Let \(p\in\mathcal{M}\) be a point in a manifold \(\mathcal{M}\). We write \(V_p\) for the tangent space at \(p\). In the previous lecture we constructed tensor products of \(V_p\) purely algebraically, ignoring the manifold structure entirely. But a manifold provides extra data: not just one vector space but a whole field of vector spaces, one at each point, linked by coordinate charts. Our goal in this lecture is to study how vectors \(v\in V_p\), covectors \(\omega\in V_p^*\), and more general tensors transform under a change of coordinates on \(\mathcal{M}\).
The cotangent space at p is the dual vector space Vp* ≡ Tp*ℳ . Elements of Vp* are called covectors (or one-forms).
Recall that \(V_p\) has a coordinate basis \(\{e_\mu\} = \{\partial/\partial x^\mu|_p\}\). We formally define the dual basis \(\{e^\mu\} = \{\mathrm{d} x^\mu\}\) by the requirement \[\label{eq:dual-basis-def} e^\mu(e_\nu) = \delta^\mu{}_\nu\,.\] In coordinate notation this reads \[\label{eq:dual-basis-coord} \boxed{\mathrm{d} x^\mu\!\left(\frac{\partial }{\partial x^\nu}\bigg|_p\right) = \delta^\mu{}_\nu}\,.\] Thus \(\mathrm{d} x^\mu\) is a linear function of tangent vectors, fully determined by [eq:dual-basis-coord].
At this stage, dxμ is just a symbol for a particular covector—a linear function on tangent vectors defined by the Kronecker-delta condition [eq:dual-basis-coord]. It has nothing (yet) to do with derivatives or integration measures. Later, dxμ will acquire an interpretation as a differential form and integration measure, but for now resist the temptation to read more into the notation than is given by the definition.
A tangent vector v specifies a direction and magnitude at p. A covector ω is a linear machine that “measures” tangent vectors: it eats a vector and returns a real number. The dual basis element dxμ measures the μ-th component of a vector.
Recall the contravariant vector transformation law (Lecture 4): \[\label{eq:vector-transform-recall} v'^{\mu} = \sum_{\nu} v^\nu\, \frac{\partial x'^\mu}{\partial x^\nu}\,.\]
Now let \(\omega\in V_p^*\). In the coordinate basis, \[\omega = \sum_\mu \omega_\mu\,\mathrm{d} x^\mu\,.\] We ask: how are the components \(\omega'_{\mu'}\) in the new coordinate system \(x'^\mu\) related to the old components \(\omega_\mu\)?
Apply \(\omega\) to an arbitrary vector \(v\in V_p\): \[\begin{aligned} \omega(v) &= \omega\!\left(\sum_\mu v^\mu\,\frac{\partial }{\partial x^\mu}\bigg|_p\right) = \sum_\mu v^\mu\, \omega\!\left(\frac{\partial }{\partial x^\mu}\bigg|_p\right) = \sum_\mu \omega_\mu\, v^\mu\,. \label{eq:omega-v-old} \end{aligned}\] In the primed coordinates the same contraction gives \[\begin{aligned} \omega(v) &= \sum_{\mu'} \omega'_{\mu'}\, v'^{\mu'} = \sum_{\mu',\mu} \omega'_{\mu'}\, \frac{\partial x'^{\mu'}}{\partial x^\mu}\, v^\mu\,. \label{eq:omega-v-new} \end{aligned}\] Since [eq:omega-v-old] and [eq:omega-v-new] must agree for all \(v^\mu\), we read off \[\label{eq:covariant-transform} \boxed{\omega_\mu = \sum_{\mu'} \omega'_{\mu'}\, \frac{\partial x'^{\mu'}}{\partial x^\mu}} \qquad\Longleftrightarrow\qquad \boxed{\omega'_{\mu'} = \sum_{\mu} \frac{\partial x^\mu}{\partial x'^{\mu'}}\,\omega_\mu}\,.\] This is the covariant vector transformation law.
Compare the two transformation laws:
Contravariant (vectors): v′μ′ = (∂x′μ′/∂xμ) vμ.
Covariant (covectors): ω′μ′ = (∂xμ/∂x′μ′) ωμ.
