Timelike vs. null vs. spacelike four-current

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In summary, the conversation discusses the relationship between electric charge, four-current, and reference frames. It is mentioned that a ball of static electric charge has a timelike four-current, while a wire carrying current has a spacelike four-current. It is also noted that a spacelike vector can be decomposed into two timelike vectors, and that the concept of null vectors is intriguing. The conversation ends with questions about the behavior of null vectors and their interaction with light.
  • #1
djy
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I've looked in several sources but haven't seen this addressed specifically.

If you have a ball of static electric charge, then the four-current is timelike. If the ball is moving, then you can transform to its rest frame, in which the four-current's spacelike components are all zero.

If you have a wire carrying current and it has no net charge density, then the four-current is spacelike. If the whole wire is moving, one can transform into its rest frame, in which the four-current's timelike component is zero.

It seems that one can always decompose a spacelike vector into two timelike vectors: for example, the four-current of the moving electrons and the four-current of the stationary protons.

As for null four-current, I wonder if there is any special significance, other than the fact that it is null in all reference frames? (Even by itself, this seems pretty intriguing.)
 
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  • #2
I have no idea. I'll add it to my list of "100 or so physics questions to ponder." It's good enough that you're talking about massive stuff with null vectors.

Maybe I do have some ideas...

1. If we fail to attach mass to the divergence of the electric field (charge densty), then charge density will have a second order solutions that propagates at a velocity of c. But you've come up with this quantity that already has null nature, and doesn't require such a nonphysical charge attribute. Given all that, I'm curious as to what obtains when we put waves of null charge/current density in motion.

2. Extending your idea a little further, can we also consider null vectors in the energy and momentum of a combination of massive particles, so that we can also consider waves in this vector. Do these wave propagate at c?

3. How would light interact with null, or near null, electric current density?
 
  • #3
For #2, I don't think it's possible, because while electric charge can be positive or negative, mass can never be negative.
 
  • #4
In trying to understand your question, I got stuck on your paragraph 4. In what sense can you decompose a spacelike vector into two timelike ones? I thought decomposing a vector meant taking its projections along the axes of a frame - but a spacelike vector cannot be decomposed into two timelike ones in this sense.
 
  • #5
yossell said:
In trying to understand your question, I got stuck on your paragraph 4. In what sense can you decompose a spacelike vector into two timelike ones? I thought decomposing a vector meant taking its projections along the axes of a frame - but a spacelike vector cannot be decomposed into two timelike ones in this sense.

Maybe 'decompose' is the wrong word. I mean that if you have a spacelike vector [itex]{\bf j}[/itex] then you can find two timelike vectors [itex]{\bf j}_1, {\bf j}_2[/itex] such that [itex]{\bf j}_1 + {\bf j}_2 = {\bf j}[/itex], as long as one is allowed to point "backward" in time--which is exactly what happens with negative charge density.
 
  • #6
Ah - I think I see now. But then, in this sense, aren't null vectors also the sum of two timelike vectors, one forward and one backward in time?
 
  • #7
Yes, that too.
 

1. What is the definition of four-current in spacetime?

The four-current is a four-vector quantity that describes the flow of a conserved quantity, such as charge or energy, in spacetime. It is represented by the symbol J, and has a time component (denoted by J0) and three spatial components (denoted by J1, J2, and J3).

2. What is the difference between timelike, null, and spacelike four-currents?

A timelike four-current is one in which the magnitude of the four-vector is greater than the speed of light, indicating a flow of energy or charge in the time direction. A null four-current has a magnitude equal to the speed of light, representing a flow in the direction of a light ray. A spacelike four-current has a magnitude less than the speed of light, indicating a flow in a spatial direction.

3. How are timelike, null, and spacelike four-currents related to causality?

Timelike four-currents are associated with events that can influence each other causally, as they are within each other's light cones. Null four-currents are associated with events that can influence each other but not causally, as they are on each other's light cones. Spacelike four-currents are associated with events that are too far apart to influence each other causally, as they are outside each other's light cones.

4. Can the direction of a four-current change in spacetime?

Yes, the four-current can change direction in spacetime, representing a change in the flow of a conserved quantity. This can occur due to accelerations, interactions with other particles, or changes in the reference frame.

5. How is the four-current related to the electromagnetic field tensor?

The four-current is related to the electromagnetic field tensor through Maxwell's equations, which describe the interaction between electric and magnetic fields and charged particles. The four-current appears in the equations as a source term for the electromagnetic field, indicating that the flow of charge is responsible for the creation of the field.

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