Relativistic current density components

[Imagine]

Bonjour,

Relativistic current density components are the vector "J" with the scalar "j*c*rho".

My questions are:

Since it's a scalar and not a vector, what is the meaning of "c" times "rho"?

Merci

 

[marcus]

Originally posted by Imagine
Bonjour,

Relativistic current density components are the vector "J" with the scalar "j*c*rho".

My questions are:

Since it's a scalar and not a vector, what is the meaning of "c" times "rho"?

Merci

the ordinary current density vector J has units "coulombs per square meter per second" C/m² s

more correctly:

Cm^-2 s^-1

the amount of charge flowing thru a sq meter of area in one second

the ordinary spatial charge density ρ has units "coulombs per cubic meter"

Cm^-3

If one multiplies by c = 3E8 m s^-1

one gets cρ also with units of current density Cm^-2 s^-1

[[[impressionistically one can think of some charge which simply by existing is "moving at the speed c along the time axis" and so
could intuitively represent a "current" in the time direction]]]

the main thing is that now since the charge density has been changed to something the same units as the rest, it can be put
together with them to make a 4-vector!

(cρ, j1, j2, j3)

to be combined into a 4-vector all the quantities must have the same units!

-------------
Be careful about the different formats which people use for writing 4-vectors. Sometimes they use the "imaginary number" i, also sometimes written j, which is sqrt(-1) and they write that with the time component so that the metric can be of the form

(ict)² +(x1)² + (x2)² + (x3)²

Other people, perhaps the majority, do not use this "imaginary time" convention and simply introduce a minus sign into the metric

- (ct)² +(x1)² + (x2)² + (x3)²

or they may write the ct component as x0 so that the metric is
- (x0)² +(x1)² + (x2)² + (x3)²

because there is some variation in the conventions, I was not
sure I understood your notation. You would need to explain in
more detail for me to be sure I understand the symbol "j" in your expression jcρ

 

[Imagine]

Merci Marcus,

The "j" shall be "i".

I understand that to convert and to fit the right units, "rho" is multiply my "c". That's all? Is the forth part (rho component) a vector or a scalar, or some hidden component?

Merci encore Marcus

 

[marcus]

I understand that to convert and to fit the right units, "rho" is multiply by "c". That's all? Is the fourth part (rho component) a vector or a scalar, or some hidden component?

You say the rho component is the FOURTH part. So we have yet another format! Is this what it looks like?

(j1, j2, j3, icρ)

In that case the metric being used requires an "i" in the fourth component of the vector

(x1)² + (x2)² + (x3)² + (ix4)²

A more common approach is not to bother with the "i" in the fourth component but simply have a minus sign in the metric:

(x1)² + (x2)² + (x3)² - (x4)²

However written, the charge density is assimilated as one component of a 4-vector.

I would not consider it a scalar, but rather as a PART of a vector.

these 4-vectors are cleverly constructed so that they TRANSFORM correctly when there is a change of observer.
what is static charge density (rho) for one person can be flowing and be a current density (j1...etc) for another person, if that
other person is moving. All these things must be packaged together because they mix around during a change of coordinates. So there is an organic reason one MUST multiply
the rho by c and incorporate it into a 4-vector.

If information is packaged poorly some may get lost in "transmission", that is, in the change to a different observer's coordinates.

Notice, BTW, how conventions differ!
For example I suppose that some particle's energy-momentum vector which I would write

(E/c, p1, p2, p3)

is something that you would write with energy in the FOURTH place, namely

(p1, p2, p3, iE/c)

I have put the "i" in because I suspect you like the "imaginary time" formalism---perhaps formality is a better word, the difference is only superficial.

what matters most is to pack the information into a 4-vector where everything has the same units and so that it will transform correctly into what the other person wants to know

(the other person may be moving relative to you but may also
want to know the momentum and energy of the particle---only transformed!)
------------------------

I still am not confident that I have identified the formalism you are using. could you describe it? do you write a point in spacetime as

(x1, x2, x3, x4), where x4 = ct

or as

(x1, x2, x3, x4) where x4 = ict

or as

(x0, x1, x2, x3) where x0 = ct

If you tell me which convention you are using I will be able to
reply more surely.
[

 

[Imagine]

Posing I use this one : (x1, x2, x3, x4) where x4 = ict, I don't think it's relevant to my question. I am not pointing to the imaginary time part here.