The Jacobian matrices that appear are inverse transposes of each other.
Let \(T\in\mathcal{T}(k,l)\) be a tensor of type \((k,l)\). In a coordinate basis, \(T\) is expanded as \[\label{eq:tensor-expansion} T = \sum T^{\mu_1\cdots\mu_k}{}_{\nu_1\cdots\nu_l}\; e_{\mu_1}\otimes\cdots\otimes e_{\mu_k} \otimes e^{\nu_1}\otimes\cdots\otimes e^{\nu_l}\,,\] where \(e_\mu = \partial/\partial x^\mu|_p\) and \(e^\nu = \mathrm{d} x^\nu\).
Under a change of coordinates \(x^\mu\to x'^{\mu'}\), the components transform as \[\label{eq:tensor-transform} \boxed{T'^{\mu'_1\cdots\mu'_k}{}_{\nu'_1\cdots\nu'_l} = \sum T^{\mu_1\cdots\mu_k}{}_{\nu_1\cdots\nu_l}\, \frac{\partial x'^{\mu'_1}}{\partial x^{\mu_1}}\cdots \frac{\partial x'^{\mu'_k}}{\partial x^{\mu_k}}\, \frac{\partial x^{\nu_1}}{\partial x'^{\nu'_1}}\cdots \frac{\partial x^{\nu_l}}{\partial x'^{\nu'_l}}}\] This is the tensor transformation law.
Classically, a tensor of type (k,l) at a point p ∈ ℳ is a collection of numbers Tμ1⋯μkν1⋯νl(p) that transforms under coordinate changes via [eq:tensor-transform].
Not every collection of numbers indexed by coordinate labels is a tensor. One can associate nk + l numbers to each point p ∈ ℳ that do not obey [eq:tensor-transform] under coordinate changes (the Christoffel symbols, which we shall meet in Lecture 7, are a prominent example). Tensor fields are therefore quite special: they are the coordinate-indexed quantities whose transformation law is purely algebraic and involves only Jacobian factors.
A smooth tensor field T of type (k,l) on ℳ is an assignment of a tensor T|p ∈ 𝒯(k,l)(Vp) to each point p ∈ ℳ, such that for all smooth covector fields ω1, …, ωk and all smooth vector fields v1, …, vl, the function p ↦ T|p(ω1|p,…,ωk|p; v1|p,…,vl|p) is C∞ on ℳ.
A covector field ω is smooth if and only if ω(v) is a C∞ function on ℳ for every smooth vector field v.
The preceding lectures have built up an abstract formalism for multilinear objects on manifolds. We now put it to work by re-examining several familiar physical quantities—the four-current, the energy-momentum tensor, and the metric—through the lens of tensor analysis, all set in Minkowski spacetime \(\mathcal{M} = \mathbb{R}^{1,3}\).
Consider \(N\) particles with worldlines \(x_j(t)\) and charges \(q_j\), \(j = 1,\dots,N\).
The charge density is $$\label{eq:charge-density} \varepsilon(\boldsymbol{x},t) = \sum_{j=1}^{N} q_j\, \delta^{(3)}\!\bigl(\boldsymbol{x} - \boldsymbol{x}_j(t)\bigr)\,,$$ and the current density is $$\label{eq:current-density} \boldsymbol{J}(\boldsymbol{x},t) = \sum_{j=1}^{N} q_j\, \delta^{(3)}\!\bigl(\boldsymbol{x} - \boldsymbol{x}_j(t)\bigr)\, \frac{d\boldsymbol{x}_j(t)}{dt}\,.$$
Define a four-current \(J^\mu\) by setting \[\label{eq:four-current} J^\mu = (\varepsilon,\;\boldsymbol{J})\,.\]
Argue that Jμ is a vector (i.e. an element of Vp) field under Lorentz transformations x′ν = Λνμ xμ.
Consider \(N\) particles with energy-momentum four-vectors \(p_j^\mu\), \(j = 1,\dots,N\).
