Einstein notation

From Wikipedia, the free encyclopedia

 

In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving notational brevity. As part of mathematics it is a notational subset of Ricci calculus; however, it is often used in applications in physics that do not distinguish between tangent and cotangent spaces. It was introduced to physics by Albert Einstein in 1916.

Introduction

Statement of convention

According to this convention, when an index variable appears twice in a single term and is not otherwise defined, it implies summation of that term over all the values of the index. So where the indices can range over the set {1, 2, 3},

${\displaystyle y=\sum _{i=1}^{3}c_{i}x^{i}=c_{1}x^{1}+c_{2}x^{2}+c_{3}x^{3}}$

is simplified by the convention to:

${\displaystyle y=c_{i}x^{i}}$

.

The upper indices are not exponents but are indices of coordinates, coefficients or basis vectors. That is, in this context x² should be understood as the second component of x rather than the square of x (this can occasionally lead to ambiguity). The upper index position in xⁱ is because, typically, an index occurs once in an upper (superscript) and once in a lower (subscript) position in a term (see 'Application' below). And typically (x¹, x², x³) would be equivalent to the traditional (x, y, z).
In general relativity, a common convention is that

• the Greek alphabet is used for space and time components, where indices take on values 0, 1, 2, or 3 (frequently used letters are μ, ν, ...),
• the Latin alphabet is used for spatial components only, where indices take on values 1, 2, or 3 (frequently used letters are i, j, ...),

In general, indices can range over any indexing set, including an infinite set. This should not be confused with a typographically similar convention used to distinguish between tensor index notation and the closely related but distinct basis-independent abstract index notation.

An index that is summed over is a summation index, in this case "i". It is also called a dummy index since any symbol can replace "i" without changing the meaning of the expression provided that it does not collide with index symbols in the same term.

An index that is not summed over is a free index and should appear only once per term. If such an index does appear, it usually also appears in terms belonging to the same sum, with the exception of special values such as zero.

Application

Einstein notation can be applied in slightly different ways. Typically, each index occurs once in an upper (superscript) and once in a lower (subscript) position in a term; however, the convention can be applied more generally to any repeated indices within a term. When dealing with covariant and contravariant vectors, where the position of an index also indicates the type of vector, the first case usually applies; a covariant vector can only be contracted with a contravariant vector, corresponding to summation of the products of coefficients. On the other hand, when there is a fixed coordinate basis (or when not considering coordinate vectors), one may choose to use only subscripts.

Vector representations

Superscripts and subscripts versus only subscripts

In terms of covariance and contravariance of vectors,

• upper indices represent components of contravariant vectors (vectors),
• lower indices represent components of covariant vectors (covectors).

They transform contravariantly or covariantly, respectively, with respect to change of basis.
In recognition of this fact, the following notation uses the same symbol both for a (co)vector and its components, as in:

${\displaystyle v=v^{i}e_{i}={\begin{bmatrix}e_{1}&e_{2}&\cdots &e_{n}\end{bmatrix}}{\begin{bmatrix}v^{1}\\v^{2}\\\vdots \\v^{n}\end{bmatrix}}\qquad w=w_{i}e^{i}={\begin{bmatrix}w_{1}&w_{2}&\cdots &w_{n}\end{bmatrix}}{\begin{bmatrix}e^{1}\\e^{2}\\\vdots \\e^{n}\end{bmatrix}}}$

where v is the vector and vⁱ are its components (not the ith covector v), w is the covector and w_i are its components.

In the presence of a non-degenerate form (an isomorphism V → V^∗, for instance a Riemannian metric or Minkowski metric), one can raise and lower indices.

A basis gives such a form (via the dual basis), hence when working on ℝⁿ with a Euclidean metric and a fixed orthonormal basis, one has the option to work with only subscripts.

However, if one changes coordinates, the way that coefficients change depends on the variance of the object, and one cannot ignore the distinction; see covariance and contravariance of vectors.

Mnemonics

In the above example, vectors are represented as n × 1 matrices (column vectors), while covectors are represented as 1 × n matrices (row covectors).

