Covariant derivative

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https://en.wikipedia.org/wiki/Covariant_derivative

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component (dependent on the embedding) and the intrinsic covariant derivative component.

The name is motivated by the importance of changes of coordinate in physics: the covariant derivative transforms covariantly under a general coordinate transformation, that is, linearly via the Jacobian matrix of the transformation.

This article presents an introduction to the covariant derivative of a vector field with respect to a vector field, both in a coordinate-free language and using a local coordinate system and the traditional index notation. The covariant derivative of a tensor field is presented as an extension of the same concept. The covariant derivative generalizes straightforwardly to a notion of differentiation associated to a connection on a vector bundle, also known as a Koszul connection.

History

Historically, at the turn of the 20th century, the covariant derivative was introduced by Gregorio Ricci-Curbastro and Tullio Levi-Civita in the theory of Riemannian and pseudo-Riemannian geometry. Ricci and Levi-Civita (following ideas of Elwin Bruno Christoffel) observed that the Christoffel symbols used to define the curvature could also provide a notion of differentiation which generalized the classical directional derivative of vector fields on a manifold. This new derivative – the Levi-Civita connection – was covariant in the sense that it satisfied Riemann's requirement that objects in geometry should be independent of their description in a particular coordinate system.

It was soon noted by other mathematicians, prominent among these being Hermann Weyl, Jan Arnoldus Schouten, and Élie Cartan, that a covariant derivative could be defined abstractly without the presence of a metric. The crucial feature was not a particular dependence on the metric, but that the Christoffel symbols satisfied a certain precise second-order transformation law. This transformation law could serve as a starting point for defining the derivative in a covariant manner. Thus the theory of covariant differentiation forked off from the strictly Riemannian context to include a wider range of possible geometries.

In the 1940s, practitioners of differential geometry began introducing other notions of covariant differentiation in general vector bundles which were, in contrast to the classical bundles of interest to geometers, not part of the tensor analysis of the manifold. By and large, these generalized covariant derivatives had to be specified ad hoc by some version of the connection concept. In 1950, Jean-Louis Koszul unified these new ideas of covariant differentiation in a vector bundle by means of what is known today as a Koszul connection or a connection on a vector bundle. Using ideas from Lie algebra cohomology, Koszul successfully converted many of the analytic features of covariant differentiation into algebraic ones. In particular, Koszul connections eliminated the need for awkward manipulations of Christoffel symbols (and other analogous non-tensorial objects) in differential geometry. Thus they quickly supplanted the classical notion of covariant derivative in many post-1950 treatments of the subject.

Motivation

The covariant derivative is a generalization of the directional derivative from vector calculus. As with the directional derivative, the covariant derivative is a rule, ${\displaystyle {\mathrm{\nabla }}_{\mathbf{u}}\mathbf{v}}$, which takes as its inputs: (1) a vector, u, defined at a point P, and (2) a vector field v defined in a neighborhood of P. The output is the vector ${\displaystyle {\mathrm{\nabla }}_{\mathbf{u}}\mathbf{v}(P)}$, also at the point P. The primary difference from the usual directional derivative is that ${\displaystyle {\mathrm{\nabla }}_{\mathbf{u}}\mathbf{v}}$ must, in a certain precise sense, be independent of the manner in which it is expressed in a coordinate system.

A vector may be described as a list of numbers in terms of a basis, but as a geometrical object the vector retains its identity regardless of how it is described. For a geometric vector written in components with respect to one basis, when the basis is changed the components transform according to a change of basis formula, with the coordinates undergoing a covariant transformation. The covariant derivative is required to transform, under a change in coordinates, by a covariant transformation in the same way as a basis does (hence the name).

In the case of Euclidean space, one usually defines the directional derivative of a vector field in terms of the difference between two vectors at two nearby points. In such a system one translates one of the vectors to the origin of the other, keeping it parallel, then takes their difference within the same vector space. With a Cartesian (fixed orthonormal) coordinate system "keeping it parallel" amounts to keeping the components constant. This ordinary directional derivative on Euclidean space is the first example of a covariant derivative.

Next, one must take into account changes of the coordinate system. For example, if the Euclidean plane is described by polar coordinates, "keeping it parallel" does not amount to keeping the polar components constant under translation, since the coordinate grid itself "rotates". Thus, the same covariant derivative written in polar coordinates contains extra terms that describe how the coordinate grid itself rotates, or how in more general coordinates the grid expands, contracts, twists, interweaves, etc.

Consider the example of a particle moving along a curve γ(t) in the Euclidean plane. In polar coordinates, γ may be written in terms of its radial and angular coordinates by γ(t) = (r(t), θ(t)). A vector at a particular time t (for instance, a constant acceleration of the particle) is expressed in terms of ${\displaystyle ({\mathbf{e}}_r,{\mathbf{e}}_{\theta })}$, where ${\displaystyle {\mathbf{e}}_r}$ and ${\displaystyle {\mathbf{e}}_{\theta }}$ are unit tangent vectors for the polar coordinates, serving as a basis to decompose a vector in terms of radial and tangential components. At a slightly later time, the new basis in polar coordinates appears slightly rotated with respect to the first set. The covariant derivative of the basis vectors (the Christoffel symbols) serve to express this change.

In a curved space, such as the surface of the Earth (regarded as a sphere), the translation of tangent vectors between different points is not well defined, and its analog, parallel transport, depends on the path along which the vector is translated. A vector on a globe on the equator at point Q is directed to the north. Suppose we transport the vector (keeping it parallel) first along the equator to the point P, then drag it along a meridian to the N pole, and finally transport it along another meridian back to Q. Then we notice that the parallel-transported vector along a closed circuit does not return as the same vector; instead, it has another orientation. This would not happen in Euclidean space and is caused by the curvature of the surface of the globe. The same effect occurs if we drag the vector along an infinitesimally small closed surface subsequently along two directions and then back. This infinitesimal change of the vector is a measure of the curvature, and can be defined in terms of the covariant derivative.

Remarks

• The definition of the covariant derivative does not use the metric in space. However, for each metric there is a unique torsion-free covariant derivative called the Levi-Civita connection such that the covariant derivative of the metric is zero.
• The properties of a derivative imply that ${\displaystyle {\mathrm{\nabla }}_{\mathbf{v}}\mathbf{u}}$ depends on the values of u on an arbitrarily small neighborhood of a point p in the same way as e.g. the derivative of a scalar function f along a curve at a given point p depends on the values of f in an arbitrarily small neighborhood of p.
• The information on the neighborhood of a point p in the covariant derivative can be used to define parallel transport of a vector. Also the curvature, torsion, and geodesics may be defined only in terms of the covariant derivative or other related variation on the idea of a linear connection.

Informal definition using an embedding into Euclidean space

Suppose an open subset ${\displaystyle U}$ of a ${\displaystyle d}$-dimensional Riemannian manifold ${\displaystyle M}$ is embedded into Euclidean space ${\displaystyle ({\mathbb{R}}^n,⟨\cdot ,\cdot ⟩)}$ via a twice continuously-differentiable (C²) mapping ${\displaystyle \overrightarrow{\mathrm{\Psi }}:{\mathbb{R}}^d\supset U\to {\mathbb{R}}^n}$ such that the tangent space at ${\displaystyle \overrightarrow{\mathrm{\Psi }}(p)}$ is spanned by the vectors $${\displaystyle \left\{{\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^i}|}_p:i\in \{1,\dots ,d\}\right\}}$$ and the scalar product ${\displaystyle ⟨\cdot ,\cdot ⟩}$ on ${\displaystyle {\mathbb{R}}^n}$ is compatible with the metric on M: $${\displaystyle g_{ij}=⟨\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^i},\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^j}⟩.}$$

(Since the manifold metric is always assumed to be regular, the compatibility condition implies linear independence of the partial derivative tangent vectors.)