The density of the \(\mu\)-th component of four-momentum is \[\label{eq:Tmu0} T^{\mu 0}(\boldsymbol{x},t) = \sum_{j=1}^{N} p_j^\mu(t)\, \delta^{(3)}\!\bigl(\boldsymbol{x} - \boldsymbol{x}_j(t)\bigr)\,.\] The corresponding current (the flux of \(p^\mu\) in the \(x^k\)-direction) is \[\label{eq:Tmuk} T^{\mu k}(\boldsymbol{x},t) = \sum_{j=1}^{N} p_j^\mu(t)\, \frac{dx_j^k(t)}{dt}\, \delta^{(3)}\!\bigl(\boldsymbol{x} - \boldsymbol{x}_j(t)\bigr)\,.\] Combining into a single formula (with \(x^0(t) = t\)): \[\label{eq:Tmunu-dt} T^{\mu\nu}(x) = \sum_{j=1}^{N} p_j^\mu\, \frac{dx_j^\nu(t)}{dt}\, \delta^{(3)}\!\bigl(\boldsymbol{x} - \boldsymbol{x}_j(t)\bigr)\,.\]
Since \(p_j^\nu = E_j\, dx_j^\nu/dt\), this may be rewritten as \[\label{eq:Tmunu-symmetric} \boxed{T^{\mu\nu}(x) = \sum_{j=1}^{N} \frac{p_j^\mu\, p_j^\nu}{E_j}\, \delta^{(3)}\!\bigl(\boldsymbol{x} - \boldsymbol{x}_j(t)\bigr)}\,.\] Hence \(T^{\mu\nu}\) is symmetric: \(T^{\mu\nu} = T^{\nu\mu}\).
In covariant (proper-time) form: \[\label{eq:Tmunu-covariant} T^{\mu\nu}(x) = \sum_{j=1}^{N} \int\!\mathrm{d}\tau\; p_j^\mu\,\frac{dx_j^\nu}{d\tau}\, \delta^{(4)}\!\bigl(x - x_j(\tau)\bigr)\,.\]
Under Lorentz transformations x′ν = Λνμ xμ (with Λμ′μ = ∂x′μ′/∂xμ), show that T′μ′ν′ = Λμ′μ Λν′ν Tμν . Conclude that Tμν is a tensor of type (2,0). Hint: the covariant form [eq:Tmunu-covariant] makes the argument more transparent, since the four-dimensional delta function and proper-time integral are manifestly Lorentz-invariant.
The energy-momentum tensor plays an extraordinarily important role in general relativity: it is the object that encodes the complete stress-energy content of matter and radiation, and will appear as the source term on the right-hand side of Einstein’s field equations.
A metric tensor $\metric$ on a manifold ℳ is a smooth tensor field of type (0,2) that is
symmetric: $\metric(v_1,v_2) = \metric(v_2,v_1)$ for all v1, v2 ∈ Vp, and
nondegenerate: $\metric(v,w) = 0$ for all w ∈ Vp implies v = 0.
A manifold by itself has no notion of distance or angle. A metric is the extra datum that supplies an infinitesimal length: an infinitesimal displacement is modelled by a tangent vector, and the “infinitesimal squared distance” is a quadratic function of that tangent vector—precisely what $\metric(v,v)$ provides.
In a coordinate basis \(\{\partial/\partial x^\mu|_p\}\), expand the metric as \[\label{eq:metric-components} \metric = \sum_{\mu,\nu} g_{\mu\nu}\, \mathrm{d} x^\mu\otimes\mathrm{d} x^\nu\,.\] This is often written (omitting the tensor product sign) as the line element \[\label{eq:line-element} \boxed{\mathrm{d} s^2 = \sum_{\mu,\nu} g_{\mu\nu}\, \mathrm{d} x^\mu\,\mathrm{d} x^\nu}\,.\]
The metric defines an inner product on \(V_p\) for each \(p\in\mathcal{M}\): \[\begin{aligned} (v,w)_p &= \sum_{\mu,\nu} g_{\mu\nu}\, (\mathrm{d} x^\mu\otimes\mathrm{d} x^\nu)(v,w) = \sum_{\mu,\nu} g_{\mu\nu}\, \mathrm{d} x^\mu(v)\,\mathrm{d} x^\nu(w) \notag\\ &= \sum_{\mu,\nu} g_{\mu\nu}\, v^\mu\, w^\nu\,. \label{eq:inner-product} \end{aligned}\]
By the Gram–Schmidt procedure, there exists an orthonormal basis \(\{v_{\hat\mu}\}\) for \(V_p\) such that \[\label{eq:orthonormal-basis} \metric(v_{\hat\mu},\, v_{\hat\nu}) = s_\mu\,\delta_{\mu\nu}\,, \qquad s_\mu \in \{+1,\,-1\}\,.\]
Prove that an orthonormal basis satisfying [eq:orthonormal-basis] exists.