When using the column vector convention

• "Upper indices go up to down; lower indices go left to right."
• "Covariant tensors are row vectors that have indices that are below (co-below-row)."
• Covectors are row vectors:

${\displaystyle {\begin{bmatrix}w_{1}&\cdots &w_{k}\end{bmatrix}}.}$

Hence the lower index indicates which column you are in.

• Contravariant vectors are column vectors:

${\displaystyle {\begin{bmatrix}v^{1}\\\vdots \\v^{k}\end{bmatrix}}}$

Hence the upper index indicates which row you are in.

Abstract description

The virtue of Einstein notation is that it represents the invariant quantities with a simple notation.
In physics, a scalar is invariant under transformations of basis. In particular, a Lorentz scalar is invariant under a Lorentz transformation. The individual terms in the sum are not. When the basis is changed, the components of a vector change by a linear transformation described by a matrix. This led Einstein to propose the convention that repeated indices imply the summation is to be done.

As for covectors, they change by the inverse matrix. This is designed to guarantee that the linear function associated with the covector, the sum above, is the same no matter what the basis is.
The value of the Einstein convention is that it applies to other vector spaces built from V using the tensor product and duality. For example, V ⊗ V, the tensor product of V with itself, has a basis consisting of tensors of the form e_ij = e_i ⊗ e_j. Any tensor T in V ⊗ V can be written as:

${\displaystyle \mathbf {T} =T^{ij}\mathbf {e} _{ij}}$.

V*, the dual of V, has a basis e¹, e², ..., eⁿ which obeys the rule

${\displaystyle \mathbf {e} ^{i}(\mathbf {e} _{j})=\delta _{j}^{i}.}$

where δ is the Kronecker delta. As

${\displaystyle \operatorname {Hom} (V,W)=V^{*}\otimes W}$

the row/column coordinates on a matrix correspond to the upper/lower indices on the tensor product.

Common operations in this notation

In Einstein notation, the usual element reference A_mn for the mth row and nth column of matrix A becomes A^m_n. We can then write the following operations in Einstein notation as follows.

Inner product (hence also vector dot product)

Using an orthogonal basis, the inner product is the sum of corresponding components multiplied together:

${\displaystyle \mathbf {u} \cdot \mathbf {v} =u_{j}v^{j}}$

This can also be calculated by multiplying the covector on the vector.

Vector cross product

Again using an orthogonal basis (in 3 dimensions) the cross product intrinsically involves summations over permutations of components:

${\displaystyle \mathbf {u} \times \mathbf {v} =\varepsilon ^{i}{}_{jk}u^{j}v^{k}\mathbf {e} _{i}}$

where

${\displaystyle \varepsilon ^{i}{}_{jk}=\delta ^{il}\varepsilon _{ljk}}$

ε_ijk is the Levi-Civita symbol, and δ^il is the generalized Kronecker delta. Based on this definition of ε, there is no difference between εⁱ_jk and ε_ijk but the position of indices.

Matrix-Vector Multiplication

The product of a matrix A_ij with a column vector v_j is :

${\displaystyle \mathbf {u} _{i}=(\mathbf {A} \mathbf {v} )_{i}=\sum _{j=1}^{N}A_{ij}v_{j}}$

equivalent to

${\displaystyle u^{i}=A^{i}{}_{j}v^{j}}$

This is a special case of matrix multiplication.

Matrix multiplication

The matrix product of two matrices A_ij and B_jk is:

${\displaystyle \mathbf {C} _{ik}=(\mathbf {A} \mathbf {B} )_{ik}=\sum _{j=1}^{N}A_{ij}B_{jk}}$

equivalent to

${\displaystyle C^{i}{}_{k}=A^{i}{}_{j}B^{j}{}_{k}}$

Trace

For a square matrix Aⁱ_j, the trace is the sum of the diagonal elements, hence the sum over a common index Aⁱ_i.

Outer product

The outer product of the column vector uⁱ by the row vector v_j yields an m × n matrix A:

${\displaystyle A^{i}{}_{j}=u^{i}v_{j}=(uv)^{i}{}_{j}}$

Since i and j represent two different indices, there is no summation and the indices are not eliminated by the multiplication.

