|PHY 514||Fields, Symmetries, Action, and Conservation Laws||Spring 2012|
|This page: Home ⇒ Week 03|
References for relativistic tensors, Lagrangian, and conservation laws: Jackson chapter 6 sections 7,10, chapter 11 sections 5,6,7, and chapter 12 sections 1,7. Landau-Lifshitz (Fields) chapter 2 sections 8,9, chapter 3, sections 23,24,25, chapter 4 sections 27,31,32. Misner-Thorne-Wheeler chapter 3. Panofsky-Phillips chapter 18 sections 1,2, chapter 21 section 1,3. Schwartz chapter 3 section 5, chapter 5 sections 2,3.
References for differential forms: Misner-Thorne-Wheeler chapter 3 section 4, chapter 4. Math for physicists: differential forms by S.S. Gubser. Applications of Differential Geometry to Physics by G.W. Gibbons.
The Third (2002) edition of Classical Mechanics by Goldstein-Poole-Safko has been revised considerably to include differential forms in chapter 7, and classical electromagnetic fields and Noether's theorem in chapter 13. I learned classical mechanics from Professor Goldstein
The Maxwell equations were originally published as partial differential equations in vector component form, for example in Maxwell's 1860 article On Physical Lines of Force in the Philosophical Magazine.
The vector calculus notation used in most physics textbooks was introduced by J.W. Gibbs and by O. Heaviside towards the end of the nineteenth century. This greatly simplified writing Maxwell's equations: compare Maxwell's component notation (left column) with modern vector notation (right column)
Equations F,G,H,I and L in the left column in Gothic letters look like vector notation, but actually represent W.R. Hamilton's Quaternions, which were used by Maxwell in his 1873 textbook A Treatise on Electricity and Magnetism.
We have seen that using 4-vector notation in Minkowski space simplifies writing the scalar and vector potential fields, and the charge and current densites, in terms of a relativistic photon field and a 4-vector electromagnetic current density
Minkowski spacetime with 3 spatial and 1 time dimension is a flat pseudo-Riemannian 4-manifold with metric tensor
which defines the inner product of two 4-vectors
The Metric on the space defines distances between points in the space. The adjective "flat" refers to the fact that the components of the metric tensor are constant independent of spacetime point . The prefix "pseudo" refers to the fact that the inner product of a non-zero 4-vector with itself can be positive, zero, or negative. This squared norm represents the "length" of the vector in the space.
The electromagnetic field in relativistic notation is a second rank tensor with two spacetime indices. Its covariant components are given by
Note: covariant indices in the numerator imply contravariant indices in the denominators of the partial derivatives.
To relate to and , apply the definition to typical components, for example
The explicit result is
To obtain contravariant components raise each index using the metric tensor, which can also be understood as matrix multiplication:
Note that each raised (or lowered) "0" index contributes a negative sign.
From the electromagnetic tensor we can construct 0-rank tensor by multiplying corresponding contra- and covariant components and summing over all indices:
The totally antisymmetric 3-index Levi-Civita tensor
is useful, for example, in defining and manipulating vector products
where is the metric tensor in 3-dimensional Euclidean space.
The rank-4 Levi-Civita tensor density in Minkowski space is totally antisymmetric and defined by
Using instead would give an equivalent tensor. The choice better matches the definition of the 3-dimensional symbol
The Levi-Civita tensor can be used to construct the dual electromagnetic tensor in which the electric and magnetic components exchange roles:
Its components can be related to and by evaluating typical components, for example
The explicit result is
The contravariant components are
From the electromagnetic tensor and its dual we can construct a second 0-rank tensor
In the absence of sources, the Maxwell equations are symmetric under a duality transformation, which interchanges electric and magnetic fields: this symmetry is discussed in Jackson Section 6.11 On the Question of Magnetic Monopoles.
The field and dual tensors each have two 4-vector indices. From each tensor, a 4-vector can be constructed by taking the 4-divergence with respect to any one of the two indices.
Consider the divervence of the field tensor
These are the two Maxwell equations with sources.
The divergence of the dual tensor is even simpler to compute
because derivatives commute while the Levi-Civita tensor is antisymmetric in each pair of indices. This equation contains the two source-free Maxwell equations.