For a tangent vector field, ${\displaystyle \overrightarrow{V}=v^j\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^j}}$, one has $${\displaystyle \frac{\mathrm{\partial }\overrightarrow{V}}{\mathrm{\partial }{x}^i}=\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^i}\left(v^j\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^j}\right)=\frac{\mathrm{\partial }{v}^j}{\mathrm{\partial }{x}^i}\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^j}+v^j\frac{{\mathrm{\partial }}^2\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^i\mathrm{\partial }{x}^j}.}$$

The last term is not tangential to M, but can be expressed as a linear combination of the tangent space base vectors using the Christoffel symbols as linear factors plus a vector orthogonal to the tangent space: $${\displaystyle v^j\frac{{\mathrm{\partial }}^2\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^i\mathrm{\partial }{x}^j}=v^j{{\mathrm{\Gamma }}^k}_{ij}\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^k}+\overrightarrow{n}.}$$

In the case of the Levi-Civita connection, the covariant derivative ${\displaystyle {\mathrm{\nabla }}_{{\mathbf{e}}_i}\overrightarrow{V}}$, also written ${\displaystyle {\mathrm{\nabla }}_i\overrightarrow{V}}$, is defined as the orthogonal projection of the usual derivative onto tangent space: $${\displaystyle {\mathrm{\nabla }}_{{\mathbf{e}}_i}\overrightarrow{V}:=\frac{\mathrm{\partial }\overrightarrow{V}}{\mathrm{\partial }{x}^i}-\overrightarrow{n}=\left(\frac{\mathrm{\partial }{v}^k}{\mathrm{\partial }{x}^i}+v^j{{\mathrm{\Gamma }}^k}_{ij}\right)\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^k}.}$$

From here it may be computationally convenient to obtain a relation between the Christoffel symbols for the Levi-Civita connection and the metric. To do this we first note that, since the vector ${\displaystyle \overrightarrow{n}}$ in the previous equation is orthogonal to the tangent space, $${\displaystyle ⟨\frac{{\mathrm{\partial }}^2\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^i\mathrm{\partial }{x}^j},\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^l}⟩=⟨{{\mathrm{\Gamma }}^k}_{ij}\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^k}+\overrightarrow{n},\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^l}⟩=⟨\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^k},\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^l}⟩{{\mathrm{\Gamma }}^k}_{ij}=g_{kl}{{\mathrm{\Gamma }}^k}_{ij}.}$$

Then, since the partial derivative of a component ${\displaystyle g_{ab}}$ of the metric with respect to a coordinate ${\displaystyle x^c}$ is $${\displaystyle \frac{\mathrm{\partial }{g}_{ab}}{\mathrm{\partial }{x}^c}=\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^c}⟨\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^a},\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^b}⟩=⟨\frac{{\mathrm{\partial }}^2\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^c\mathrm{\partial }{x}^a},\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^b}⟩+⟨\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^a},\frac{{\mathrm{\partial }}^2\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^c\mathrm{\partial }{x}^b}⟩,}$$

any triplet ${\displaystyle i,j,k}$ of indices yields a system of equations $${\displaystyle \{\begin{array}{rlrl}\frac{\mathrm{\partial }{g}_{jk}}{\mathrm{\partial }{x}^i}=& & ⟨\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^j},\frac{{\mathrm{\partial }}^2\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^k\mathrm{\partial }{x}^i}⟩& +⟨\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^k},\frac{{\mathrm{\partial }}^2\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^i\mathrm{\partial }{x}^j}⟩\\ \frac{\mathrm{\partial }{g}_{ki}}{\mathrm{\partial }{x}^j}=& ⟨\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^i},\frac{{\mathrm{\partial }}^2\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^j\mathrm{\partial }{x}^k}⟩& & +⟨\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^k},\frac{{\mathrm{\partial }}^2\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^i\mathrm{\partial }{x}^j}⟩\\ \frac{\mathrm{\partial }{g}_{ij}}{\mathrm{\partial }{x}^k}=& ⟨\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^i},\frac{{\mathrm{\partial }}^2\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^j\mathrm{\partial }{x}^k}⟩& +⟨\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^j},\frac{{\mathrm{\partial }}^2\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^k\mathrm{\partial }{x}^i}⟩& & .\end{array}}$$ (Here the symmetry of the scalar product has been used and the order of partial differentiations have been swapped.)

Adding the first two equations and subtracting the third, we obtain $${\displaystyle \frac{\mathrm{\partial }{g}_{jk}}{\mathrm{\partial }{x}^i}+\frac{\mathrm{\partial }{g}_{ki}}{\mathrm{\partial }{x}^j}-\frac{\mathrm{\partial }{g}_{ij}}{\mathrm{\partial }{x}^k}=2⟨\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^k},\frac{{\mathrm{\partial }}^2\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^i\mathrm{\partial }{x}^j}⟩.}$$

Thus the Christoffel symbols for the Levi-Civita connection are related to the metric by $${\displaystyle g_{kl}{{\mathrm{\Gamma }}^k}_{ij}=\frac{1}{2}\left(\frac{\mathrm{\partial }{g}_{jl}}{\mathrm{\partial }{x}^i}+\frac{\mathrm{\partial }{g}_{li}}{\mathrm{\partial }{x}^j}-\frac{\mathrm{\partial }{g}_{ij}}{\mathrm{\partial }{x}^l}\right).}$$

If ${\displaystyle g}$ is nondegenerate then ${\displaystyle {{\mathrm{\Gamma }}^k}_{ij}}$ can be solved for directly as

$${\displaystyle {{\mathrm{\Gamma }}^k}_{ij}=\frac{1}{2}{g}^{kl}\left(\frac{\mathrm{\partial }{g}_{jl}}{\mathrm{\partial }{x}^i}+\frac{\mathrm{\partial }{g}_{li}}{\mathrm{\partial }{x}^j}-\frac{\mathrm{\partial }{g}_{ij}}{\mathrm{\partial }{x}^l}\right).}$$

For a very simple example that captures the essence of the description above, draw a circle on a flat sheet of paper. Travel around the circle at a constant speed. The derivative of your velocity, your acceleration vector, always points radially inward. Roll this sheet of paper into a cylinder. Now the (Euclidean) derivative of your velocity has a component that sometimes points inward toward the axis of the cylinder depending on whether you're near a solstice or an equinox. (At the point of the circle when you are moving parallel to the axis, there is no inward acceleration. Conversely, at a point (1/4 of a circle later) when the velocity is along the cylinder's bend, the inward acceleration is maximum.) This is the (Euclidean) normal component. The covariant derivative component is the component parallel to the cylinder's surface, and is the same as that before you rolled the sheet into a cylinder.

Formal definition

A covariant derivative is a (Koszul) connection on the tangent bundle and other tensor bundles: it differentiates vector fields in a way analogous to the usual differential on functions. The definition extends to a differentiation on the dual of vector fields (i.e. covector fields) and to arbitrary tensor fields, in a unique way that ensures compatibility with the tensor product and trace operations (tensor contraction).

