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A contravariant tensor transforms with one forward coordinate Jacobian for each upper index. A Lorentz four-vector is the rank-one example; the electromagnetic field tensor with two upper indices is a rank-two example. Lower-index covariant tensors transform with inverse Jacobians, and full contractions yield scalars. Index position specifies a transformation law rather than implying that the numerical components are invariant.

A metric tensor is a nondegenerate symmetric covariant tensor that defines squared line elements by $d{s}^2=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$.

Let $A_b^{\prime }=J^d{}_b{A}_d$, where $J^d{}_b=\partial x^d/\partial x^{\prime b}$. ThenThe second term prevents the ordinary derivative from being a generally covariant tensor. It vanishes for affine coordinate changes, including Cartesian Lorentz transformations, so the restricted claim of Lorentz covariance is valid. Antisymmetrization cancels the same term for arbitrary smooth coordinate changes and gives the exterior derivative of a one-form.

Use upper indices for the defining representation of SU(3) and lower indices for its dual representation. A Young diagram with row lengths $({\ell }_1,{\ell }_2)$ has highest weight Dynkin labels $(p,q)=({\ell }_1-{\ell }_2,{\ell }_2)$; a full column of height three is a determinant representation and can be removed for SU(3). The Weyl dimension formula reduces toThe six irreducible representations of smallest dimensions are $1,3,\overline{3},6,\overline{6},8$. The following box arrays draw their Young diagrams; filling boxes by tensor-slot labels gives representative Young tableaux:Their associated tensors, in the same order, can be taken as follows:

These low-dimensional SU3 tensor representations have dimensions $1,3,3,6,6,8$ by direct component counting. To check that none were missed, if $p+q\ge 3$ then $(p+1)(q+1)\ge p+q+1$, so the displayed dimension is at least $(p+q+1)(p+q+2)/2\ge 10$. The pairs with $p+q\le 2$ are precisely the six listed pairs.

Normal coordinates at an event have $g_{ab}={\eta }_{ab}$ and first metric derivatives, equivalently connection coefficients, zero there. They describe freely falling frames in which nongravitational laws take their special-relativistic form to first order, expressing the Equivalence principle. Curvature and tidal effects cannot generally be removed throughout a neighbourhood.

For a covector, ${\nabla }_a{V}_b={\partial }_a{V}_b-{\Gamma }_{ab}^c{V}_c$. The Levi-Civita connection is torsion-free, sois an antisymmetric covariant tensor. Next expand ${\nabla }_a{K}_b+{\nabla }_b{K}_a$ using $K_b=g_{bc}{K}^c$ and $2g_{cd}{\Gamma }_{ab}^d={\partial }_a{g}_{bc}+{\partial }_b{g}_{ac}-{\partial }_c{g}_{ab}$. Cancelling the first two metric derivatives leavesIf $K={\partial }_4$ has constant coordinate components, $Q_{ab}={\partial }_4{g}_{ab}$. Thus $Q=0$ implies the metric is independent of $x^4$ and $K$ is a Killing vector. Its derivative is antisymmetric, yieldingFor an affinely parametrized geodesic with tangent $u$, differentiation givesThe first term vanishes by antisymmetry and the second by the geodesic equation, proving conservation in every coordinate system. A stationary metric requires the Killing vector to be timelike in the region concerned. For particle energy, the tangent should be future timelike for a massive particle, with affine parameter proportional to proper time, or future null with appropriate momentum normalization for a photon; normalize the future timelike Killing vector relative to the chosen stationary observers or at infinity. Then the conserved quantity is proportional to their Killing energy. A spacelike Killing vector instead represents momentum, not stationary time translation.

In Questions 3–4, use primes for conformal-time derivatives and $\mathcal{H}=a^{\prime }/a$; proper-time derivatives will be stated separately. Let ${\xi }^{\mu }=(T,L^i)$. The metric is a covariant tensor, so under the passive coordinate change ${\tilde{x}}^{\mu }=x^{\mu }+{\xi }^{\mu }$,The inverse Jacobian is ${\delta }_{\mu }^{\alpha }-{\partial }_{\mu }{\xi }^{\alpha }$ to first order. At the same new coordinate argument, the homogeneous background satisfies ${\bar{g}}_{\alpha \beta }(x)={\bar{g}}_{\alpha \beta }(\tilde{x})-T{\bar{g}}_{\alpha \beta }^{\prime }$. Shifting the argument of $\delta g$ or multiplying it by a Jacobian perturbation produces only second-order terms, which we discard. Subtracting the unperturbed metric therefore givesThis is the printed Jacobian expression expanded to linear order, equivalently $\delta \tilde{g}=\delta g-{\mathcal{L}}_{\xi }\bar{g}$. It is a gauge transformation in cosmological perturbation theory, not a change of the physical spacetime.

A type $(0,s)$ covariant tensor is a multilinear form on $s$ vectors. For arbitrary $V_1,\dots ,V_s\in T_p\mathcal{M}$, the pullback of a covariant tensor isProjection of all covariant slots means $(P\omega )(X_1,\dots ,X_s)=\omega (P{X}_1,\dots ,P{X}_s)$. Since every pushed-forward vector is already tangential,Equality on all arguments proves ${\varphi }^{\ast }\omega ={\varphi }^{\ast }(P\omega )$. No antisymmetry is needed: this holds for every covariant tensor, not only for differential forms.

For Cartesian tensors, let $R$ be the orthogonal matrix taking old vector components to new ones, $v_i^{\prime }=R_{ij}{v}_j$. A rank-$n$ tensor transforms by one copy of $R$ on each index:Repeated indices are summed according to Einstein notation. In general coordinates a contravariant tensor index instead carries the corresponding Jacobian matrix $\partial x^{\prime i}/\partial x^j$, and a covariant tensor index carries its inverse; in orthonormal bases of Cartesian axes their component laws coincide with the displayed form.

An isotropic tensor has identical components after every proper orthogonal transformation, so $T^{\prime }=T$ for every $R$ with $R^{T}R=I$ and $\det R=1$. Requiring reflection invariance as well is a stronger convention for odd tensor ranks. The Kronecker delta is invariant under all orthogonal transformations, while the Levi-Civita symbol is invariant under proper rotations and acquires a determinant sign under reflections.

Use coordinates $x^{\mu }=(ct,\mathbf{x})$, signature $(+---)$, and the convention $F^{0i}=-E_i/c$, $F^{ij}=-{ϵ}_{ijk}{B}_k$. The antisymmetric electromagnetic field tensor combines six field components:It transforms as $F^{\prime \mu \nu }={\Lambda }^{\mu }{}_{\alpha }{\Lambda }^{\nu }{}_{\beta }{F}^{\alpha \beta }$. Lowering indices changes the time–space signs: $F_{0i}=E_i/c$, while $F_{ij}=F^{ij}$. The covariant tensor instead uses the inverse coordinate Jacobian on each lower index.

For a frame moving with velocity $\mathbf{v}$ relative to the original one,Electric and magnetic fields therefore mix under a boost; they are frame-dependent parts of one tensor. For example, a purely transverse electric field acquires a magnetic field in a boosted frame. The electromagnetic field invariants $B^2-E^2/c^2$ and $\mathbf{E}\cdot \mathbf{B}$ restrict which fields can be eliminated by a frame change. The covariant force law $d{p}^{\mu }/d\tau =q{F}^{\mu }{}_{\nu }{U}^{\nu }$ guarantees consistent particle dynamics in all inertial frames.