The number of \(+1\)’s and \(-1\)’s is independent of the choice of orthonormal basis. This invariant is the signature of \(\metric\).
A metric $\metric$ with sμ = + 1 for all μ is called Riemannian (positive definite).
The metric of spacetime has signature (−1,+1,+1,+1)—this is called a Lorentzian metric.
A metric tensor \(\metric\) can be viewed simultaneously as:
a \((0,2)\) tensor,
a bilinear map \(V_p\times V_p\to\mathbb{R}\), and
a linear map \(V_p\to V_p^*\).
The third interpretation is induced by the map \[\label{eq:metric-map} v \;\longmapsto\; \metric(\cdot\,,v) \;=:\; \omega\,.\] Here \(\metric(\cdot\,,v): V_p\to\mathbb{R}\), so indeed \(\metric(\cdot\,,v)\in V_p^*\).
If $\metric$ is nondegenerate, then the map [eq:metric-map] is a linear isomorphism $V_p\xrightarrow{\;\sim\;} V_p^*$.
Proof. The map \(v\mapsto\metric(\cdot\,,v)\) is linear by bilinearity of \(\metric\). Suppose \(\metric(\cdot\,,v) = 0\), i.e. \(\metric(w,v) = 0\) for all \(w\in V_p\). By nondegeneracy, \(v = 0\). Hence the kernel is trivial and the map is injective. Since \(\dim V_p = \dim V_p^*\), it is also surjective. ◻
The metric provides a canonical, basis-independent correspondence between vectors and covectors. In components: vμ = ∑νgμν vν (lowering) , ωμ = ∑νgμν ων (raising) , where gμν is the matrix inverse of gμν.
This canonical isomorphism is so convenient that many authors identify vectors with covectors from the outset, never introducing the cotangent space as a separate object. We prefer to keep the distinction explicit: the metric is extra data that may not always be present (e.g. in symplectic geometry or contact geometry), and understanding the difference between Vp and Vp* clarifies which structures depend on that extra data and which do not.
Suppose \(T\in\mathcal{T}(k,l)\). Think of \(T\) as a multilinear map \[T: \underbrace{V_p^*\otimes\cdots\otimes V_p^*}_{k} \;\otimes\; \underbrace{V_p\otimes\cdots\otimes V_p}_{l} \;\longrightarrow\; \mathbb{R}\,.\] One can specify \(T\) by its components \(T^{\mu_1\cdots\mu_k}{}_{\nu_1\cdots\nu_l}\) in some basis. However, it is often sufficient to know merely which arguments of \(T\) take vectors and which take covectors, without committing to a particular basis.
In the abstract index notation, each argument of a tensor is labelled by a lower-case Latin letter.
A superscript (upper) index labels a contravariant slot (takes a covector).
A subscript (lower) index labels a covariant slot (takes a vector).
A vector is written va—one contravariant slot.
A covector is written ωa—one covariant slot.
Tabcd denotes a tensor of type (2,2).
Here the lower-case Latin letters a, b, c, d, … label arguments and their type, not components with respect to a basis (for which we use Greek indices μ, ν, ρ, …).
The abstract index notation (due to R. Penrose) is a compromise between the coordinate-free and component-based approaches. It carries the structural information of index placement (contravariant vs. covariant) without fixing a basis, yet retains the computational convenience of the index calculus. In practice, any physical calculation eventually requires choosing a basis—there is no such thing as doing physics entirely without coordinates. But one can, and should, carry out as many manipulations as possible in a basis-free manner; the abstract index notation is designed to facilitate exactly this.