Functions

Given a point ${\displaystyle p\in M}$ of the manifold ${\displaystyle M}$, a real function ${\displaystyle f:M\to \mathbb{R}}$ on the manifold and a tangent vector ${\displaystyle \mathbf{v}\in T_{p}M}$, the covariant derivative of f at p along v is the scalar at p, denoted ${\displaystyle {\left({\mathrm{\nabla }}_{\mathbf{v}}f\right)}_p}$, that represents the principal part of the change in the value of f when the argument of f is changed by the infinitesimal displacement vector v. (This is the differential of f evaluated against the vector v.) Formally, there is a differentiable curve ${\displaystyle \varphi :[-1,1]\to M}$ such that ${\displaystyle \varphi (0)=p}$ and ${\displaystyle {\varphi }^{\prime }(0)=\mathbf{v}}$, and the covariant derivative of f at p is defined by $${\displaystyle {\left({\mathrm{\nabla }}_{\mathbf{v}}f\right)}_p={\left(f\circ \varphi \right)}^{\prime }\left(0\right)=\underset{t\to 0}{lim}\frac{f(\varphi \left(t\right))-f(p)}{t}.}$$

When ${\displaystyle \mathbf{v}:M\to T_{p}M}$ is a vector field on ${\displaystyle M}$, the covariant derivative ${\displaystyle {\mathrm{\nabla }}_{\mathbf{v}}f:M\to \mathbb{R}}$ is the function that associates with each point p in the common domain of f and v the scalar ${\displaystyle {\left({\mathrm{\nabla }}_{\mathbf{v}}f\right)}_p}$.

For a scalar function f and vector field v, the covariant derivative ${\displaystyle {\mathrm{\nabla }}_{\mathbf{v}}f}$ coincides with the Lie derivative ${\displaystyle L_v(f)}$, and with the exterior derivative ${\displaystyle df(v)}$.

Vector fields

Given a point ${\displaystyle p}$ of the manifold ${\displaystyle M}$, a vector field ${\displaystyle \mathbf{u}:M\to T_{p}M}$ defined in a neighborhood of p and a tangent vector ${\displaystyle \mathbf{v}\in T_{p}M}$, the covariant derivative of u at p along v is the tangent vector at p, denoted ${\displaystyle ({\mathrm{\nabla }}_{\mathbf{v}}\mathbf{u}{)}_p}$, such that the following properties hold (for any tangent vectors v, x and y at p, vector fields u and w defined in a neighborhood of p, scalar values g and h at p, and scalar function f defined in a neighborhood of p):

1. ${\displaystyle {\left({\mathrm{\nabla }}_{\mathbf{v}}\mathbf{u}\right)}_p}$ is linear in ${\displaystyle \mathbf{v}}$ so $${\displaystyle {\left({\mathrm{\nabla }}_{g\mathbf{x}+h\mathbf{y}}\mathbf{u}\right)}_p=g(p){\left({\mathrm{\nabla }}_{\mathbf{x}}\mathbf{u}\right)}_p+h(p){\left({\mathrm{\nabla }}_{\mathbf{y}}\mathbf{u}\right)}_p}$$
2. ${\displaystyle {\left({\mathrm{\nabla }}_{\mathbf{v}}\mathbf{u}\right)}_p}$ is additive in ${\displaystyle \mathbf{u}}$ so: $${\displaystyle {\left({\mathrm{\nabla }}_{\mathbf{v}}\left[\mathbf{u}+\mathbf{w}\right]\right)}_p={\left({\mathrm{\nabla }}_{\mathbf{v}}\mathbf{u}\right)}_p+{\left({\mathrm{\nabla }}_{\mathbf{v}}\mathbf{w}\right)}_p}$$
3. ${\displaystyle ({\mathrm{\nabla }}_{\mathbf{v}}\mathbf{u}{)}_p}$ obeys the product rule; i.e., where ${\displaystyle {\mathrm{\nabla }}_{\mathbf{v}}f}$ is defined above, $${\displaystyle {\left({\mathrm{\nabla }}_{\mathbf{v}}\left[f\mathbf{u}\right]\right)}_p=f(p){\left({\mathrm{\nabla }}_{\mathbf{v}}\mathbf{u}{)}_p+({\mathrm{\nabla }}_{\mathbf{v}}f\right)}_p{\mathbf{u}}_p.}$$

Note that ${\displaystyle {\left({\mathrm{\nabla }}_{\mathbf{v}}\mathbf{u}\right)}_p}$ depends not only on the value of u at p but also on values of u in an infinitesimal neighborhood of p because of the last property, the product rule.

If u and v are both vector fields defined over a common domain, then ${\displaystyle {\mathrm{\nabla }}_{\mathbf{v}}\mathbf{u}}$ denotes the vector field whose value at each point p of the domain is the tangent vector ${\displaystyle {\left({\mathrm{\nabla }}_{\mathbf{v}}\mathbf{u}\right)}_p}$.

Covector fields

Given a field of covectors (or one-form) ${\displaystyle \alpha }$ defined in a neighborhood of p, its covariant derivative ${\displaystyle ({\mathrm{\nabla }}_{\mathbf{v}}\alpha {)}_p}$ is defined in a way to make the resulting operation compatible with tensor contraction and the product rule. That is, ${\displaystyle ({\mathrm{\nabla }}_{\mathbf{v}}\alpha {)}_p}$ is defined as the unique one-form at p such that the following identity is satisfied for all vector fields u in a neighborhood of p $${\displaystyle {\left({\mathrm{\nabla }}_{\mathbf{v}}\alpha \right)}_p\left({\mathbf{u}}_p\right)={\mathrm{\nabla }}_{\mathbf{v}}{\left[\alpha \left(\mathbf{u}\right)\right]}_p-{\alpha }_p\left[{\left({\mathrm{\nabla }}_{\mathbf{v}}\mathbf{u}\right)}_p\right].}$$

The covariant derivative of a covector field along a vector field v is again a covector field.

Tensor fields

Once the covariant derivative is defined for fields of vectors and covectors it can be defined for arbitrary tensor fields by imposing the following identities for every pair of tensor fields ${\displaystyle \phi }$ and ${\displaystyle \psi }$ in a neighborhood of the point p: $${\displaystyle {\mathrm{\nabla }}_{\mathbf{v}}{\left(\phi \otimes \psi \right)}_p={\left({\mathrm{\nabla }}_{\mathbf{v}}\phi \right)}_p\otimes \psi (p)+\phi (p)\otimes {\left({\mathrm{\nabla }}_{\mathbf{v}}\psi \right)}_p,}$$ and for ${\displaystyle \phi }$ and ${\displaystyle \psi }$ of the same valence $${\displaystyle {\mathrm{\nabla }}_{\mathbf{v}}(\phi +\psi {)}_p=({\mathrm{\nabla }}_{\mathbf{v}}\phi {)}_p+({\mathrm{\nabla }}_{\mathbf{v}}\psi {)}_p.}$$ The covariant derivative of a tensor field along a vector field v is again a tensor field of the same type.

Explicitly, let T be a tensor field of type (p, q). Consider T to be a differentiable multilinear map of smooth sections α¹, α², ..., α^q of the cotangent bundle T^∗M and of sections X₁, X₂, ..., X_p of the tangent bundle TM, written T(α¹, α², ..., X₁, X₂, ...) into R. The covariant derivative of T along Y is given by the formula

$${\displaystyle \begin{array}{rl}({\mathrm{\nabla }}_{Y}T)\left({\alpha }_1,{\alpha }_2,\dots ,X_1,X_2,\dots \right)=& {\mathrm{\nabla }}_Y\left(T\left({\alpha }_1,{\alpha }_2,\dots ,X_1,X_2,\dots \right)\right)\\ & -T\left({\mathrm{\nabla }}_Y{\alpha }_1,{\alpha }_2,\dots ,X_1,X_2,\dots \right)-T\left({\alpha }_1,{\mathrm{\nabla }}_Y{\alpha }_2,\dots ,X_1,X_2,\dots \right)-\cdots \\ & -T\left({\alpha }_1,{\alpha }_2,\dots ,{\mathrm{\nabla }}_Y{X}_1,X_2,\dots \right)-T\left({\alpha }_1,{\alpha }_2,\dots ,X_1,{\mathrm{\nabla }}_Y{X}_2,\dots \right)-\cdots \end{array}}$$

Coordinate description

This section uses the Einstein summation convention.

