Covariance and contravariance of vectors 3 for some invertible n. The expansion in orthonormal basis is mathematically simple. The dual space is the space of linear functions that assigns a vector a number. What is contravariant and covariant vector, tensor in. It need to be expanded in some basis for practical calculation and its components do depend on the chosen basis. In particular, the topic of a constantcurvature space is examined. Vectors with an upper index are called contravariant, and those with a lower index. Basic fourmomentum kinematics as lunds universitet. This fact that a covariant vector like w can convert any contravariant vector like v into a scale invariant value like f is summarized by saying that w is a linear functional. A contravariant 4 vector is an object defined as x x0. Scalars, vectors and tensors a scalar is a physical quantity that it represented by a dimensional number at a particular point in space and time.
We now derive some basic fourvector identities in complex fourdimensional spacetime frame, which generalize standard vector identities in threedimensional euclidean space 5 6. When studying tensor calculus the distinction between covariance and contravariance may be obscure and is rarely explained visually. The relation between the potential a and the fields e and b given in section 4. These and other pictorial examples of visualizing contravariant and covariant vectors are discussed in am. Our definition of a contravariant 4 vector in 1 whist easy to understand is not the whole story. Intuitively, by a parallel vector field, we mean a vector field with the property that the vectors at different points are parallel. A contravariant vector or tangent vector often abbreviated simply as vector, such as a direction vector or velocity vector has components that contravary with a change of basis to compensate. Obviously, our space coordinates and time variable join to form a contravariant four vector x. For a contravariant vector to be coordinate system invariant, the components of the vector must transform oppositely contra as in against, like contrary to the change of the basis vectors the axes to cancel the change. Since the space parts form the time component of a four vector, e must be the timespace part of a tensor of rank two. One is called the contravariant vector or just the vector, and the other one is called the covariant vector or dual vector or. Lorentz transformations expressed in 4 vector notation. The basis for these names will be explained in the next section, but at this stage it is just a name used to distinguish two types of vector.
Relativistic basic question four vector, lorentz matrix. The upper index is the row and the lower index is the column, so for contravariant transformations, is the row and is the column of the matrix. You can also imagine a line segment drawn between two events. What we have been using thus far in these lecture notes are contravariant 4 vectors a, denoted by the superscript aaaaa 01 2 3,, contravariant 4 vector. Notice how the contravariant basis vector g is not differentiated. It gives me great pleasure to write the foreword to dr. Mar 31, 2020 the covariant derivative in electromagnetism. So you see that you have various errors in your index notation.
Here, each vector y j of the f basis is a linear combination of the vectors x i of the f basis, so that contravariant transformation a vector v in v is expressed uniquely as a linear combination of the elements of the f basis as 2. I know the author as a research scholar who has worked with me for several years. For spacetime, the derivative represents a four by four matrix of partial derivatives. So whether a vector is treated as contravariant or covariant is a matter of choice.
Vector is a physical quantity and it does not depend on any coordinate system. I will go over the needed concepts briefly but you are encouraged to consult a general relativity textbook for this. Physics 221a fall 2019 appendixe introduction to tensor analysis 1. A contravariant 4vector am is a set of four numbers. This material is covered thoroughly in chapters one and seven in our. Normally we think about threedimensional vectors, but here we have to consider fourvectors. From this we can deduce the 4tensor form for the electromagnetic field. Is there a notion of a parallel field on a manifold. A velocity v in one system of coordinates may be transformed into v0in a new system of coordinates. There seems to be a real difference in the nature of these vectors, as opposed to simply the components and bases used to express them. We regard the inner product as being a bilinear form g. Introduction using the equivalence principle, we have studied the trajectories of free test particles.
Chapter 10 vectors and tensors georgia institute of. Covariant formulation of classical electromagnetism wikipedia. In index notation, the contravariant and covariant components transform according to, respectively. Vectors, metric and the connection 1 contravariant and.
Introduction in a recent derivation 1, the present author identified the unit wave vector k. Covariant and contravariant vectors can be interconverted with use of the metric tensor, given by9. It helps to know some basic differential geometry and to be able to think in a coordinateindependent manner. Tensors, contravariant and covariant components of a vector. Give and example of a contravariant vector field that is.
Vectors and lorentz transformations wiley online library. For the contravariant components it is customary to use superscripts for the indices such as j and k. And so many physicists have the sloppy tendency to say that vectors transform contravariantly. However, in physics, we tend do ignore the basis vectors and only look at the components.
The g term accounts for the change in the coordinates. Vectors and tensors in curved space time asaf peer1 may 20, 2015 this part of the course is based on refs. In special relativity, a fourvector is an object with four components, which transform in a. For a scalar function f in ndimensional euclidean space, we have from multivariate calculus df. This book is a humble step of efforts made by him to prove him to be a dedicated and striving teacher who has worked relentlessly in this field. The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism in particular, maxwells equations and the lorentz force in a form that is manifestly invariant under lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems. Now what bothers me is this lowering and rising index stuff where one can transform a contravariant vector to a covariant one and viceversa by multiplying by the minkowski metric tensor in the special relativity case. Introduction these notes contain an introduction to tensor analysis as it is commonly used in physics, but mostly limited to the needs of this course.
