## Abstract

According to the standard realistic interpretation of classical electrodynamics, the electromagnetic field is conceived as a real physical entity existing in space and time. The problem we address in this paper is how to understand this spatiotemporal existence, that is, how to describe the persistence of a field-like physical entity like electromagnetic field. First, we provide a formal description of the notion of persistence: we derive an “equation of persistence” constituting a necessary condition that the spatiotemporal distributions of the fundamental attributes of a persisting physical entity must satisfy. We then prove a theorem according to which the vast majority of the solutions of Maxwell’s equations, describing possible spatiotemporal distributions of the fundamental attributes of the electromagnetic field, violate the equation of persistence. Finally, we discuss the consequences of this result for the ontology of the electromagnetic field.

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## Notes

- 1.
In the sense of Reichenbach (1965).

- 2.
In a mathematical sense the equation of persistence (6) is of the same form as a continuity equation without source and conductive current densities,

$$\begin{aligned} \partial _{t}f_{i}({\mathbf {r}},t)+\nabla \left( f_{i}({\mathbf {r}},t){\mathbf {v}}({\mathbf {r}},t)\right) =0 \end{aligned}$$in the particular case when \(\nabla {\mathbf {v}}({\mathbf {r}},t)=0\), that is the velocity field describes an “incompressible” flow. It must be emphasized however that the two equations have different contents. Conceptually, the equation of persistence is about the quantities tracking the object in question. Such a quantity is not necessarily a density-like quantity, in the sense that its volume integral is not necessarily a meaningful physical quantity, especially not a conserved one. Moreover, the equation of persistence and the continuity equation are independent: for a given set of quantities, one equation may hold without the other. The coarse-grained density of the spreading gas we will discuss in Sect. 5 is an example where the continuity equation holds but not the equation of persistence. In contrast, the whiteness and blackness of an inflating spotted ball (Fig. 4) are quantities that satisfy the equation of persistence but not the continuity equation (the velocity field describing an inflating object is not divergence free).

- 3.
In the present analysis we restrict ourselves to classical (Galileo covariant) and special relativistic physics, and set aside the possible generalization for general relativity.

- 4.
For example, consider \(\left\{ t,x,y,z,E_{x},E_{y},E_{z},B_{x},B_{y},B_{z}\right\}\) where \(E_{x},E_{y},E_{z}\) and \(B_{x},B_{y},B_{z}\) are the electric and magnetic field strengths in

*K*. Now, for example, \(\left\{ E_{x},E_{y},E_{z}\right\}\) is not closed against the Lorentz transformation, while subset \(\left\{ E_{x},E_{y},E_{z},B_{x},B_{y},B_{z}\right\}\) is closed; the values of \(E_{x},E_{y},E_{z},B_{x},B_{y},B_{z}\) in*K*uniquely determine the values of \(E'_{x},E'_{y},E'_{z},B'_{x},B'_{y},B'_{z}\). Similarly, \(\left\{ t,x,y,z\right\}\) is a closed subset, while \(\left\{ x,y,z\right\}\) is not. - 5.
For a more precise formulation see Gömöri–Szabó 2015.

- 6.
It must be pointed out that velocity \({\mathbf {V}}\) conceptually differs from the speed of light

*c*. Basically,*c*is a constant of nature in the Maxwell–Lorentz equations, which can emerge in the solutions of the equations; and, in some cases, it can be interpreted as the velocity of propagation of changes in the electromagnetic field. For example, in our case, the stationary field of a uniformly moving point charge, in collective motion with velocity \({\mathbf {V}},\) can be constructed from the superposition of retarded potentials, in which the retardation is calculated with velocity*c*. Nevertheless, the two velocities are different concepts. To illustrate the difference, consider the fields of a charge at rest (16), and in motion (17). The speed of light*c*plays the same role in both cases. Both fields can be constructed from the superposition of retarded potentials in which the retardation is calculated with velocity*c*. Also, in both cases, a small local perturbation in the field configuration would propagate with velocity*c*. But still there is a consensus to say that the system described by (16) is at rest while the one described by (17) is moving with velocity \({\mathbf {V}}\) (relative to*K*.) A good analogy would be a Lorentz contracted moving rod: \({\mathbf {V}}\) is the velocity of the rod, which differs from the speed of sound in the rod. - 7.
In \({\mathsf {D}}{\mathbf {E}}({\mathbf {r}},t)\) and \({\mathsf {D}}{\mathbf {B}}({\mathbf {r}},t)\), \({\mathsf {D}}\) denotes the spatial derivative operator (Jacobian for variables