Given coordinate functions $${\displaystyle x^i,i=0,1,2,\dots ,}$$ any tangent vector can be described by its components in the basis $${\displaystyle {\mathbf{e}}_i=\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^i}.}$$

The covariant derivative of a basis vector along a basis vector is again a vector and so can be expressed as a linear combination ${\displaystyle {\mathrm{\Gamma }}^k{\mathbf{e}}_k}$. To specify the covariant derivative it is enough to specify the covariant derivative of each basis vector field ${\displaystyle {\mathbf{e}}_i}$ along ${\displaystyle {\mathbf{e}}_j}$. $${\displaystyle {\mathrm{\nabla }}_{{\mathbf{e}}_j}{\mathbf{e}}_i={{\mathrm{\Gamma }}^k}_{ij}{\mathbf{e}}_k,}$$

the coefficients ${\displaystyle {\mathrm{\Gamma }}_{ij}^k}$ are the components of the connection with respect to a system of local coordinates. In the theory of Riemannian and pseudo-Riemannian manifolds, the components of the Levi-Civita connection with respect to a system of local coordinates are called Christoffel symbols.

Then using the rules in the definition, we find that for general vector fields ${\displaystyle \mathbf{v}=v^j{\mathbf{e}}_j}$ and ${\displaystyle \mathbf{u}=u^i{\mathbf{e}}_i}$ we get $${\displaystyle \begin{array}{rl}{\mathrm{\nabla }}_{\mathbf{v}}\mathbf{u}& ={\mathrm{\nabla }}_{v^j{\mathbf{e}}_j}{u}^i{\mathbf{e}}_i\\ & =v^j{\mathrm{\nabla }}_{{\mathbf{e}}_j}{u}^i{\mathbf{e}}_i\\ & =v^j{u}^i{\mathrm{\nabla }}_{{\mathbf{e}}_j}{\mathbf{e}}_i+v^j{\mathbf{e}}_i{\mathrm{\nabla }}_{{\mathbf{e}}_j}{u}^i\\ & =v^j{u}^i{{\mathrm{\Gamma }}^k}_{ij}{\mathbf{e}}_k+v^j\frac{\mathrm{\partial }{u}^i}{\mathrm{\partial }{x}^j}{\mathbf{e}}_i\end{array}}$$

so $${\displaystyle {\mathrm{\nabla }}_{\mathbf{v}}\mathbf{u}=\left(v^j{u}^i{{\mathrm{\Gamma }}^k}_{ij}+v^j\frac{\mathrm{\partial }{u}^k}{\mathrm{\partial }{x}^j}\right){\mathbf{e}}_k.}$$

The first term in this formula is responsible for "twisting" the coordinate system with respect to the covariant derivative and the second for changes of components of the vector field u. In particular $${\displaystyle {\mathrm{\nabla }}_{{\mathbf{e}}_j}\mathbf{u}={\mathrm{\nabla }}_j\mathbf{u}=\left(\frac{\mathrm{\partial }{u}^i}{\mathrm{\partial }{x}^j}+u^k{{\mathrm{\Gamma }}^i}_{kj}\right){\mathbf{e}}_i}$$

In words: the covariant derivative is the usual derivative along the coordinates with correction terms which tell how the coordinates change.

For covectors similarly we have $${\displaystyle {\mathrm{\nabla }}_{{\mathbf{e}}_j}\theta =\left(\frac{\mathrm{\partial }{\theta }_i}{\mathrm{\partial }{x}^j}-{\theta }_k{{\mathrm{\Gamma }}^k}_{ij}\right){{\mathbf{e}}^{\ast }}^i}$$

where ${\displaystyle {{\mathbf{e}}^{\ast }}^i({\mathbf{e}}_j)={{\delta }^i}_j}$.

The covariant derivative of a type (r, s) tensor field along ${\displaystyle e_c}$ is given by the expression:

$${\displaystyle \begin{array}{rl}{({\mathrm{\nabla }}_{e_c}T{)}^{a_1\dots a_r}}_{b_1\dots b_s}=& \frac{\mathrm{\partial }}{\mathrm{\partial }{x}^c}{T^{a_1\dots a_r}}_{b_1\dots b_s}\\ & +{{\mathrm{\Gamma }}^{a_1}}_{dc}{T^{d{a}_2\dots a_r}}_{b_1\dots b_s}+\cdots +{{\mathrm{\Gamma }}^{a_r}}_{dc}{T^{a_1\dots a_{r-1}d}}_{b_1\dots b_s}\\ & -{{\mathrm{\Gamma }}^d}_{b_{1}c}{T^{a_1\dots a_r}}_{d{b}_2\dots b_s}-\cdots -{{\mathrm{\Gamma }}^d}_{b_{s}c}{T^{a_1\dots a_r}}_{b_1\dots b_{s-1}d}.\end{array}}$$

Or, in words: take the partial derivative of the tensor and add: ${\displaystyle +{{\mathrm{\Gamma }}^{a_i}}_{dc}}$ for every upper index ${\displaystyle a_i}$, and ${\displaystyle -{{\mathrm{\Gamma }}^d}_{b_{i}c}}$ for every lower index ${\displaystyle b_i}$.

If instead of a tensor, one is trying to differentiate a tensor density (of weight +1), then one also adds a term $${\displaystyle -{{\mathrm{\Gamma }}^d}_{dc}{T^{a_1\dots a_r}}_{b_1\dots b_s}.}$$ If it is a tensor density of weight W, then multiply that term by W. For example, $\sqrt{-g}$ is a scalar density (of weight +1), so we get: $${\displaystyle {\left(\sqrt{-g}\right)}_{;c}={\left(\sqrt{-g}\right)}_{,c}-\sqrt{-g}{{\mathrm{\Gamma }}^d}_{dc}}$$

where semicolon ";" indicates covariant differentiation and comma "," indicates partial differentiation. Incidentally, this particular expression is equal to zero, because the covariant derivative of a function solely of the metric is always zero.

Notation

In textbooks on physics, the covariant derivative is sometimes simply stated in terms of its components in this equation.