Take an other quadri-vector, relativistic momentum: P = (p,E/c), with imaginary part or not. P and p are vectors, E/c is scalar.

May I consider the partial derivative of (E/c) over time as scalar pointing toward the positive direction of time and, as per, an implicit vector?

Or should I consider P, quadri-vector, being expressed as tri-vector plus scalar? IMHO, that would means special interpretation about quadri-vector? Right?

Note: If I remember correctly, quadri-vector was introduced consequently to Lorentz transformation where, depending on the observer's relative velocity, the "scalar component" becomes the "tri-vector" component (and vice-versa) when conservation law applied.

 

[marcus]

Imagine, where you are involved I do not like to say "No!" or to occupy a post of authority, so I think now would be a good time for one of the others to answer.

In the middle ages, if you would suggest that one can make a vector by putting a scalar in the fourth position, to make it "point" in the "time direction", then your house would be visited by the officers of the Inquisition.
And if you persisted in this opinion and told them that one of the components of a quadrivector is a "scalar" you would be burnt at the stake.
My child, you are in great danger of heresy!

A "scalar" quantity is one which is not changed by passage to a different system of coordinates

but each of the components of a quadrivector may change (it may also not change depending on which transformation is used but it MAY change) and so it is not a scalar.

I must defer to a PF mentor or one of several other competent authorities who can instruct you

Originally posted by Imagine
Posing I use this one : (x1, x2, x3, x4) where x4 = ict, I don't think it's relevant to my question. I am not pointing to the imaginary time part here.

Take an other quadri-vector, relativistic momentum: P = (p,E/c), with imaginary part or not. P and p are vectors, E/c is scalar.

May I consider the partial derivative of (E/c) over time as scalar pointing toward the positive direction of time and, as per, an implicit vector?

Or should I consider P, quadri-vector, being expressed as tri-vector plus scalar? IMHO, that would means special interpretation about quadri-vector? Right?

Note: If I remember correctly, quadri-vector was introduced consequently to Lorentz transformation where, depending on the observer's relative velocity, the "scalar component" becomes the "tri-vector" component (and vice-versa) when conservation law applied.

 

[Imagine]

Sorry Marcus to have been heretical. My be I losted something during translation. I'll check that.

Returning to the relativistic current density question, the quadri-vector is composed of a non-relativistic vector (tri-vector) and "ic*rho", which is not by itself a vector.

Shall I consider "c*rho" to be at current density at rest as per "E/c" is at momentum at rest?

 

[marcus]

Originally posted by Imagine
Sorry Marcus to have been heretical. My be I losted something during translation. I'll check that.

Returning to the relativistic current density question, the quadri-vector is composed of a non-relativistic vector (tri-vector) and "ic*rho", which is not by itself a vector.

Shall I consider "c*rho" to be at current density at rest as per "E/c" is at momentum at rest?

Imagine, I was trying to be funny about the issue of
heresy. It wasnt very intelligent of me---now I see it
was not very amusing after all.

We must find someone else to help----a mentor, or perhaps
"Pete" who signs himself "pmb". There are plenty of people
here at PF who can give you a little relativity lecture.

I actually enjoy talking to you, but not explaining to you! I like your
tone of voice, style, whatever je-ne-sais-quoi.
Pardon me for not being useful!