Raising and lowering indices

Given a tensor, one can raise an index or lower an index by contracting the tensor with the metric tensor, g_μν. For example, take the tensor T^α_β, one can raise an index:

${\displaystyle T^{\mu \alpha }=g^{\mu \sigma }T_{\sigma }{}^{\alpha }}$

Or one can lower an index:

${\displaystyle T_{\mu \beta }=g_{\mu \sigma }T^{\sigma }{}_{\beta }}$

Stress–energy tensor

From Wikipedia, the free encyclopedia

 

Contravariant components of the stress–energy tensor.

The stress–energy tensor, sometimes stress–energy–momentum tensor or energy–momentum tensor, is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields. The stress–energy tensor is the source of the gravitational field in the Einstein field equations of general relativity, just as mass density is the source of such a field in Newtonian gravity.

Definition

The stress–energy tensor involves the use of superscripted variables (not exponents). If Cartesian coordinates in SI units are used, then the components of the position four-vector are given by: x⁰ = t, x¹ = x, x² = y, and x³ = z, where t is time in seconds, and x, y, and z are distances in meters.

The stress–energy tensor is defined as the tensor T^αβ of order two that gives the flux of the αth component of the momentum vector across a surface with constant x^β coordinate. In the theory of relativity, this momentum vector is taken as the four-momentum. In general relativity, the stress–energy tensor is symmetric,

${\displaystyle T^{\alpha \beta }=T^{\beta \alpha }.}$

In some alternative theories like Einstein–Cartan theory, the stress–energy tensor may not be perfectly symmetric because of a nonzero spin tensor, which geometrically corresponds to a nonzero torsion tensor.

Identifying the components of the tensor

Because the stress–energy tensor is of order two, its components can be displayed in 4 × 4 matrix form:

${\displaystyle (T^{\mu \nu })_{\mu ,\nu =0,1,2,3}={\begin{pmatrix}T^{00}&T^{01}&T^{02}&T^{03}\\T^{10}&T^{11}&T^{12}&T^{13}\\T^{20}&T^{21}&T^{22}&T^{23}\\T^{30}&T^{31}&T^{32}&T^{33}\end{pmatrix}}.}$

In the following, i and k range from 1 through 3.

The time–time component is the density of relativistic mass, i.e. the energy density divided by the speed of light squared. Its components have a direct physical interpretation. In the case of a perfect fluid this component is

${\displaystyle T^{00}=\rho ,}$

where ${\displaystyle \rho }$ is the relativistic mass per unit volume, and for an electromagnetic field in otherwise empty space this component is

${\displaystyle T^{00}={1 \over c^{2}}\left({\frac {1}{2}}\epsilon _{0}E^{2}+{\frac {1}{2\mu _{0}}}B^{2}\right),}$

where E and B are the electric and magnetic fields, respectively.

The flux of relativistic mass across the xⁱ surface is equivalent to the density of the ith component of linear momentum,

${\displaystyle T^{0i}=T^{i0}.}$

The components

${\displaystyle T^{ik}}$

represent flux of ith component of linear momentum across the x^k surface. In particular,

${\displaystyle T^{ii}}$

(not summed) represents normal stress, which is called pressure when it is independent of direction. The remaining components

${\displaystyle T^{ik}\quad i\neq k}$

represent shear stress (compare with the stress tensor).

In solid state physics and fluid mechanics, the stress tensor is defined to be the spatial components of the stress–energy tensor in the proper frame of reference. In other words, the stress energy tensor in engineering differs from the stress–energy tensor here by a momentum convective term.

Covariant and mixed forms

In most of this article we work with the contravariant form, T^μν of the stress–energy tensor. However, it is often necessary to work with the covariant form,

${\displaystyle T_{\mu \nu }=T^{\alpha \beta }g_{\alpha \mu }g_{\beta \nu },}$

or the mixed form,

${\displaystyle T^{\mu }{}_{\nu }=T^{\mu \alpha }g_{\alpha \nu },}$

or as a mixed tensor density

${\displaystyle {\mathfrak {T}}^{\mu }{}_{\nu }=T^{\mu }{}_{\nu }{\sqrt {-g}}\,.}$

In this article we use the spacelike sign convention (−+++) for the metric signature.