This book has been presented in such a clear and easy way that the students will have no difficulty. Raising and lowering indices suppose we have a contravariant vector v i at some point x of the manifold, and a covariant vector wi at the same. The presentation is based on how various quantities transform under coordinate transformations, and is fairly standard. Then any vector a can be expressed with respect to this coordinate system in terms of its base vectors as where c 1, c 2, c 3 are called the contravariant components of a. Mathematical operations, such as converting one sort of vector into another sort of vector, are done on vector spaces. Contrary to the threedimensional euclidean case, the norm of a four vector. Which of these four terms does the figure illustrate. From this we can deduce the 4 tensor form for the electromagnetic field. Covariant formulation of classical electromagnetism. Nazrul islams book entitled tensors and their applications. Show that x i is identically zero in any coordinate system. Curl of gradient fourvector a gradient fourvector generated through application of the spacetime derivative fourvector.
The vector space or linear space, mve4 space, or just space of all kcontravariant, covariant tensors tensors of valence k at the point p in a manifold m will be denoted tk mp, with tmp and t. Pictorial representation of the inner product between a contravariant vector and a covariant vector. Fourvectors and lorentz transformations 477 the derivative with respect to a contravariant coordinate transforms as a covariant 4vector. A is a fixed object usually called a vector that may be described in various coordinate systems. You should only sum over one upper and one lower index. Introduction to tensors contravariant and covariant vectors rotation in 2space. In general an upper index inside a derivative works i. If one does this operation on a 4 position contravariant it will just change some sign of the 4 position but not the. Physics 221a fall 2019 appendixe introduction to tensor. From this we see that a covariant 4 vector transforms as by chain rule remark.
But it is also customary to call the collection of coefficients a i a vector more specifically, a contravariant vector, while we have already called. However, you will read all over the literature that vectors are contravariant. Complex spacetime frame, four vector identities, contravariant and covariant forms, complex tensors 1. In the relativistic field theories, we must use lorentz scalars to express the iiinteractions. Lorentz invariance lorentz invariance of the laws of physics is satisfied if the laws are cast in terms of four.
For example, if the axes were rotated in one direction, the component representation of the vector. Suppose that x j is a contravariant vector field on the manifold m with the following property. Section of covariant and contravariant vectors explains this. Transformation of contravariant covariant components. Vectors, metric and the connection 1 contravariant and covariant vectors 1. General relativitycontravariant and covariant indices. The temporal direction is specified as an imaginary axis with unit.
Apr 26, 2019 videos 1 and 2 of this playlist concern a simple geometric example in which covariant and contravariant components of a two dimensional vector are constructed. Vectors and tensors will be specialized to fourvectors and fourtensors. A lorentz matrix should not turn a covariant vector to a contravariant vector, only the metric does. If it transforms as coordinates, and therefore in the opposite direction as basis vectors, its a contravariant vector, or just a vector. The contravariant vector is a \stick whose magnitude is the length of the stick, while the. If c i is covariant and v j is contravariant, then c k v k is a scalar. A covariant 4 vector a is denoted by its subscript aaaaa 01 2 3,, covariant 4 vector. May 05, 20 yet, elsewhere i read that the gradient is a covariant vector, and that velocity is a contravariant vector, because of the way the components transform. But in many physical situations we have to choose an nonorthogonal basis or oblique coordinate system. There is another type of vector called a covariant vector that has a di.
We wish to write the velocity vector at some time t 0. Given two inertial or rotated frames of reference, a fourvector is defined as a quantity which transforms according to the lorentz transformation matrix. Lorentz invariance and the 4vector dot product uspas. The covariant derivative is the derivative that under a general coordinate transformation transforms covariantly. Vectors and covectors in special relativity jim napolitano march 12, 2010 these notes are meant to accompany the course electromagnetic theory for the spring 2010 term at rpi. Were talking blithely about derivatives, but its not obvious how to define a derivative in the context of general relativity in such a way that taking a derivative results in wellbehaved tensor. First we will explain the distinction between the covariant and contravariant components of vectors, thinking of vector fields where a vector is defined at a. In other words, the 0th component is the timelike component. The covariant derivative of the r component in the q direction is the regular derivative plus another term. Tensors have their applications to riemannian geometry, mechanics, elasticity, theory of relativity, electromagnetic theory and many other disciplines of science and engineering.
Tensors, contravariant and covariant components of a. Gubkina,8,moscow,119991,russia 20 september 2016 abstract some basic theorems on killing vector. Covariant and contravariant components of a vector youtube. Covariant versus contravariant vectors physics forums. Verify the following claim in the proof of proposition 4. Covariance and contravariance of vectors wikipedia. The vector space or linear space, mve4 space, or just space of all k contravariant, covariant tensors tensors of valence k at the point p in a manifold m will be denoted tk mp, with tmp and t.
The \stick is imbedded in the \lasagna and the inner product is equal to the. As transforms as a contravariant 4 vector and as a scalar indeed transforms as a contravariant 4 vector, so the notation makes sense. How do contravariant and covariant fourvectors multiply. The length of a physical vector must have physical units. Contravariant and covariant vectors physics forums. Useful expressions for the contravariant coordinates of ov are, using the sine rule, 4 covariant coordinates the covariant components of a vector are defined by the transformation 5. Also this implies that there is the covariant four vector. The contravariant vector inverts the sign of the spacelike component, and is indicated by using a superscript for the components. Four vector dot products are said to be lorentz scalars.
That is, the matrix that transforms the vector components must be the inverse of the matrix that transforms the basis vectors. Tk mn rp is a mixed tensor with contravariant rank 4 and covariant rank 2. If we apply the same correction to the derivatives of other secondrank contravariant tensors, we will get. The idea of a covariant derivative of a vector field a in the. Related to the examples above that are given as contravariant vectors, there are also the. A 1form can also be thought of geometrically as a series of infinitestimal planes placed parallel to each other at some interval.
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