 

[Imagine]

Bonjour Marcus,

I wouldn't like to discourage your explanation, I always appreciate your discussions. I used to be funny in french but that's another story in english. It's seams that I have two language-oriented brain's instances, a french emotional and an english rational. IMHO, being heretically stamped was very french funny and a compliment for my english mind-side.

Shall I reformulate my mind and interrogation to have precious helps and aimed answers from PF members?

For what I know (FWIK), physics have conservation laws for Energy, Electric charge, Momentum and Angular momentum. (Did I missed something?) FWIK, these physical properties are fundamentally major due to these laws. Also, I just red about quadri-vector (4-V) current density and 4-V momentum. I can understand that, by definition and consequences, 4-V have a 3-V and a non-vector as components (I didn't said scalar . Thanks Marcus! And didn't talk about vectorized-time component. Wouldn't like to be burnt at the stake)

To help me further understandings:

1) What are the physical properties where 4-V applies? Electric current, Momentum, Flows, ...? All properties that are 3-V shall become 4-V when relativistic is considered?

2) For 4-V considerations, is it better to handle the property's density instead of the property itself?

3) When I looked at the 4-V current density, I saw charge density times "c". When I looked at the 4-V momentum, I saw "E/c" which is mass times "c". I was surprised since I don't have, in my memories, any thing that looks like "E=qc²" and don't have mass conservation law. That's why I try to understand the fourth component of 4-V.

Everyone is welcome to help me orient or clarify my mind. I can understand that my interrogations could be considered junior stuff for full-time physicists but for me these are extremely important.

Answers always appreciated and therefore thanks everyone in advance.

 

[dhris]

Hi Imagine,

I'll throw in my two cents if it helps. I believe that it would help you to consider what is actually meant by the terms "vector" and "scalar". A scalar, for example, it not just some real number, and a vector is not simply a collection of 3 (or 4) numbers.

A scalar is, as Marcus pointed out, a quantity that is invariant under coordinate transformations. In Special Relativity, the coordinate transformations are Lorentz transformations in 4d spacetime. So the only quantities that can be properly called "scalar" are those which have the same value regardless of the reference frame in which they are measured. Charge, for example, is a scalar; the charge of a moving electron is the same as one at rest. Charge DENSITY, on the other hand is most definitely not a scalar, since volumes change (due to length contraction) depending on the frame of reference.

In the case of vectors, consider 3-vectors for a moment where our coordinate transformations are simply rotations of the axes. An arbitrary 3-vector has components (x1,x2,x3). However, it is possible to change to a coordinate system in which one or even two of these components are zero. In particular, I can take my new 3-axis to point in the direction of the vector. The new components are then (0,0,x3') where x3' is equal in magnitude to the length of the vector. From this example you can see that none of the components can be called a scalar since none of them was invariant under the coordinate transformation. What DIDN'T change, however, is the (squared) length of the vector, given by L^2 = x1^2 + x2^2 + x3^2 which implies that L^2 = (x3')^2. So, while none of the components of the vector is a scalar, its magnitude IS a scalar. A set of 3 numbers must have this property before it can be called a vector.

The same is true of relativistic 4-vectors. Note, however, that we have some room to define what is meant by "length of a vector". In the above example, I used the Pythagorean Theorem to define length.

 

[dhris]

cont...(sorry for being long-winded)

Consider a 4-d displacement in spacetime. Its components can be given as (ct, x1, x2, x3), where these components measure displacement from the origin. But I need some notion of their length, which furthermore is a scalar. It turns out that in SR, a quantity called the spacetime interval, defined (for displacements from the origin) as:
s^2 = -(ct)^2 + (x1)^2 + (x2)^2 + (x3)^2 IS invariant under Lorentz transformations! If for an arbitrary 4-vector (v0,v1,v2,v3), I define length to be L^2 = -(v0)^2 + (v1)^2 + (v2)^2 + (v3)^2, then my above displacement can properly be called a vector, with the magnitude given by the spacetime interval. It is important to realize that without the invariant spacetime interval, there would be no good notion of length, and therefore no displacement 4-vector.