Conservation law

In special relativity

The stress–energy tensor is the conserved Noether current associated with spacetime translations.

The divergence of the non-gravitational stress–energy is zero. In other words, non-gravitational energy and momentum are conserved,

${\displaystyle 0=T^{\mu \nu }{}_{;\nu }=\nabla _{\nu }T^{\mu \nu }{}.\!}$

When gravity is negligible and using a Cartesian coordinate system for spacetime, this may be expressed in terms of partial derivatives as

${\displaystyle 0=T^{\mu \nu }{}_{,\nu }=\partial _{\nu }T^{\mu \nu }.\!}$

The integral form of this is

${\displaystyle 0=\int _{\partial N}T^{\mu \nu }\mathrm {d} ^{3}s_{\nu }\!}$

where N is any compact four-dimensional region of spacetime; ${\displaystyle \partial N}$ is its boundary, a three-dimensional hypersurface; and ${\displaystyle \mathrm {d} ^{3}s_{\nu }}$ is an element of the boundary regarded as the outward pointing normal.

In flat spacetime and using Cartesian coordinates, if one combines this with the symmetry of the stress–energy tensor, one can show that angular momentum is also conserved:

${\displaystyle 0=(x^{\alpha }T^{\mu \nu }-x^{\mu }T^{\alpha \nu })_{,\nu }.\!}$

In general relativity

When gravity is non-negligible or when using arbitrary coordinate systems, the divergence of the stress–energy still vanishes. But in this case, a coordinate free definition of the divergence is used which incorporates the covariant derivative

${\displaystyle 0=\operatorname {div} T=T^{\mu \nu }{}_{;\nu }=\nabla _{\nu }T^{\mu \nu }=T^{\mu \nu }{}_{,\nu }+\Gamma ^{\mu }{}_{\sigma \nu }T^{\sigma \nu }+\Gamma ^{\nu }{}_{\sigma \nu }T^{\mu \sigma }}$

where ${\displaystyle \Gamma ^{\mu }{}_{\sigma \nu }}$ is the Christoffel symbol which is the gravitational force field.

Consequently, if ${\displaystyle \xi ^{\mu }}$ is any Killing vector field, then the conservation law associated with the symmetry generated by the Killing vector field may be expressed as

${\displaystyle 0=\nabla _{\nu }(\xi ^{\mu }T_{\mu }^{\nu })={\frac {1}{\sqrt {-g}}}\partial _{\nu }({\sqrt {-g}}\ \xi ^{\mu }T_{\mu }^{\nu })}$

The integral form of this is

${\displaystyle 0=\int _{\partial N}{\sqrt {-g}}\ \xi ^{\mu }T_{\mu }^{\nu }\ \mathrm {d} ^{3}s_{\nu }=\int _{\partial N}\xi ^{\mu }{\mathfrak {T}}_{\mu }^{\nu }\ \mathrm {d} ^{3}s_{\nu }}$

In general relativity

In general relativity, the symmetric stress–energy tensor acts as the source of spacetime curvature, and is the current density associated with gauge transformations of gravity which are general curvilinear coordinate transformations. (If there is torsion, then the tensor is no longer symmetric. This corresponds to the case with a nonzero spin tensor in Einstein–Cartan gravity theory.)

In general relativity, the partial derivatives used in special relativity are replaced by covariant derivatives. What this means is that the continuity equation no longer implies that the non-gravitational energy and momentum expressed by the tensor are absolutely conserved, i.e. the gravitational field can do work on matter and vice versa. In the classical limit of Newtonian gravity, this has a simple interpretation: energy is being exchanged with gravitational potential energy, which is not included in the tensor, and momentum is being transferred through the field to other bodies. In general relativity the Landau–Lifshitz pseudotensor is a unique way to define the gravitational field energy and momentum densities. Any such stress–energy pseudotensor can be made to vanish locally by a coordinate transformation.