Often a notation is used in which the covariant derivative is given with a semicolon, while a normal partial derivative is indicated by a comma. In this notation we write the same as: $${\displaystyle {\mathrm{\nabla }}_{e_j}\mathbf{v}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{v^s}_{;j}{\mathbf{e}}_s{v^i}_{;j}={v^i}_{,j}+v^k{{\mathrm{\Gamma }}^i}_{kj}}$$ In case two or more indexes appear after the semicolon, all of them must be understood as covariant derivatives: $${\displaystyle {\mathrm{\nabla }}_{e_k}\left({\mathrm{\nabla }}_{e_j}\mathbf{v}\right)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{v^s}_{;jk}{\mathbf{e}}_s}$$

In some older texts (notably Adler, Bazin & Schiffer, Introduction to General Relativity), the covariant derivative is denoted by a double pipe and the partial derivative by single pipe: $${\displaystyle {\mathrm{\nabla }}_{e_j}\mathbf{v}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{v^i}_{||j}={v^i}_{|j}+v^k{{\mathrm{\Gamma }}^i}_{kj}}$$

Covariant derivative by field type

For a scalar field ${\displaystyle \varphi }$, covariant differentiation is simply partial differentiation: $${\displaystyle {\varphi }_{;a}\equiv {\mathrm{\partial }}_a\varphi }$$

For a contravariant vector field ${\displaystyle {\lambda }^a}$, we have: $${\displaystyle {{\lambda }^a}_{;b}\equiv {\mathrm{\partial }}_b{\lambda }^a+{{\mathrm{\Gamma }}^a}_{bc}{\lambda }^c}$$

For a covariant vector field ${\displaystyle {\lambda }_a}$, we have: $${\displaystyle {\lambda }_{a;c}\equiv {\mathrm{\partial }}_c{\lambda }_a-{{\mathrm{\Gamma }}^b}_{ca}{\lambda }_b}$$

For a type (2,0) tensor field ${\displaystyle {\tau }^{ab}}$, we have: $${\displaystyle {{\tau }^{ab}}_{;c}\equiv {\mathrm{\partial }}_c{\tau }^{ab}+{{\mathrm{\Gamma }}^a}_{cd}{\tau }^{db}+{{\mathrm{\Gamma }}^b}_{cd}{\tau }^{ad}}$$

For a type (0,2) tensor field ${\displaystyle {\tau }_{ab}}$, we have: $${\displaystyle {\tau }_{ab;c}\equiv {\mathrm{\partial }}_c{\tau }_{ab}-{{\mathrm{\Gamma }}^d}_{ca}{\tau }_{db}-{{\mathrm{\Gamma }}^d}_{cb}{\tau }_{ad}}$$

For a type (1,1) tensor field ${\displaystyle {{\tau }^a}_b}$, we have: $${\displaystyle {{\tau }^a}_{b;c}\equiv {\mathrm{\partial }}_c{{\tau }^a}_b+{{\mathrm{\Gamma }}^a}_{cd}{{\tau }^d}_b-{{\mathrm{\Gamma }}^d}_{cb}{{\tau }^a}_d}$$

The notation above is meant in the sense $${\displaystyle {{\tau }^{ab}}_{;c}\equiv {\left({\mathrm{\nabla }}_{{\mathbf{e}}_c}\tau \right)}^{ab}}$$

Properties

In general, covariant derivatives do not commute. By example, the covariant derivatives of vector field ${\displaystyle {\lambda }_{a;bc}\ne {\lambda }_{a;cb}}$. The Riemann tensor ${\displaystyle {R^d}_{abc}}$ is defined such that: $${\displaystyle {\lambda }_{a;bc}-{\lambda }_{a;cb}={R^d}_{abc}{\lambda }_d}$$

or, equivalently, $${\displaystyle {{\lambda }^a}_{;bc}-{{\lambda }^a}_{;cb}=-{R^a}_{dbc}{\lambda }^d}$$

The covariant derivative of a (2,0)-tensor field fulfills: $${\displaystyle {{\tau }^{ab}}_{;cd}-{{\tau }^{ab}}_{;dc}=-{R^a}_{ecd}{\tau }^{eb}-{R^b}_{ecd}{\tau }^{ae}}$$

The latter can be shown by taking (without loss of generality) that ${\displaystyle {\tau }^{ab}={\lambda }^a{\mu }^b}$.

Derivative along a curve

Since the covariant derivative ${\displaystyle {\mathrm{\nabla }}_{X}T}$ of a tensor field ${\displaystyle T}$ at a point ${\displaystyle p}$ depends only on the value of the vector field ${\displaystyle X}$ at ${\displaystyle p}$ one can define the covariant derivative along a smooth curve ${\displaystyle \gamma (t)}$ in a manifold: $${\displaystyle D_{t}T={\mathrm{\nabla }}_{\dot{\gamma }(t)}T.}$$ Note that the tensor field ${\displaystyle T}$ only needs to be defined on the curve ${\displaystyle \gamma (t)}$ for this definition to make sense.

In particular, ${\displaystyle \dot{\gamma }(t)}$ is a vector field along the curve ${\displaystyle \gamma }$ itself. If ${\displaystyle {\mathrm{\nabla }}_{\dot{\gamma }(t)}\dot{\gamma }(t)}$ vanishes then the curve is called a geodesic of the covariant derivative. If the covariant derivative is the Levi-Civita connection of a positive-definite metric then the geodesics for the connection are precisely the geodesics of the metric that are parametrized by arc length.

The derivative along a curve is also used to define the parallel transport along the curve.

Sometimes the covariant derivative along a curve is called absolute or intrinsic derivative.

Relation to Lie derivative

A covariant derivative introduces an extra geometric structure on a manifold that allows vectors in neighboring tangent spaces to be compared: there is no canonical way to compare vectors from different tangent spaces because there is no canonical coordinate system.

There is however another generalization of directional derivatives which is canonical: the Lie derivative, which evaluates the change of one vector field along the flow of another vector field. Thus, one must know both vector fields in an open neighborhood, not merely at a single point. The covariant derivative on the other hand introduces its own change for vectors in a given direction, and it only depends on the vector direction at a single point, rather than a vector field in an open neighborhood of a point. In other words, the covariant derivative is linear (over C^∞(M)) in the direction argument, while the Lie derivative is linear in neither argument.

Note that the antisymmetrized covariant derivative ∇_uv − ∇_vu, and the Lie derivative L_uv differ by the torsion of the connection, so that if a connection is torsion free, then its antisymmetrization is the Lie derivative.

Wheeler–Feynman absorber theory

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Wheeler%E2%80%93Feynman_absorber_theory

The Wheeler–Feynman absorber theory (also called the Wheeler–Feynman time-symmetric theory), named after its originators, the physicists Richard Feynman and John Archibald Wheeler, is a theory of electrodynamics based on a relativistic correct extension of action at a distance electron particles. The theory postulates no independent electromagnetic field. Rather, the whole theory is encapsulated by the Lorentz-invariant action ${\displaystyle S}$ of particle trajectories ${\displaystyle a^{\mu }(\tau ),b^{\mu }(\tau ),\cdots }$ defined as

${\displaystyle S=-\sum _a{m}_{a}c\int \sqrt{-d{a}_{\mu }d{a}^{\mu }}+\sum _{a<b}\frac{e_a{e}_b}{c}\int \int d{a}_{\nu }d{b}^{\nu }{\delta }^4(a{b}_{\mu }a{b}^{\mu }),}$

where ${\displaystyle a{b}_{\mu }\equiv a_{\mu }-b_{\mu }}$.

The absorber theory is invariant under time-reversal transformation, consistent with the lack of any physical basis for microscopic time-reversal symmetry breaking. Another key principle resulting from this interpretation, and somewhat reminiscent of Mach's principle and the work of Hugo Tetrode, is that elementary particles are not self-interacting. This immediately removes the problem of electron self-energy giving an infinity in the energy of an electromagnetic field.

Motivation

Wheeler and Feynman begin by observing that classical electromagnetic field theory was designed before the discovery of electrons: charge is a continuous substance in the theory. An electron particle does not naturally fit in to the theory: should a point charge see the effect of its own field? They reconsider the fundamental problem of a collection of point charges, taking up a field-free action at a distance theory developed separately by Karl Schwarzschild, Hugo Tetrode, and Adriaan Fokker. Unlike instantaneous action at a distance theories of the early 1800s these "direct interaction" theories are based on interaction propagation at the speed of light. They differ from the classical field theory in three ways 1) no independent field is postulated; 2) the point charges do not act upon themselves; 3) the equations are time symmetric. Wheeler and Feynman propose to develop these equations into a relativistically correct generalization of electromagnetism based on Newtonian mechanics.