The same is true of all SR 4-vectors. Their length, defined in the above way, MUST form a spacetime invariant. In your example, we want to turn the current density into a 4-vector. It so happens that an invariant IS formed by J^2 - (c rho)^2 = (c rho_0)^2, where J is the set of 3 spatial components of the current density, rho is the charge density and (rho_0) is the rest density (which is then a scalar). You can verify this by applying the appropriate transformations. This means that (c rho, J1,J2,J3) forms a proper 4-vector. Furthermore, c rho is the ONLY choice for the fourth component, since any other choice will not lead to invariant length.

So you can see that 4-vectors do not arise arbitrarily by tacking on any old extra component with the right units. Rather, the existence of a 4-vector reflects the existence of a corresponding spacetime invariant. There are also other ways of defining vectors, one of which Marcus mentioned, but they turn out to be equivalent.

Hope this helps,
dhris

 

[Imagine]

Merci beaucoup dhris !

Now, I really understand 4-V and scalar. Particularly when I red the zero in "J^2 - (c rho)^2 = (c rho_0)^2". This help me to recover some losted neurons.

You wrote that electric charge is a scalar and charge density is not a scalar (due to length contraction), that's right. The length of the 4-V current density is invariant, in Lorentz transformation, but not its components. Ok!

Here, I have electric charge and current-density's length that are invariant but the first has a position (4-D) and the second has also a direction (4-D).

In regards to "mass" domain instead of electric domain, my newbie following-quest question is: What "properties" are invariant? Mass density? Momentum length? Energy density?

 

[dhris]

Well, in that case, it's the rest mass (or rest energy). You have:

m^2 c^4 = -E^2 + c^2 p^2

meaning that we can form a 4-vector out of energy and momentum. You can verify this relation by applying the Lorentz transformations for energy and momentum.

dhris

 

[Imagine]

You means rest energy or rest momentum like:

(m₀ c²)² = - (E)² + (c p)²
(m₀ c)² = - (E/c)² + (p)²

 

[dhris]

Well, those are the same thing (they are both invariant), they just differ by a factor of c^2. I suppose, though, that the momentum 4-vector is (E/c,p1,p2,p3) so the magnitude of this particular vector is given by your second expression. Multiplying the components by c does yield a different 4-vector with magnitude given by the first expression, but this difference is trivial. In fact, in many theoretical applications units are scaled such that c=1.

 

[Imagine]

Do you have an equation expressing the four-vector current similar to four-vector current density?

 

[Imagine]

Originally posted by Imagine
Do you have an equation expressing the four-vector current similar to four-vector current density?

The real question is: Do you know an invariant four-current expression?

 

[Imagine]

Originally posted by Imagine
The real question is: Do you know an invariant four-current expression?

I will guest !
I² - (c q)² = (c q₀)²
I₄ = ( I₃ , icq )
Is it right or the volume changed somethings?

 

[marcus]

Originally posted by Imagine
I will guest !
I² - (c q)² = (c q₀)²
I₄ = ( I₃ , icq )
Is it right or the volume changed somethings?

Hello Imagine,
dhris was providing some very welcome explanations
(aussi-bien lucides que gentiles) but he seems temporarily absent
his last post was of the 18th and now it is 22nd!

the units of conventional electric current are "Coulombs per second"

that is the same as "Amperes"-----measuring the passage of charge-per-unit-time.

Is what you mean by the symbol "I" an ordinary current with units coulomb/second?