*x*,*y*and*z*). That is, in components we have:$$\begin{aligned} -\partial _{t}E_{x}({\mathbf {r}},t)&= {} V_{x}\partial _{x}E_{x}({\mathbf {r}},t)+V_{y}\partial _{y}E_{x}({\mathbf {r}},t)+V_{z}\partial _{z}E_{x}({\mathbf {r}},t)\\ -\partial _{t}E_{y}({\mathbf {r}},t)&= {} V_{x}\partial _{x}E_{y}({\mathbf {r}},t)+V_{y}\partial _{y}E_{y}({\mathbf {r}},t)+V_{z}\partial _{z}E_{y}({\mathbf {r}},t)\\&\vdots \\ -\partial _{t}B_{z}({\mathbf {r}},t)&= {} V_{x}\partial _{x}B_{z}({\mathbf {r}},t)+V_{y}\partial _{y}B_{z}({\mathbf {r}},t)+V_{z}\partial _{z}B_{z}({\mathbf {r}},t)\\ -\partial _{t}\varrho ({\mathbf {r}},t)&= {} V_{x}\partial _{x}\varrho ({\mathbf {r}},t)+V_{y}\partial _{y}\varrho ({\mathbf {r}},t)+V_{z}\partial _{z}\varrho ({\mathbf {r}},t) \end{aligned}$$ - 8.
Without entering into the details, it must be noted that the Maxwell–Lorentz equations (11)–(15), exactly in this form, have

*no*solution. The reason is that the field is singular at precisely the points where the coupling happens: on the trajectories of the particles. The generally accepted answer to this problem is that the real source densities are some “smoothed out” Dirac deltas, determined by the physical laws of the internal worlds of the particles—which are, supposedly, outside of the scope of classical electrodynamics. With this explanation, for the sake of simplicity we leave the Dirac deltas in the equations. Since our considerations here focus on the electromagnetic field, satisfying the four Maxwell equations, we only have to assume that there is a coupled dynamics—approximately described by equations (11)–(15)—and that it constitutes an initial value problem. In fact, Theorem 2 could be stated in a weaker form, by leaving the concrete form and dynamics of the source densities unspecified. - 9.
\({\mathbf {E}}_{\lambda }^{\#}({\mathbf {r}})\) and \({\mathbf {B}}_{\lambda }({\mathbf {r}},t_{*})\) can be regarded as the initial configurations at time \(t_{*}\); we do not need to specify a particular choice of initial values for the sources.

- 10.
Notice that our investigation has been concerned with the general laws of Maxwell–Lorentz electrodynamics of a coupled particles + electromagnetic field system. The proof was essentially based on the presumption that all solutions of the Maxwell–Lorentz equations, determined by

*any*initial state of the particles + electromagnetic field system, corresponded to physically possible configurations of the electromagnetic field. It is sometimes claimed, however, that the solutions must be restricted by the so called retardation condition, according to which all physically admissible field configurations must be generated from the retarded potentials belonging to some pre-histories of the charged particles (Jánossy 1971, 171; Frisch 2005, 145). There is no obvious answer to the question of how Theorem 2 is altered under such additional condition. - 11.
In point

**III**the equations of persistence were based on the metaphysical intuition that an extended object can be conceived as the mereological sum of its local parts, each of which itself being a persisting entity. One might object that in case of the electromagnetic field this intuition is not justified: the electromagnetic field should rather be seen as one single indivisible entity spreading over the whole of space, whose persistence simply means that the field, as a whole, is present at all instants of time. This fact then might be translated as the condition that the field strengths take*some*values in all spatiotemporal regions, which is clearly respected by all solutions of the Maxwell–Lorentz equations.We believe nonetheless that this is not the way we usually think about the electromagnetic field, and in fact one has good physical grounds to talk about the local parts of the field as entities themselves. We make three observations: (1) In electrodynamics we attribute properties to the local parts of the electromagnetic field—the parts of the field occupying certain spatial regions—that we attribute to

*entities*in other cases. Such properties, for example, are energy and momentum. (2) Part of the reason why one believes that the electromagnetic field is a real physical entity is that it makes manifest the idea of*local*action—that of the continuous propagation of physical actions in space and time. The idea of local action makes no sense unless there exists a local entity, the local part of the field, that mediates the physical action. (3) Another aspect of locality in electrodynamics is that the state of the electromagnetic field given on a segment of Cauchy surface determines the state of the field in the future dependence domain of the surface in question. Clearly, this idea requires that we must be able to assign local states of the electromagnetic field to spatiotemporal regions (to the surface and domain of dependence in question). Such an assignment only makes sense if there exists something, a local entity, that is capable of being in those local states.

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## Acknowledgements

Funding was provided by Hungarian National Research, Development and Innovation Office (Grant Nos. K100715, K115593).

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Gömöri, M., Szabó, L.E. On the Persistence of the Electromagnetic Field.
*J Gen Philos Sci* **50, **43–61 (2019). https://doi.org/10.1007/s10838-018-9430-3

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### Keywords

- Classical field
- Persistence
- Velocity
- Covariance
- Identity
- Qualitative similarity