Problems with previous direct-interaction theories

The Tetrode-Fokker work left unsolved two major problems. First, in a non-instantaneous action at a distance theory, the equal action-reaction of Newton's laws of motion conflicts with causality. If an action propagates forward in time, the reaction would necessarily propagate backwards in time. Second, existing explanations of radiation reaction force or radiation resistance depended upon accelerating electrons interacting with their own field; the direct interaction models explicitly omit self-interaction.

Absorber and radiation resistance

Wheeler and Feynman postulate the "universe" of all other electrons as an absorber of radiation to overcome these issues and extend the direct interaction theories. Rather than considering an unphysical isolated point charge, they model all charges in the universe with a uniform absorber in a shell around a charge. As the charge moves relative to the absorber, it radiates into the absorber which "pushes back", causing the radiation resistance.

Key result

Feynman and Wheeler obtained their result in a very simple and elegant way. They considered all the charged particles (emitters) present in our universe and assumed all of them to generate time-reversal symmetric waves. The resulting field is

${\displaystyle E_{\text{tot}}(\mathbf{x},t)=\sum _n\frac{E_n^{\text{ret}}(\mathbf{x},t)+E_n^{\text{adv}}(\mathbf{x},t)}{2}.}$

Then they observed that if the relation

${\displaystyle E_{\text{free}}(\mathbf{x},t)=\sum _n\frac{E_n^{\text{ret}}(\mathbf{x},t)-E_n^{\text{adv}}(\mathbf{x},t)}{2}=0}$

holds, then ${\displaystyle E_{\text{free}}}$, being a solution of the homogeneous Maxwell equation, can be used to obtain the total field

${\displaystyle E_{\text{tot}}(\mathbf{x},t)=\sum _n\frac{E_n^{\text{ret}}(\mathbf{x},t)+E_n^{\text{adv}}(\mathbf{x},t)}{2}+\sum _n\frac{E_n^{\text{ret}}(\mathbf{x},t)-E_n^{\text{adv}}(\mathbf{x},t)}{2}=\sum _n{E}_n^{\text{ret}}(\mathbf{x},t).}$

The total field is then the observed pure retarded field.

The assumption that the free field is identically zero is the core of the absorber idea. It means that the radiation emitted by each particle is completely absorbed by all other particles present in the universe. To better understand this point, it may be useful to consider how the absorption mechanism works in common materials. At the microscopic scale, it results from the sum of the incoming electromagnetic wave and the waves generated from the electrons of the material, which react to the external perturbation. If the incoming wave is absorbed, the result is a zero outgoing field. In the absorber theory the same concept is used, however, in presence of both retarded and advanced waves.

Arrow of time ambiguity

The resulting wave appears to have a preferred time direction, because it respects causality. However, this is only an illusion. Indeed, it is always possible to reverse the time direction by simply exchanging the labels emitter and absorber. Thus, the apparently preferred time direction results from the arbitrary labelling. Wheeler and Feynman claimed that thermodynamics picked the observed direction; cosmological selections have also been proposed.

The requirement of time-reversal symmetry, in general, is difficult to reconcile with the principle of causality. Maxwell's equations and the equations for electromagnetic waves have, in general, two possible solutions: a retarded (delayed) solution and an advanced one. Accordingly, any charged particle generates waves, say at time ${\displaystyle t_0=0}$ and point ${\displaystyle x_0=0}$, which will arrive at point ${\displaystyle x_1}$ at the instant ${\displaystyle t_1=x_1/c}$ (here ${\displaystyle c}$ is the speed of light), after the emission (retarded solution), and other waves, which will arrive at the same place at the instant ${\displaystyle t_2=-x_1/c}$, before the emission (advanced solution). The latter, however, violates the causality principle: advanced waves could be detected before their emission. Thus the advanced solutions are usually discarded in the interpretation of electromagnetic waves.

In the absorber theory, instead charged particles are considered as both emitters and absorbers, and the emission process is connected with the absorption process as follows: Both the retarded waves from emitter to absorber and the advanced waves from absorber to emitter are considered. The sum of the two, however, results in causal waves, although the anti-causal (advanced) solutions are not discarded a priori.

Alternatively, the way that Wheeler/Feynman came up with the primary equation is: They assumed that their Lagrangian only interacted when and where the fields for the individual particles were separated by a proper time of zero. So since only massless particles propagate from emission to detection with zero proper time separation, this Lagrangian automatically demands an electromagnetic like interaction.

New interpretation of radiation damping

One of the major results of the absorber theory is the elegant and clear interpretation of the electromagnetic radiation process. A charged particle that experiences acceleration is known to emit electromagnetic waves, i.e., to lose energy. Thus, the Newtonian equation for the particle (${\displaystyle F=ma}$) must contain a dissipative force (damping term), which takes into account this energy loss. In the causal interpretation of electromagnetism, Hendrik Lorentz and Max Abraham proposed that such a force, later called Abraham–Lorentz force, is due to the retarded self-interaction of the particle with its own field. This first interpretation, however, is not completely satisfactory, as it leads to divergences in the theory and needs some assumptions on the structure of charge distribution of the particle. Paul Dirac generalized the formula to make it relativistically invariant. While doing so, he also suggested a different interpretation. He showed that the damping term can be expressed in terms of a free field acting on the particle at its own position:

${\displaystyle E^{\text{damping}}({\mathbf{x}}_j,t)=\frac{E_j^{\text{ret}}({\mathbf{x}}_j,t)-E_j^{\text{adv}}({\mathbf{x}}_j,t)}{2}.}$

However, Dirac did not propose any physical explanation of this interpretation.

A clear and simple explanation can instead be obtained in the framework of absorber theory, starting from the simple idea that each particle does not interact with itself. This is actually the opposite of the first Abraham–Lorentz proposal. The field acting on the particle ${\displaystyle j}$ at its own position (the point ${\displaystyle x_j}$) is then

${\displaystyle E^{\text{tot}}({\mathbf{x}}_j,t)=\sum _{n\ne j}\frac{E_n^{\text{ret}}({\mathbf{x}}_j,t)+E_n^{\text{adv}}({\mathbf{x}}_j,t)}{2}.}$

If we sum the free-field term of this expression, we obtain

${\displaystyle E^{\text{tot}}({\mathbf{x}}_j,t)=\sum _{n\ne j}\frac{E_n^{\text{ret}}({\mathbf{x}}_j,t)+E_n^{\text{adv}}({\mathbf{x}}_j,t)}{2}+\sum _n\frac{E_n^{\text{ret}}({\mathbf{x}}_j,t)-E_n^{\text{adv}}({\mathbf{x}}_j,t)}{2}}$

and, thanks to Dirac's result,

${\displaystyle E^{\text{tot}}({\mathbf{x}}_j,t)=\sum _{n\ne j}{E}_n^{\text{ret}}({\mathbf{x}}_j,t)+E^{\text{damping}}({\mathbf{x}}_j,t).}$

Thus, the damping force is obtained without the need for self-interaction, which is known to lead to divergences, and also giving a physical justification to the expression derived by Dirac.

Developments since original formulation

Gravity theory

Main article: Hoyle–Narlikar theory of gravity

Inspired by the Machian nature of the Wheeler–Feynman absorber theory for electrodynamics, Fred Hoyle and Jayant Narlikar proposed their own theory of gravity in the context of general relativity. This model still exists in spite of recent astronomical observations that have challenged the theory. Stephen Hawking had criticized the original Hoyle-Narlikar theory believing that the advanced waves going off to infinity would lead to a divergence, as indeed they would, if the universe were only expanding.