It is useful to write vectors in 3-dimensional Euclidean space in terms of an orthonormal system of 3 unit vectors, for example
This concept can be extended to 4-vectors in Minkowski space, but it must be done carefully to distinguish contravariant and covariant components. The contravariant gradient of the gauge function is
The minus sign is due to
The set of contravariant 4-vectors provides a basis for the Tangent space to the manifold. The concept is very general and can be applied to arbitrary Riemannian manifolds. In General Relativity such a basis depends on location and is called a Tetrad or vierbein (4-legs) or frame field.
An Integral is the weighted sum of the values of a function over a range of values :
The measure specifies the weight to be assigned to the value of of at .
A Differential form on a submanifold of a differentiable manifold is everything under the integral sign in an integral of a function over the submanifold:
Differential forms on -dimensional submanifolds of an -dimensional manifold are called -forms on . The degree (also called grade) of the differential form can take values , corresponding to points, curves, area, volumes, hypervolumes, .
Electrodynamics is defined on spacetime with . Electrodynamics objects in spacetime can defined in terms of -forms with . There are independent differential forms.
A 0-dimensional submanifold of is a set of one or more points in . A differential 0-form is defined to be a smooth (differentiable) function on .
The gauge function , which transforms the gauge of the 4-vector potential according to
is an example of a 0-form on spacetime.
Each component of points in spacetime is a 0-form. These component 0-forms are associated with a special set of 1-forms. Consider, for example, a segment along the axis with length
This defines a 1-form , where 2 denotes second or component, because its integral over a 1-dimensional subspace is the line integral along the 2-axis with integrand .
The coordinate 1-forms
are defined to give the corresponding line integrals
and provide a natural basis for constructing -forms on .
Note that the superscript is not a relativistic 4-vector index. It simply labels the direction in spacetime of the integration axis.
The electromagnetic 4-vector potential is associated with the potential 1-form
The potential 1-form has a simple geometrical interpretation: it represents the 3-dimensional hypersurface that is perpendicular to the direction of the contravariant 4-vector .
In flat Euclidean space perpendicular directions differ by . Perpendicular directions in spacetime are defined by zero inner product using the metric .
The potential 1-form can be visualized as a family of - dimensional hypersurfaces in . It provides a line-integral measure at each spacetime point along the normal direction to this family of hypersurfaces.
Differenential forms on a manifold represent volume elements of submanifolds.
Areas and volumes in 3-dimensional Euclidean space can be constructed using the vector Cross product introduced by Gibbs. Vector algebra based on the cross product works only in flat 3-dimensional space.
The generalization to manifolds of any dimension with non-Euclidean geometry is the Exterior algebra based on the wedge product.
A simple example of a wedge product is the 2-form
The wedge product of two 1-forms is antisymmetric. There six different 2-forms constructed from the four basis 1-forms provide a basis for 2-forms in Minkowski space.
The electromagnetic field tensor can be represented by the field 2-form
The Exterior derivative is a differential operator on a -form that produces a form.
A 0-form is just a continuous function on space time. The exterior derivative of a 0-form is the 1-form (covariant derivative)
corresponding to the 4-gradient of .
The exterior derivative of a -form is defined to satisfy two properties
These requirements determine the exterior derivative of the potential 1-form:
and imply that the exterior derivative of the electromagnetic field 2-form
The Hodge dual or Hodge star operator converts a -form into an -form, where is the dimension of the manifold.
Its action on a basis -form is defined to be
where is an even permutation of .
The Hodge dual of a function or 0-form is a form
which represents the function as integrand and the volume measure so
The Hodge dual of the current 1-form is a form
The Hodge dual of the electromagnetic field 2-form is also an form
and represents the dual field tensor .
The Maxwell equations can be generated as a system of differential forms starting from the potential and current 1-forms and applying the exterior derivative operator and the Hodge dual operator repeatedly.
The derivative of the potential 1-form gives the electromagnetic field 2-form
Applying derivation and Hodge star to the field 1-form
gives the source-free Maxwell equations and the dual field.
Applying derivation and Hodge star to the dual electromagnetic field 2-form
gives the Maxwell equations with sources and recovers the field 2-form.
Finally, applying derivation to the current 3-form
gives the current conservation equation.
In a relativistic theory, the squared norm of spacetime separation of any two events at and
must have the same numerical value when measured in any inertial reference frame.
A transformation is function that maps variables in one inertial reference frame to those in another. This is the "passive" view of transformations: the manifold of events in spacetime does not change. Different observers make measurements on the same universe of events.
In the "active" view, the same observer measures different states of the same system that are related by a transformation of variables.
