What Is an Electric Field 4-Vector?

[pmb_phy]

I made a new web page to describe/define the electric field 4-vector to someone. I thought I'd post a link to that page here. Some of you might find it interesting since most people only think of the electric field as being only the components of a 2-tensor. See

http://www.geocities.com/physics_world/sr/efield_4vector.htm

This is similar to the energy measured by observer defined as follows

Let U_obs be the 4-velocity of an observer. Let P be the 4-momentum of a particle. Then the scalar product of the two is called E_obs = energy measured by observer

E_obs = P*U_obs

In some sense of the term this can be viewed as an invariant scalar. In any case E_obs is a tensor of rank zero and is therefore a scalar by definition. Like wise E^a = F^ab U_a is the electric field 4-vector as measured by the observer whose 4-velocity is U.

Pete

 

[Stingray]

It might also be helpful to note that since F is antisymmetric, E_a U^a = 0. So E is adequately described by only three components, which makes it more clear how it is related to the electric field vector used in introductory classes.

 

[pmb_phy]

It might also be helpful to note that since F is antisymmetric, E_a U^a = 0. So E is adequately described by only three components, which makes it more clear how it is related to the electric field vector used in introductory classes.

I do't follow you. Why do you hold that since E*U = 0 that E can be described by three numbers? Take 4-force - F*U = 0 for a rest mass preserving force, yet 4-force requires 4 quantities to be defined.

Anyway ...

FYI - Due to recent observations elsewhere it appears that the notion of an electric field 4-vector is quite difficult, if not impossible, for some people to grasp. They continuosly confuse the electric field 4-vector with the electric field. These are two distinct quantities which are related to each other.

Its similar to 4-momentum. Had one only known of the term "momentum" as in 3-momentum and known of P = m₀U only as the energy-momentum 4-vector then these same people might also be confused had they heard the term "momentum 4-vector". Momentum 4-vector is a 4-vector and is a different quantity than "momentum" (i.e. 3-momentum). A similar thing applies to the electric field 4-vector. This is a 4-vector but is a verfy different thing that the electric field 3-vector.


Pete

 

[Stingray]

I do't follow you. Why do you hold that since E*U = 0 that E can be described by three numbers? Take 4-force - F*U = 0 for a rest mass preserving force, yet 4-force requires 4 quantities to be defined.

In a general coordinate system, it does of course require 4 numbers at each point. But in the frame of the observer, it might as well be a 3-vector. His (naturally defined) time component is always zero. So there are really only three functions. The 4-vector in an arbitrary coordinate system can be written down in terms of these three components by using a coordinate transformation.

Alternatively, E has four unknowns, and E*U=0 is one equation that is given for them. There are therefore three degrees of freedom still remaining once you take into account that constraint.

The same remarks apply for your (rest) mass-preserving force.