Transactional interpretation of quantum mechanics

Main article: Transactional interpretation

Again inspired by the Wheeler–Feynman absorber theory, the transactional interpretation of quantum mechanics (TIQM) first proposed in 1986 by John G. Cramer, describes quantum interactions in terms of a standing wave formed by retarded (forward-in-time) and advanced (backward-in-time) waves. Cramer claims it avoids the philosophical problems with the Copenhagen interpretation and the role of the observer, and resolves various quantum paradoxes, such as quantum nonlocality, quantum entanglement and retrocausality.

Attempted resolution of causality

T. C. Scott and R. A. Moore demonstrated that the apparent acausality suggested by the presence of advanced Liénard–Wiechert potentials could be removed by recasting the theory in terms of retarded potentials only, without the complications of the absorber idea. The Lagrangian describing a particle (${\displaystyle p_1}$) under the influence of the time-symmetric potential generated by another particle (${\displaystyle p_2}$) is

${\displaystyle L_1=T_1-\frac{1}{2}\left((V_R{)}_1^2+(V_A{)}_1^2\right),}$

where ${\displaystyle T_i}$ is the relativistic kinetic energy functional of particle ${\displaystyle p_i}$, and ${\displaystyle (V_R{)}_i^j}$ and ${\displaystyle (V_A{)}_i^j}$ are respectively the retarded and advanced Liénard–Wiechert potentials acting on particle ${\displaystyle p_i}$ and generated by particle ${\displaystyle p_j}$. The corresponding Lagrangian for particle ${\displaystyle p_2}$ is

${\displaystyle L_2=T_2-\frac{1}{2}\left((V_R{)}_2^1+(V_A{)}_2^1\right).}$

It was originally demonstrated with computer algebra and then proven analytically that

${\displaystyle (V_R{)}_j^i-(V_A{)}_i^j}$

is a total time derivative, i.e. a divergence in the calculus of variations, and thus it gives no contribution to the Euler–Lagrange equations. Thanks to this result the advanced potentials can be eliminated; here the total derivative plays the same role as the free field. The Lagrangian for the N-body system is therefore

${\displaystyle L=\sum _{i=1}^N{T}_i-\frac{1}{2}\sum _{i\ne j}^N(V_R{)}_j^i.}$

The resulting Lagrangian is symmetric under the exchange of ${\displaystyle p_i}$ with ${\displaystyle p_j}$. For ${\displaystyle N=2}$ this Lagrangian will generate exactly the same equations of motion of ${\displaystyle L_1}$ and ${\displaystyle L_2}$. Therefore, from the point of view of an outside observer, everything is causal. This formulation reflects particle-particle symmetry with the variational principle applied to the N-particle system as a whole, and thus Tetrode's Machian principle. Only if we isolate the forces acting on a particular body do the advanced potentials make their appearance. This recasting of the problem comes at a price: the N-body Lagrangian depends on all the time derivatives of the curves traced by all particles, i.e. the Lagrangian is infinite-order. However, much progress was made in examining the unresolved issue of quantizing the theory. Also, this formulation recovers the Darwin Lagrangian, from which the Breit equation was originally derived, but without the dissipative terms. This ensures agreement with theory and experiment, up to but not including the Lamb shift. Numerical solutions for the classical problem were also found. Furthermore, Moore showed that a model by Feynman and Albert Hibbs is amenable to the methods of higher than first-order Lagrangians and revealed chaotic-like solutions. Moore and Scott showed that the radiation reaction can be alternatively derived using the notion that, on average, the net dipole moment is zero for a collection of charged particles, thereby avoiding the complications of the absorber theory.

This apparent acausality may be viewed as merely apparent, and this entire problem goes away. An opposing view was held by Einstein.

Alternative Lamb shift calculation

As mentioned previously, a serious criticism against the absorber theory is that its Machian assumption that point particles do not act on themselves does not allow (infinite) self-energies and consequently an explanation for the Lamb shift according to quantum electrodynamics (QED). Ed Jaynes proposed an alternate model where the Lamb-like shift is due instead to the interaction with other particles very much along the same notions of the Wheeler–Feynman absorber theory itself. One simple model is to calculate the motion of an oscillator coupled directly with many other oscillators. Jaynes has shown that it is easy to get both spontaneous emission and Lamb shift behavior in classical mechanics. Furthermore, Jaynes' alternative provides a solution to the process of "addition and subtraction of infinities" associated with renormalization.

This model leads to the same type of Bethe logarithm (an essential part of the Lamb shift calculation), vindicating Jaynes' claim that two different physical models can be mathematically isomorphic to each other and therefore yield the same results, a point also apparently made by Scott and Moore on the issue of causality.

Relationship to quantum field theory

This universal absorber theory is mentioned in the chapter titled "Monster Minds" in Feynman's autobiographical work Surely You're Joking, Mr. Feynman! and in Vol. II of the Feynman Lectures on Physics. It led to the formulation of a framework of quantum mechanics using a Lagrangian and action as starting points, rather than a Hamiltonian, namely the formulation using Feynman path integrals, which proved useful in Feynman's earliest calculations in quantum electrodynamics and quantum field theory in general. Both retarded and advanced fields appear respectively as retarded and advanced propagators and also in the Feynman propagator and the Dyson propagator. In hindsight, the relationship between retarded and advanced potentials shown here is not so surprising in view of the fact that, in quantum field theory, the advanced propagator can be obtained from the retarded propagator by exchanging the roles of field source and test particle (usually within the kernel of a Green's function formalism). In quantum field theory, advanced and retarded fields are simply viewed as mathematical solutions of Maxwell's equations whose combinations are decided by the boundary conditions.

Superluminal communication

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Superluminal_communication

Superluminal communication is a hypothetical process in which information is conveyed at faster-than-light speeds. The current scientific consensus is that faster-than-light communication is not possible, and to date it has not been achieved in any experiment.

Superluminal communication other than possibly through wormholes is likely impossible because, in a Lorentz-invariant theory, it could be used to transmit information into the past. This would complicate causality, but no theoretical arguments conclusively preclude this possibility.

A number of theories and phenomena related to superluminal communication have been proposed or studied, including tachyons, neutrinos, quantum nonlocality, wormholes, and quantum tunneling.

Proposed mechanisms

Spacetime diagram showing that moving faster than light implies time travel in the context of special relativity. A spaceship departs from Earth from A to C slower than light. At B, Earth emits a tachyon, particle that travels faster than light but forward in time in Earth's reference frame. It reaches the spaceship at C. The spaceship then sends another tachyon back to Earth from C to D. This tachyon also travels forward in time in the spaceship's reference frame. This effectively allows Earth to send a signal from B to D, back in time.

Tachyons

Tachyonic particles are hypothetical particles that travel faster than light, which could conceivably allow for superluminal communication. Because such a particle would violate the known laws of physics, many scientists reject the idea that they exist. By contrast, tachyonic fields – quantum fields with imaginary mass – do exist and exhibit superluminal group velocity under some circumstances. However, such fields have luminal signal velocity and do not allow superluminal communication.

Quantum nonlocality

Quantum mechanics is non-local in the sense that distant systems can be entangled. Entangled states lead to correlations in the results of otherwise random measurements, even when the measurements are made nearly simultaneously and at far distant points. The impossibility of superluminal communication led Einstein, Podolsky, and Rosen to propose that quantum mechanics must be incomplete (see EPR paradox).

However, it is now well understood that quantum entanglement does not allow any influence or information to propagate superluminally.

Practically, any attempt to force one member of an entangled pair of particles into a particular quantum state, breaks the entanglement between the two particles. That is to say, the other member of the entangled pair is completely unaffected by this "forcing" action, and its quantum state remains random; a preferred outcome cannot be encoded into a quantum measurement.

