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In learning about relativity I've noticed that in the construction of Lorentz covariants (specifically four-vectors) two physical quantities that were previously considered distinct are instead treated as a single object.

Examples include position and time (spacetime), electric and magnetic fields, electric and magnetic potential, energy and momentum, charge and current density, and so on.

I was interested in learning more about this I tried googling my best guess, "Lorentz Pairs", to no avail. My question is whether there is more specific terminology referring to these pairs of physical quantities outside of the general term "Lorentz Invariant".

Additionally, why do we so often see two physical quantities being combined instead of, say, three or some other number?

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    $\begingroup$ It is not completely true that two physical quantities are combined instead of three or more. The "combination" of for example energy and momentum is just because we thought before that the world was best described by 3-d theories with time as an independent thing and thus we gave different names to quantities that are parts of the same thing in a covariant formalism (relativity) $\endgroup$
    – Prastt
    Oct 21, 2013 at 5:54
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    $\begingroup$ How many "things" get combined depends on the tensor structure of the covariant object you eventually get. To give an example of five things commonly thought to be separate which are combined in relativity: energy density, energy flux, momentum density, pressure and anisotropic stress. These get combined into the relativistic stress-energy (or equivalently energy-momentum) tensor $T_{\mu\nu}$. It just happens that you see simpler objects (rank one tensors like $p_\mu$) more often than more complicated ones (rank two tensors like $T_{\mu\nu}$). $\endgroup$
    – Michael
    Oct 21, 2013 at 7:40
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    $\begingroup$ It's important to note that electric and magnetic fields are quite different from all the other pairs you mention. Those match a (newtonian) scalar with a (newtonian) vector and construct a Lorentz-invariant four-vector out of it. Electric and magnetic fields follow a different scheme, in which a Lorentz two-tensor gets broken down into a newtonian vector and a pseudovector. $\endgroup$ Oct 21, 2013 at 11:43
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    $\begingroup$ That said, though, I really like your term, "Lorentz pair". Unfortunately it's not in use and I'm not aware of similar terminology, but it would be quite useful to be able to say, simply, "energy and momentum are a Lorentz pair". $\endgroup$ Oct 21, 2013 at 11:45
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    $\begingroup$ The Stress-Energy-Momentum Tensor from General Relativity is an interesting example. Are there any other examples of Relativistic Tensors that are commonly used that combine many (4-5) different Physical Quantities? $\endgroup$
    – jcelios
    Oct 21, 2013 at 18:34

3 Answers 3

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Clifford algebra offers many interesting Lorentz pairs. There is no standard nomenclature. Here are my suggestions for the Clifford algebra $Cl_{4,0}$ applied to electromagnetic theory:

spacetime position: $\hat r = ct + \mathbf r$,

event: $\hat r\mathbf e_0$,

spacetime derivative: $\frac{\partial}{\partial\hat r}=\frac{1}{c}\frac{\partial}{\partial t} + \nabla$,

event derivative: $\frac{\partial}{\partial \hat r\mathbf e_0} = \mathbf e_0\frac{\partial}{\partial\hat r}$,

electromagnetic field: $\mathbf E + i\zeta\mathbf H$,

charge-current density: $\hat j = \rho c + \mathbf j$

event current density: $\hat j\mathbf e_0$

energy-momentum density: $U + \mathbf S/c$

Source: Electromagnetic energy-momentum equation without tensors: a geometric algebra approach.

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  • $\begingroup$ thanks to Ignacio and Ruslan for the latex note and edits :) $\endgroup$
    – qsugon
    Oct 30, 2013 at 14:55
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I'm not sure if there's a term for these Lorentz Pairs, but great food for thought!

I find it interesting, they seem so often to correspond to some conservation principle.

Consider charge conservation, $\nabla \cdot \vec{J}+\partial \rho /\partial t=0=\nabla \cdot \vec{J}+\partial \rho c/\partial ct$. And we have the charge 4-vector $J^\alpha=(\rho c,\vec{J})$. The Lorentz Gauge gives us $\nabla \cdot \vec{A}+(1/c^2)\partial V/\partial t=0 $ and the 4-potential $A^\alpha=(V/c,\vec{A})$.

We have also $\nabla \cdot \vec{S} +\partial u/\partial t=-\vec{J}\cdot\vec{E}$. The term on the right is zero in free space with no current density and so we have a proper conservation equation. This suggests $S^\alpha=(uc,\vec{S})$, but I've never seen mention of a Poynting 4-vector, but wouldn't that be redundant with the Maxwell Stress Energy Tensor?

The prototypical 4-vector, $(ct,\vec{x})$ corresponds to $\nabla \cdot \vec{x}+\partial ct/\partial ct=4. $ Not quite a conservation equation, but very similar. For conservation laws

Uncertainly principles popup in similar pairs.

Does every conservation law imply a Lorentz Pair? An uncertainty relation?

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  • $\begingroup$ You might want to look up Noether's theorem. $\endgroup$ Jul 27, 2022 at 6:24
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What you might be looking for is the concept of Lorentz covariance. A Lorentz covariant quantity is a (finite collection of) quantities which are taken to linear combinations of themselves under Lorentz transformation. An subcase of this is what one might call 'rotational covariance'. For example, one does not consider, say, the momentum in the x-axis by itself since if you rotate the system a bit, this quantity becomes some linear combination of the x, y, and z-axis momenta. Energy is by itself a rotationally covariant quantity since under rotations it is unchanged. So already, you have a rotational 'triplet' in momentum and energy is all by itself.

In order to get Lorentz covariance, though you also need to consider boosts. When you boost, energy and momentum mix together, so they are combined into a single four-momentum. Charge density and current mix together, and so they are combined into the four-current. The E and B fields are combined into the electromagnetic tensor. Three quantities, the energy density, the momentum density, and the stress tensor have to be combined in order to form the covariant stress-energy tensor. Any less, and the result would not be Lorentz covariant.

The reason they combine this way is that if you only require invariance under rotation, Lorentz covariant quantities split. For example, four-vectors, like the position four-vector and the momentum four-vector, have the time component invariant under rotations and the spatial components are taken to each other. So usually four-vectors come from the combination of a rotational scalar and a three-vector (see time and spatial position, energy and momentum, charge density and current density, etc.). The electromagnetic tensor $F_{\mu\nu}$ is a anti-symmetric matrix (one that equals the negative of its transpose). This means that it has six independent components, since the entries below the diagonal are fixed to be the negative of the entries above the diagonal. When considering only rotations, this splits into three $F_{0 i}$ components and three $F_{ij}$ components, giving the electric and magnetic three-vectors.

A simpler illustration of this kind of thing is from restricting rotations. Imagine a land with such strong gravity that vertical motion is completely different from horizontal. If you only require covariance under rotations about the z-axis, then 3-vectors like position and momentum split into a 2-vector, the horizontal component, and a scalar, the vertical component. So creatures living in this land might come up with different concepts for height and horizontal position, and horizontal and vertical momentum, until someone comes up with a theory of 'very special relativity' to unite them. Similarly, nature started with Lorentz invariant quantities, but we move so slowly compared to the speed of light that energy and momentum seem completely independent of each other until they are united under special relativity.

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