In curved spacetime, the spacelike integral now depends on the spacelike slice, in general. There is in fact no way to define a global energy–momentum vector in a general curved spacetime.

The Einstein field equations

In general relativity, the stress tensor is studied in the context of the Einstein field equations which are often written as

${\displaystyle R_{\mu \nu }-{\tfrac {1}{2}}R\,g_{\mu \nu }={8\pi G \over c^{4}}T_{\mu \nu },}$

where ${\displaystyle R_{\mu \nu }}$ is the Ricci tensor, ${\displaystyle R}$ is the Ricci scalar (the tensor contraction of the Ricci tensor), ${\displaystyle g_{\mu \nu }\,}$ the metric tensor, and ${\displaystyle G}$ is the universal gravitational constant.

Stress–energy in special situations

Isolated particle

In special relativity, the stress–energy of a non-interacting particle with mass m and trajectory ${\displaystyle \mathbf {x} _{\text{p}}(t)}$ is:

${\displaystyle T^{\alpha \beta }(\mathbf {x} ,t)={\frac {m\,v^{\alpha }(t)v^{\beta }(t)}{\sqrt {1-(v/c)^{2}}}}\;\,\delta (\mathbf {x} -\mathbf {x} _{\text{p}}(t))={\frac {E}{c^{2}}}\;v^{\alpha }(t)v^{\beta }(t)\;\,\delta (\mathbf {x} -\mathbf {x} _{\text{p}}(t))}$

where ${\displaystyle (v^{\alpha })_{\alpha =0,1,2,3}\!}$ is the velocity vector (which should not be confused with four-velocity, since it is missing a ${\displaystyle \gamma }$)

${\displaystyle (v^{\alpha })_{\alpha =0,1,2,3}=\left(1,{\frac {d\mathbf {x} _{\text{p}}}{dt}}(t)\right)\,,}$

δ is the Dirac delta function and ${\displaystyle E={\sqrt {p^{2}c^{2}+m^{2}c^{4}}}}$ is the energy of the particle.

Stress–energy of a fluid in equilibrium

For a perfect fluid in thermodynamic equilibrium, the stress–energy tensor takes on a particularly simple form

${\displaystyle T^{\alpha \beta }\,=\left(\rho +{p \over c^{2}}\right)u^{\alpha }u^{\beta }+pg^{\alpha \beta }}$

where ${\displaystyle \rho }$ is the mass–energy density (kilograms per cubic meter), ${\displaystyle p}$ is the hydrostatic pressure (pascals), ${\displaystyle u^{\alpha }}$ is the fluid's four velocity, and ${\displaystyle g^{\alpha \beta }}$ is the reciprocal of the metric tensor. Therefore, the trace is given by

${\displaystyle T=3p-\rho c^{2}\,.}$

The four velocity satisfies

${\displaystyle u^{\alpha }u^{\beta }g_{\alpha \beta }=-c^{2}\,.}$

In an inertial frame of reference comoving with the fluid, better known as the fluid's proper frame of reference, the four velocity is

${\displaystyle (u^{\alpha })_{\alpha =0,1,2,3}=(1,0,0,0)\,,}$

the reciprocal of the metric tensor is simply

${\displaystyle (g^{\alpha \beta })_{\alpha ,\beta =0,1,2,3}\,=\left({\begin{matrix}-c^{-2}&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{matrix}}\right)\,}$

and the stress–energy tensor is a diagonal matrix

${\displaystyle (T^{\alpha \beta })_{\alpha ,\beta =0,1,2,3}=\left({\begin{matrix}\rho &0&0&0\\0&p&0&0\\0&0&p&0\\0&0&0&p\end{matrix}}\right).}$

Electromagnetic stress–energy tensor

The Hilbert stress–energy tensor of a source-free electromagnetic field is

${\displaystyle T^{\mu \nu }={\frac {1}{\mu _{0}}}\left(F^{\mu \alpha }g_{\alpha \beta }F^{\nu \beta }-{\frac {1}{4}}g^{\mu \nu }F_{\delta \gamma }F^{\delta \gamma }\right)}$

where ${\displaystyle F_{\mu \nu }}$ is the electromagnetic field tensor.

