential geometry, standard tensors, which transform linearly under coordinate changes via the Jacobian matrix without additional scaling factors, fail to preserve volumes or integrals when subjected to diffeomorphisms that distort local metrics or orientations. For instance, the volume element defined by a standard tensor, such as the Cartesian dx dy dz, alters its measure under a nonlinear coordinate transformation, leading to inconsistencies in physical or geometric quantities like mass distribution or flux across manifolds. This limitation arises because diffeomorphisms can stretch or compress regions unevenly, requiring compensation to maintain invariance of integrals over arbitrary domains.[6] The Jacobian determinant addresses this by quantifying the local volume scaling under coordinate transformations, serving as the key factor in the transformation laws for tensor densities. Specifically, it measures the determinant of the partial derivatives of the new coordinates with respect to the old ones, |det(∂x'/∂x)|, which multiplies the original volume element to yield the transformed one, ensuring that densities adjust accordingly for uniform behavior. In this way, tensor densities incorporate powers of the Jacobian determinant to counteract the distortion, allowing quantities like probability densities or mass densities to remain consistent across coordinate systems.[6] Tensor densities thus function as essential tools for achieving invariant integration over manifolds, where integrals of density fields yield coordinate-independent results, such as total mass or action in variational principles. By embedding the Jacobian scaling directly into their structure, these objects enable the construction of volume forms that are preserved under diffeomorphisms, facilitating computations in curved spaces or non-orthogonal coordinates without ad hoc adjustments. This approach ensures that geometric invariants, like those in symplectic mechanics or optics, hold regardless of the chosen chart.[6] The historical motivation for tensor densities traces back to early 20th-century differential geometry, particularly Élie Cartan's development of integral invariants in the 1920s, where densities were introduced to preserve measures under transformations in continuous media and variational problems. In his 1922 work, Cartan employed densities, such as mass per unit volume ρ, to form absolute integral invariants like ∫∫∫ ρ δx δy δz over volumes, addressing the failure of unadjusted forms to remain invariant under diffeomorphisms. This framework laid the groundwork for modern uses in invariant integration, emphasizing densities' role in handling orientation and volume preservation in geometric contexts.[7]Distinction from Standard Tensors
Standard tensors, whether contravariant or covariant, are multilinear maps that transform linearly under changes of coordinates via the partial derivatives of the coordinate functions, without any additional scaling factors.[8] This ensures that their components in different coordinate systems are related in a way that preserves the intrinsic geometric structure, independent of the specific coordinate choice.[9] In contrast, tensor densities incorporate a weight factor w, modifying the standard tensor transformation by multiplying it with the absolute value of the Jacobian determinant raised to the power w.[8] Pseudotensors form a related but distinct category, typically referring to densities that also change sign under orientation-reversing transformations, such as the Levi-Civita symbol, which behaves as a pseudotensor density of weight -1.[9] Physically, tensor densities describe quantities that scale with volume elements, such as mass density, which transforms with weight -n in n-dimensional space to maintain invariance of total mass under coordinate rescaling.[9] Similarly, probability densities in continuous distributions scale inversely with volume to preserve total probability.[8] Tensor densities generalize standard tensors by accounting for the scaling introduced by the Jacobian determinant in non-orthogonal or curvilinear coordinate systems, enabling consistent descriptions of volume-dependent quantities.[9]Formal Definition
General Transformation Law
A tensor density of weight w generalizes the transformation behavior of standard tensors by incorporating an additional factor involving the Jacobian determinant of the coordinate transformation. Under a change of coordinates from x^\mu to x'^\nu, the components of a scalar tensor density \phi transform according to \phi'(x') = \phi(x) \det \left( \frac{\partial x^\rho}{\partial x'^\sigma} \right)^w. [10][11] This law accounts for the scaling under orientation-reversing transformations, where the sign of the determinant affects densities with non-zero integer weights.[8][11] For a general (k, l)-tensor density T^{\mu_1 \dots \mu_k}_{\nu_1 \dots \nu_l} of weight w, the transformation law extends the standard tensor rule by multiplying it with the Jacobian factor: T'^{\alpha_1 \dots \alpha_k}_{\beta_1 \dots \beta_l}(x') = \left( \prod_{i=1}^k \frac{\partial x'^{\alpha_i}}{\partial x^{\mu_i}} \right) \left( \prod_{j=1}^l \frac{\partial x^{\nu_j}}{\partial x'^{\beta_j}} \right) T^{\mu_1 \dots \mu_k}_{\nu_1 \dots \nu_l}(x) \det \left( \frac{\partial x^\rho}{\partial x'^\sigma} \right)^w. [10][2] The products of partial derivatives arise directly from the chain rule applied to the multilinearity of tensor components under differentiation.[10] The determinant factor, raised to the power w, is introduced to account for the scaling of the volume element d^n x' = \det \left( \frac{\partial x'}{\partial x} \right) d^n x, which modifies the transformation beyond the pure tensor case.[2][8] This convention with the signed determinant is standard for oriented manifolds, where the sign may vary, allowing densities to reflect orientation.[11][8]Parity and Weight Classification
Tensor densities are classified by their weight w, which is typically a real number, most commonly an integer, that determines the power to which the Jacobian determinant is raised in the transformation law. Specifically, under a coordinate transformation with Jacobian matrix J, the components transform as T' = (\det J)^w \cdot (standard tensor transformation). When w = 0, this reduces to the transformation law for standard tensors, recovering their behavior without the additional determinant factor.[12] The parity of a tensor density refers to its behavior under orientation-reversing coordinate transformations, where \det J < 0. Assuming |\det J| = 1 for simplicity (as in orthogonal transformations), the factor (\det J)^w = \operatorname{sign}(\det J)^w = (-1)^w. Thus, for even integer weights (w even), the density is invariant under such reversals, exhibiting even parity similar to true tensors but with scaling by |\det J|^w. For odd integer weights (w odd), the density acquires an extra minus sign, exhibiting odd parity akin to pseudotensors, which is essential for quantities sensitive to orientation, such as oriented volumes.[13][1] A representative example is the Levi-Civita symbol \varepsilon_{i_1 \dots i_n}, which serves as a covariant pseudotensor density of weight -1 (odd) in n dimensions. It changes sign under orientation reversal, reflecting its role in defining oriented structures like determinants and volume elements. In contrast, the scalar density \sqrt{|\det g|} has weight -1 (odd) and also flips sign, while products like (\sqrt{|\det g|})^2 = |\det g| have even weight +2 and remain invariant.| Property | Even Weight (e.g., w = 0, \pm 2) | Odd Weight (e.g., w = \pm 1) |
|---|---|---|
| Behavior under orientation reversal (\det J = -1) | Invariant (factor +1) | Changes sign (factor -1) |
| Parity type | Even (true tensor-like) | Odd (pseudotensor-like) |
| Example | Determinant of metric $ | \det g |
| Implications | Suitable for absolute volumes or invariants | Essential for oriented volumes and antisymmetric forms |