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Section 6. Four vectors. Four radius vector. Coordinates of an event in 4-space are ( ct,x,y,z ). Radius vector in 4-space = 4-radius vector. Square of the “length” (interval) does not change under any rotations of 4 space. . General 4-vector.

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Section 6

Section 6

Four vectors

Four radius vector
Four radius vector

  • Coordinates of an event in 4-space are (ct,x,y,z).

  • Radius vector in 4-space = 4-radius vector.

  • Square of the “length” (interval) does not change under any rotations of 4 space.

General 4 vector
General 4-vector

  • Any set of four quantities (A0,A1,A2,A3) that transform like (x0, x1, x2, x3) under all transformations of 4 space, including Lorentz transformations, is called a four-vector.

Two types of 4 vector components
Two types of 4-vector components

  • Contravariant: Ai, with superscript index

  • Covariant: Ai, with subscript index

  • Lorentz transform changes sign for covariant components: V -> -V

Square of four vector
Square of four vector

  • Summation convention: Sum assumed if

    • Indices are repeated AND

    • One is subscript while other is superscript.

    • e.g.AiAi

  • Such repeated indices are called “dummy indices.”

  • Latin letters are reserved for 4-D indices.

  • Greek letters are reserved for 3-D indices.

Here are some properties of dummy indices
Here are some properties of dummy indices

  • We can switch upper and lower indices in any pair of dummy indices: AiBi=AiBi

  • We can rename them: AiBi=AkBk

Scalar product
Scalar Product

  • Scalar Product AiBi is a four scalar.

  • 4-scalars are invariant under rotations.

  • The interval is a 4-scalar.

The components of a general 4 vector have either time or space character
The components of a general 4-vector have either time or space character

  • A0 is the “time component”

  • A1, A2, A3 are “space components”

  • If square of A = AiAi is

    • >0, then A is time like

    • <0, then A is space like

    • =0, then A is a null vector or isotropic

A four-tensor of rank 2 is a set of sixteen quantities space characterAik that transform like products of two four vectors

  • For example, A12 transforms like x1x2 = xy

  • Contravariant: Aik

  • Covariant: Aik

  • Mixed: Aik and Aik

Here are some sign rules for 4 tensors
Here are some sign rules for 4-tensors space character

  • Raising or lowering an index 1,2,3 changes sign of component

  • Raising or lowering the index 0 does not.

  • Symmetric tensors

  • Antisymmetric tensors

Here are some rules for tensor equations
Here are some rules for tensor equations space character

  • The two sides of every tensor equation must contain identical, and identically placed, free indices.

  • Shifting of free indices must be done simultaneously on both sides.

  • Pairs of dummy indices can be renamed or raised/lowered anytime.

Contraction space character

  • From tensor Aik, we can form a scalar by contraction: Aii = “the trace” of Aik

  • The scalar product AiBi is a contraction of the tensor AiBk

  • Contraction lowers rank of tensor by 2

Some special tensors are the same in all coordinate systems
Some special space charactertensors are the same in all coordinate systems.

  • Unit 4-tensor

  • Metric tensor, obtained by raising or lowering one index of unit tensor, taking into account the sign change for the space components.

Completely antisymmetric unit tensor of 4 th rank
Completely space characterantisymmetric unit tensor of 4th rank.

E iklm is a pseudo tensor
e space characteriklm is a pseudo tensor

  • Under rotations of 4-space (Lorentz transformations) eiklm behaves like a 4-tensor.

  • But under inversions and reflections (change of sign of one or more coordinates) it does not.

  • By definition eiklm is invariant under any transformation, but components of a true rank-4 tensor (e.g. xixkxlxm) with all indices different change sign when one or 3 coordinates change sign.

Pseudo tensors including pseudo scalars
Pseudo tensors, including pseudo scalars, space character

  • Behave like tensors for all coordinate transforms except those that cannot be reduced to rotations (e.g. reflections)

  • Other invariant true tensors: formed from products of components of unit tensor.

If we lower the index of a contravariant 4 vector does the corresponding component change sign
If we lower the index of a space charactercontravariant 4-vector, does the corresponding component change sign?

  • Yes

  • No

  • Sometimes




Dual tensors
“Dual” tensors space character

3d vectors and tensors
3D vectors and tensors space character

  • Completely antisymmetric unit pseudo tensor of rank 3

  • Very useful in proving usual vector identities.

Inversion of coordinates
Inversion of coordinates space character

  • Components of ordinary vector change sign (polar)

  • Components of cross product of two polar vectors do not (axial)

  • Scalar product of polar and axial vector is a pseudo scalar

  • An axial vector is a pseudo vector dual to some antisymmetric tensor.

Antisymmetric 4 tensor a ik
Antisymmetric space character 4-tensor Aik

  • Space components (i,k, = 1,2,3) form a 3-D antisymmetric tensor with respect to spatial transforms.

  • By dual relationship, the components are expressed in terms of 3-D axial vector.

    • e.g. A12 transforms like 3rd component of an axial vector by dual relationship.

  • Components A01, A02, A03, form a 3-D polar vector with respect to spatial transforms.

    • e.g. A01tranforms like tx, which changes sign on spatial inversion.

4 gradient of a scalar function f f x 0 x 1 x 2 x 3
4-gradient of a scalar function space characterf=f(x0, x1, x2, x3)

Types of integration
Types of integration space character

  • In 3D, we have 3 types of integrals

    • Line

    • Surface

    • Volume

  • In 4D, there are 4 types

    • Curve in 4-space

    • 2D surface

    • Hypersurface (3D manifold)

    • 4-D volume

Integral over 2d surface
Integral over 2D surface space character

Integral over a hypersurface 3d manifold
Integral over a space characterhypersurface (3D manifold)

Integral over 4d volume
Integral over 4D volume space character

4d gauss theorem
4D space characterGauss theorem

4d stokes theorem
4D Stokes Theorem space character