Scalar field

The stress–energy tensor for a scalar field ${\displaystyle \phi }$ which satisfies the Klein–Gordon equation is

${\displaystyle T^{\mu \nu }={\frac {\hbar ^{2}}{m}}(g^{\mu \alpha }g^{\nu \beta }+g^{\mu \beta }g^{\nu \alpha }-g^{\mu \nu }g^{\alpha \beta })\partial _{\alpha }{\bar {\phi }}\partial _{\beta }\phi -g^{\mu \nu }mc^{2}{\bar {\phi }}\phi ,}$

and when the metric is flat (Minkowski) its components work out to be:

${\displaystyle {\begin{aligned}T^{00}&={\frac {\hbar ^{2}}{mc^{4}}}\left(\partial _{0}{\bar {\phi }}\partial _{0}\phi +c^{2}\partial _{k}{\bar {\phi }}\partial _{k}\phi \right)+m{\bar {\phi }}\phi ,\\T^{0i}=T^{i0}&=-{\frac {\hbar ^{2}}{mc^{2}}}\left(\partial _{0}{\bar {\phi }}\partial _{i}\phi +\partial _{i}{\bar {\phi }}\partial _{0}\phi \right),\ \mathrm {and} \\T^{ij}&={\frac {\hbar ^{2}}{m}}\left(\partial _{i}{\bar {\phi }}\partial _{j}\phi +\partial _{j}{\bar {\phi }}\partial _{i}\phi \right)-\delta _{ij}\left({\frac {\hbar ^{2}}{m}}\eta ^{\alpha \beta }\partial _{\alpha }{\bar {\phi }}\partial _{\beta }\phi +mc^{2}{\bar {\phi }}\phi \right).\end{aligned}}}$

Variant definitions of stress–energy

There are a number of inequivalent definitions of non-gravitational stress–energy:

Hilbert stress–energy tensor

The Hilbert stress–energy tensor is defined as the functional derivative

${\displaystyle T_{\mu \nu }={\frac {-2}{\sqrt {-g}}}{\frac {\delta ({\mathcal {L}}_{\mathrm {matter} }{\sqrt {-g}})}{\delta g^{\mu \nu }}}=-2{\frac {\delta {\mathcal {L}}_{\mathrm {matter} }}{\delta g^{\mu \nu }}}+g_{\mu \nu }{\mathcal {L}}_{\mathrm {matter} },}$

where ${\displaystyle {\mathcal {L}}_{\mathrm {matter} }}$ is the nongravitational part of the Lagrangian density of the action. This is symmetric and gauge-invariant. See Einstein–Hilbert action for more information.

Canonical stress–energy tensor

Noether's theorem implies that there is a conserved current associated with translations through space and time. This is called the canonical stress–energy tensor. Generally, this is not symmetric and if we have some gauge theory, it may not be gauge invariant because space-dependent gauge transformations do not commute with spatial translations.

In general relativity, the translations are with respect to the coordinate system and as such, do not transform covariantly. See the section below on the gravitational stress–energy pseudo-tensor.

Belinfante–Rosenfeld stress–energy tensor

In the presence of spin or other intrinsic angular momentum, the canonical Noether stress energy tensor fails to be symmetric. The Belinfante–Rosenfeld stress energy tensor is constructed from the canonical stress–energy tensor and the spin current in such a way as to be symmetric and still conserved. In general relativity, this modified tensor agrees with the Hilbert stress–energy tensor.

Gravitational stress–energy

By the equivalence principle gravitational stress–energy will always vanish locally at any chosen point in some chosen frame, therefore gravitational stress–energy cannot be expressed as a non-zero tensor; instead we have to use a pseudotensor.

In general relativity, there are many possible distinct definitions of the gravitational stress–energy–momentum pseudotensor. These include the Einstein pseudotensor and the Landau–Lifshitz pseudotensor. The Landau–Lifshitz pseudotensor can be reduced to zero at any event in spacetime by choosing an appropriate coordinate system.