For a field theory in infinite spacetime the passive view is preferred because active transformations cannot be realized. In classical physics, different observers can measure the same universe of events with negligible effect on its history.
The active and passive views should be equivalent for finite classical systems. In quantum physics, the active view is preferred because measurements disturb the system profoundly.
In Minkowski spacetime, the most important symmetries arise from global linear transformations of event coordinates:
where are the coordinates measured by observer and are the coordinates measured by observer , is a constant "translation" 4-vector and is a constant "4-rotation" of the 4-vector event coordinates.
The translation obviously does not change the squared norm of separations. The rotation will not change the squared norm if
This condition can be viewed as matrix multiplication of matrices
The squared norm is invariant under two discrete transformation: space reflection, and time reversal.
There are ten continuous transformations under which the norm is invariant.
The action for a system described by a Lagrangian function is
The Lagrangian for a field theory is the integral of a Lagrangian density
The Lagrangian density for the electromagnetic field interacting with a given current density is
This does not include the Lagrangian density of the charged matter that constitutes the current.
The Euler-Lagrange equations for a classical field theory can be obtained by treating the field as the continuum limit of mechanical degrees of freedom at each point in space.
The equations for a relativistic electromagnetic potential field are
Noether's Theorem relates continuous symmetries to conservations laws. It applies to systems with a Lagrange function and Euler-Lagrange equations of motion. It states that there is a conserved current for each one-parameter group of continuous symmetry transformations.
is form-invariant under two one-parameter continuous symmetry groups, time translation
and spatial rotation
The Lagrangian has the same functional form in terms of the primed variables as the unprimed variables because it does not depend explicitly on time or the angular coordinate .
Consider a particular solution of Lagrange's equations. The action
has the same numerical value because the same solution is integrated between the same initial and final points. The form invariance of the Lagrangian function implies
In the second equation line the difference is taken at fixed integration time . In the third equation line is the variation at fixed integration variable and commutes with . Lagrange's equation of motion has been used in the fourth equation line. The variation in the generalized coordinate has two components
a variation due to the variation in the time, and the variation at fixed time. This relation has been used in the fifth equation line.
Since the infinitesimal variations are arbitrary functions of time
is constant in time. Invariance under time translation implies conservation of energy
and invariance under rotation about the origin implies conservation of angular momentum
The Poincaré symmetries of Minkowski spacetime can be viewed as ten independent one-parameter transformations: 4 spacetime translations, 3 spatial rotations, and 3 Lorentz boosts.
Noether's theorem for the electromagnetic field is derived in detail in Section 13.7 of Goldstein-Poole-Safko. The proof follows the same steps outlined above for the 2-D oscillator with the following expression for the conservered Noether currents:
The Stress-energy tensor or stress tensor represents the conserved currents Noether currents of the continuous Poincaré symmetries of electromagnetism. This tensor is very important in other fields of science and engineering, including continuum and fluid mechanics, and it is the source of the gravitational field in the General Theory of Relativity.
Noether's theorem implies conservation of the canonical stress tensor
The symmetric stress tensor is
Please format your submission as a single PDF file according to the instructions in the syllabus and submit it before 11:59 pm on Sunday, February 12.
Problem 1: Show that (a) If and are perpendicular in one inertial reference frame, they are perpedicular in all inertial reference frames. (b) If the angle between and is acute (obtuse) in one inertial reference frame, then it is acute (or obtuse) in any other frame. (c) If is perpendicular to but , then there exists frames in which the electromagnetic field is either purely electric or purely magnetic [Note: the wording of this problem taken from Panofsky-Phillips chapter 18 means more precisely "either there exist frames in which the field is purely electric, or there exist frames in which the field is purely magnetic, but not both"].
Problem 2: Jackson Problem 11.10: For the Lorentz boost and rotation matrices
for any real 3-vectors and hence
where and is the boost parameter or rapidity defined by .
Problem 3: Rewrite the Vector Formulas from the inside front cover of Jackson using the notation of differential geometry in 3 dimensional Euclidean space .
Problem 4: Verify the Vector Formulas using Mathematica. Professor David Slavsky has a nice tutorial on Vector Analysis using Mathematica.
Problem 5: Explain and prove Stokes' Theorem in differential geometry:
Show that Gauss', Stokes' and Green's theorems of vector calculus (Assignment 0) are special cases. Can you relate Stokes' theorem to the Helmholtz decomposition?