The units of every component in a 4-vector must be the same.

but the units of qc are coulomb-meter per second!
this is not the same as coulomb-per-second!
alas

edit: two afterthoughts
hats off to you for guessing, I admire your taking initiative

the idea of current density is elegant, I urge you to be satisfied with it (Bach chorale BWV 511 "Gib dich zufrieden.")
it is defined at every point in space-----one does not need to describe ACCESSORIES like wires and particle beams and other annoying hardware, in order to talk about the current density
at every point in space there is a current density in the x-direction and in the y, and in the z, direction----so 3 current densities----the idea is very economical: once one has said those three the thing is completely described

(oh, plus the charge density to make it a quadrivector so that it
will transform to moving coordinates, but being a sedentary person I ignore moving coordinates most of the time and only think of trivectors, as you call them)

 

[Imagine]

Merci Marcus,
I revisited my EM books and discovered that, you're right, Ampere is Coulomb/sec. I was "thinking" that Ampere was Coulomb times speed. I waked up disgracefully.

Now, suppose that O₄ and O₃ are energy density vectors, and R_p is momentum density value.
Could I write:
O₄ = ( O₃, icR_p )
and therefore
(O₄)² = (O₃)² - (cR_p)²
where (O₄)² is invariant under Lorentz transformation.

The point here, from what I learned from your posts, is to write something like:
div₄(O) = div₃(O) + ^dR_p/_dt (partial)

Here,
R_p has ^(kg-m/s)/_(m³)
and
O₄ and O₃ have ^(kg-m/s)/_(m²-s)

Rem: Replacing Coulomb by Momentum in Current density quadri vector.

I would physically interpret like this:
---------------------------------------
Considering a small volume, T.
The rate at which the momentum T*R_p, enclosed in the volume T increase with time is T*^dR_p/_dt (partial)
The rate at which the enclosed momentum decrease with time is T*div₃(O) since the energy density, O, is the momentum flowing out per unit time and per unit area.
---------------------------------------

P.S.1: I red the https://www.physicsforums.com/announcement.php?s=&forumid=11 post but I wasn't able to preview these code. Some help will be appreciated.

P.S.2: Est-ce que (Bach chorale BWV 511 "Gib dich zufrieden.") signifie "À quoi cela sert-il de souffrir"?

 

[quantumdude]

Originally posted by Imagine
The real question is: Do you know an invariant four-current expression?

There is none, because both current and charge are scalars in Euclidean 3-space. Current is related to current density by:

I=∫J^.dA

which is manifestly a scalar product.

 

[Imagine]

Gib dich zufrieden
Are you satisfied?

 

[marcus]

Originally posted by Imagine
Gib dich zufrieden
Are you satisfied?

de ma part, je suis tres content
et toi?

 

[Imagine]

Français:
Pour ma part, je ne suis pas satisfait.
J'apprécierais que tu me donnes ton opinion sur l'équation que j'ai décrite concernant la densité d'énergie et la quantité de mouvement.

Anglais:
For my part, I am not so happy.
I would appreciate that you deliver your opinion to me on the equation which I described concerning the density of energy and the momentum.

Italien:
Per la mia parte, non sono così felice.¶
Apprezzerei che esprimete il vostro parere me sull'equazione che ho descritto interessare la densità di energia e della quantità di moto.¶

Espagnol:
Para mi parte, no soy tan feliz.¶
Apreciaría que usted me entregue su opinión en la ecuación que describí el referirme de la densidad de la energía y del ímpetu.¶

Portuguais:
Para minha parte, eu não sou assim feliz.¶
Eu apreciaria que você me entrega sua opinião na equação que eu descrevi concernir a densidade da energia e do momentum.¶

Allemand:
Für meinen Teil bin ich nicht. so glücklich¶
Ich würde schätzen, daß Sie Ihr Urteil an mich auf der Gleichung abgeben, der ich beschrieb, die Dichte von Energie und von Momentum zu betreffen.¶

Is it meaningfull? I used the SysTran software!

 

[marcus]

Imagine, you have Tom answering your posts
at PF it doesn't get any better
he teaches college physics, in upstate NY somewhere I think,
not far from "Trois rivieres" probably
ask a direct question if you want to learn something

 

[quantumdude]

Originally posted by marcus
Imagine, you have Tom answering your posts
at PF it doesn't get any better

Nah, I'm just a humble medium energy theorist. It's you LQG and String jocks who are the real authorities around here.

ask a direct question if you want to learn something

Preferably in English!