Technically, the microscopic causality postulate of axiomatic quantum field theory implies the impossibility of superluminal communication using any phenomena whose behavior can be described by orthodox quantum field theory. A special case of this is the no-communication theorem, which prevents communication using the quantum entanglement of a composite system shared between two spacelike-separated observers.

Wormholes

If wormholes are possible, then ordinary subluminal methods of communication could be sent through them to achieve effectively superluminal transmission speeds across non-local regions of spacetime. Considering the immense energy or exotic matter with negative mass/negative energy that current theories suggest would be required to open a wormhole large enough to pass spacecraft through, it may be that only atomic-scale wormholes would be practical to build, limiting their use solely to information transmission. Some hypotheses of wormhole formation would prevent them from ever becoming "timeholes", allowing superluminal communication without the additional complication of allowing communication with the past.

Fictional devices

Tachyon-like

The Dirac communicator features in several of the works of James Blish, notably his 1954 short story "Beep" (later expanded into The Quincunx of Time). As alluded to in the title, any active device received the sum of all transmitted messages in universal space-time, in a single pulse, so that demultiplexing yielded information about the past, present, and future.

Superluminal transmitters and ansibles

The terms "ultrawave" and "hyperwave" have been used by several authors, often interchangeably, to denote faster-than-light communications. Examples include:

• E. E. Smith used the term "ultrawave" in his Lensman series, for waves which propagated through a sub-ether and could be used for weapons, communications, and other applications.
• In Isaac Asimov's Foundation series, "ultrawave" and "hyperwave" are used interchangeably to represent a superluminal communications medium. The hyperwave relay also features.
• In the Star Trek universe, subspace carries faster-than-light communication (subspace radio) and travel (warp drive).
• The Cities in Flight series by James Blish featured ultrawave communications which used the known phenomenon of phase velocity to carry information, a property which in fact is impossible. The limitations of phase velocity beyond the speed of light later led him to develop his Dirac communicator.
• Larry Niven used hyperwave in his Known Space series as the term for a faster-than-light method of communication. Unlike the hyperdrive that moved ships at a finite superluminal speed, hyperwave was essentially instantaneous.
• In Richard K. Morgan's Takeshi Kovacs novels human colonies on distant planets maintain contact with earth and each other via hyperspatial needlecast, a technology which moves information "...so close to instantaneously that scientists are still arguing about the terminology".

A later device was the ansible coined by Ursula K. Le Guin and used extensively in her Hainish Cycle. Like Blish's device it provided instantaneous communication, but without the inconvenient beep.

The ansible is also a major plot element, nearly a MacGuffin, in Elizabeth Moon's Vatta's War series. Much of the story line revolves around various parties attacking or repairing ansibles, and around the internal politics of ISC (InterStellar Communications), a corporation which holds a monopoly on the ansible technology.

The ansible is also used as the main form of communication in Orson Scott Card's Ender's Game series. It is inhabited by an energy based, non-artificial sentient creature called an Aiúa that was placed within the ansible network and goes by the name of Jane. It was made when the humans realized that the Buggers, an alien species that attacked Earth, could communicate instantaneously and so the humans tried to do the same.

Quantum entanglement

• In Ernest Cline's novel Armada, alien invaders possess technology for instant "quantum communication" with unlimited range. Humans reverse engineer the device from captured alien technology.
• In the Mass Effect series of video games, instantaneous communication is possible using quantum-entanglement communicators placed in the communications rooms of starships.
• In the Avatar continuity, faster-than-light communication via a subtle control over the state of entangled particles is possible, but for practical purposes extremely slow and expensive: at a transmission rate of three bits of information per hour and a cost of $7,500 per bit, it is used for only the highest priority messages.
• Charles Stross's books Singularity Sky and Iron Sunrise make use of "causal channels" which use entangled particles for instantaneous two-way communication. The technique has drawbacks in that the entangled particles are expendable and the use of faster-than-light travel destroys the entanglement, so that one end of the channel must be transported below light speed. This makes them expensive and limits their usefulness somewhat.
• In Liu Cixin's novel The Three-Body Problem, the alien Trisolarans, while preparing to invade the Solar System, use a device with Ansible characteristics to communicate with their collaborators on Earth in real time. Additionally, they use spying/sabotaging devices called 'Sophons' on Earth which by penetration can access any kind of electronically saved and visual information, interact with electronics, and communicate results back to Trisolaris in real-time via quantum entanglement. The technology used is "single protons that have been unfolded from eleven space dimensions to two dimensions, programmed, and then refolded" and thus Sophons remain undetectable for humans.

Psychic links

Psychic links, belonging to pseudoscience, have been described as explainable by physical principles or unexplained, but they are claimed to operate instantaneously over large distances.

In the Stargate television series, characters are able to communicate instantaneously over long distances by transferring their consciousness into another person or being anywhere in the universe using "Ancient communication stones". It is not known how these stones operate, but the technology explained in the show usually revolves around wormholes for instant teleportation, faster-than-light, space-warping travel, and sometimes around quantum multiverses.

In Robert A. Heinlein's Time for the Stars, twin telepathy was used to maintain communication with a distant spaceship.

Peter F. Hamilton's Void Trilogy features psychic links between the multiple bodies simultaneously occupied by some characters.

In Brandon Sanderson's Skyward series, characters are able to use "Cytonics" to communicate instantaneously over any distance by sending messages via an inter-dimensional reality called "nowhere".

Other devices

Similar devices are present in the works of numerous others, such as Frank Herbert and Philip Pullman, who called his a lodestone resonator.

Anne McCaffrey's Crystal Singer series posited an instantaneous communication device powered by rare "Black Crystal" from the planet Ballybran. Black Crystals cut from the same mineral deposit could be "tuned" to sympathetically vibrate with each other instantly, even when separated by interstellar distances, allowing instantaneous telephone-like voice and data communication. Similarly, in Gregory Keyes' series The Age of Unreason, "aetherschreibers" use two-halves of a single "chime" to communicate, aided by scientific alchemy. While the speed of communication is important, so is the fact that the messages cannot be overheard except by listeners with a piece of the same original crystal.

Stephen R. Donaldson, in his Gap cycle, proposed a similar system, Symbiotic Crystalline Resonance Transmission, clearly ansible-type technology but very difficult to produce and limited to text messages.

In "With Folded Hands" (1947) and The Humanoids (1949), by Jack Williamson, instant communication and power transfer through interstellar space is possible with rhodomagnetic energy.

In Ivan Yefremov's 1957 novel Andromeda Nebula, a device for instant transfer of information and matter is made real by using "bipolar mathematics" to explore use of anti-gravitational shadow vectors through a zero field and the antispace, which enables them to make contact with the planet of Epsilon Tucanae.

In Edmond Hamilton's The Star Kings (1949), the discovery of an unknown form of electromagnetic radiation called sub-spectrum rays moves faster than light. The fastest of these are those of the Minus-42nd Octave, which allows for real time telestereo communication with anyone within the galaxy.

In Cordwainer Smith's Instrumentality novels and stories, interplanetary and interstellar communication is normally relayed from planet to planet, presumably at superluminal speed for each stage (at least between solar systems) but with a cumulative delay. For urgent communication there is the "instant message", which is effectively instantaneous but very expensive.

In Howard Taylor's web comic series Schlock Mercenary, superluminal communication is performed via the hypernet, a galaxy-spanning analogue to the Internet. Through the hypernet, communications and data are routed through nanoscopic wormholes, using conventional electromagnetic signals.