 

[quantumdude]

Originally posted by Imagine
Anglais:
For my part, I am not so happy.
I would appreciate that you deliver your opinion to me on the equation which I described concerning the density of energy and the momentum.

Are you still talking about the "current 4-vector"?

If so, then there is no need to go into the energy and momentum. All we need to do is look at your definition of the "4-current":

I² - (c q)² = (c q₀)²
I₄=(I₃ , icq )

First, electric charge is a scalar (both Euclidean and Lorentzian). Physically, that means the charge of a particle does not change when you speed it up or rotate it. So the charge cannot be the timelike piece of a 4-vector.

Second, what you call "I₃" (the current) is not a 3-vector. It is a Euclidean scalar (but not a Lorentz scalar!). That means it cannot be the spacelike components of a 4-vector.

 

[Imagine]

Bonjour Tom,
I separated the energy-momentum into another post.

Concerning the "current 4-vector", it seams that "volume" shall be assimilated to charge and current to be components of a 4-vector.

I still have to learn distinctive concepts about scalars and vectors. In my french mind, a scalar was a unidimensional vector (a scale, a metric) and a vector has more-than-one dimensions. Euclidian space have 3 dimensions (eg: O₃) and Lorentzian space have 4 dimensions (eg: O₄). Scalar was not relevant in front of invariability.

Direct question: Reading suggestion about scalar, vector (3 and 4)?

 

[quantumdude]

Originally posted by Imagine
Bonjour Tom,
I separated the energy-momentum into another post.

Ah, OK then.

Concerning the "current 4-vector", it seams that "volume" shall be assimilated to charge and current to be components of a 4-vector.

I don't understand this part.

I still have to learn distinctive concepts about scalars and vectors. In my french mind, a scalar was a unidimensional vector (a scale, a metric)

A scalar is an invariant. Whether or not something is a scalar depends on what set of transformations under which that thing does not change.

In Euclidean 3-space under the set of transformations (rotations + parity), some invariants (scalars) are charge, energy, speed, wavelength of a light pulse, etc. When we enlarge to the Minkowski 4-space and the set of transformations (rotations + parity + Lorentz boosts), then the list changes. Energy, speed, and wavelength of a light pulse are no longer invariant.

and a vector has more-than-one dimensions.

Not necessarily. A vector can have just one dimension, but in a 1D space a vector is indistinguishible from a pseudoscalar, which has a parity of -1.

In general, a vector is an object that transforms as:

v-->v'=Rv

where R is any element from the aforementioned set of transformations. In Euclidean space, it is the set of (rotations + parity). In Minkowski 4-space, it is enlarged to include Lorentz boosts.

Euclidian space have 3 dimensions (eg: O₃) and Lorentzian space have 4 dimensions (eg: O₄). Scalar was not relevant in front of invariability.

I hope you see why that is wrong now.

Direct question: Reading suggestion about scalar, vector (3 and 4)?

A 4-vector in Minkowski space is of the form

v=(v,φ)

where v represent the (three) spacelike components and φ represents the single timelike piece. Those components have the properties of being a Euclidean 3-vector and a Euclidean scalar, respectively.

Since I₃ is not a Euclidean vector, and since q is a Lorentz scalar, your "4-current" relation is invalid.

edit: typo

 

[Imagine]

Originally posted by Tom
... set of transformations (rotations + parity), ...

Parity?

 

[quantumdude]

Originally posted by Imagine
Parity?

Yes, parity is part of the group of transformations that defines the tensor character of mathematical objects in both Euclidean and Minkowski spaces. The parity operator P reverses the sign on all spatial coordinates, like so:

x-->x'=Px=-x

 

[Imagine]

Merci beaucoup Tom!