2205.14233.pdf

Absorbing the Arrow of

Electromagnetic Radiation

Mario Hubert and Charles T. Sebens

Division of the Humanities and Social Sciences

California Institute of Technology

May 27, 2022

arXiv v.1

Abstract

We argue that the asymmetry between diverging and converging electromagnetic waves

is just one of many asymmetries in observed phenomena that can be explained by a past

hypothesis and statistical postulate (together assigning probabilities to different states

of matter and field in the early universe). The arrow of electromagnetic radiation is

thus absorbed into a broader account of temporal asymmetries in nature. We give an

accessible introduction to the problem of explaining the arrow of radiation and compare

our preferred strategy for explaining the arrow to three alternatives: (i) modifying the laws

of electromagnetism by adding a radiation condition requiring that electromagnetic fields

always be attributable to past sources, (ii) removing electromagnetic fields and having

particles interact directly with one another through retarded action-at-a-distance, (iii)

adopting the Wheeler-Feynman approach and having particles interact directly through

half-retarded half-advanced action-at-a-distance. In addition to the asymmetry between

diverging and converging waves, we also consider the related asymmetry of radiation

reaction.

arXiv:2205.14233v1 [physics.hist-ph] 27 May 2022

Contents

1 Introduction

2 Waves in Classical Electromagnetism

3 Strategy 1: A Statistical Explanation

3.1 The Past Hypothesis and the Statistical Postulate . . . . . . . . . . . . . . . . .

3.2 Incorporating the Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . .

3.3 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Strategy 2: The Sommerfeld Radiation Condition

4.1 Justification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Strategy 3: Retarded Action-at-a-Distance

5.1 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 Strategy 4: The Wheeler-Feynman Theory

6.1 Time-Asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 Conclusion

1 Introduction

The equations that describe water waves, sound waves, and electromagnetic waves are

time-symmetric and allow for both diverging waves—that propagate outwards from a central

point or region—and their time-reverse, converging waves (figure 1). If you throw a stone in a

pond, shout, or switch on a lightbulb, you will produce diverging waves (of water, sound, or light).

Diverging waves are commonplace. Converging waves are allowed by the laws of the relevant

physical theories, but in our world they are rare. We never see circular waves spontaneously

form from rustlings at the edge of a pond, increasing in amplitude as they converge towards the

center (Popper, 1956; Zeh, 2007, pg. 17). Still, converging waves can occur. One way to make

such waves would be to place a large floating ring on a calm body of water and carefully pulse

it up and down (while keeping it level).

This would result in converging waves within the ring

and diverging waves outside of it.

There are a great many processes that, like diverging waves, can happen in reverse but hardly

ever do. Price has called the question as to why these processes occur in one temporal order but

not the other “the puzzle of temporal bias”:

An example like this one appears in Davies (1977, pg. 119).

Figure 1: The figure on the left shows diverging waves moving away from the center point and

the figure on the right shows the time-reverse of this: converging waves approaching the center

point.

“Late in the nineteenth century, physics noticed a puzzling conflict between the laws

of physics and what actually happens. The laws make no distinction between past

and future—if they allow a process to happen one way, they allow it in reverse. But,

many familiar processes are in practice ‘irreversible,’ common in one orientation but

unknown ‘backwards.’ Air leaks out of a punctured tire, for example, but never

leaks back in. Hot drinks cool down to room temperature, but never spontaneously

heat up. Once we start looking, these examples are all around us—that’s why films

shown in reverse look so odd. Hence the puzzle: What could be the source of this

wide-spread temporal bias in the world, if the underlying laws are so even-handed?”

(Price, 2004, pg. 219)

Among philosophers of physics, there has emerged a fairly widespread consensus on how

to solve the puzzle of temporal bias (though there is disagreement in the details), at least

for standard thermodynamic processes like air leaking out of a tire or hot drinks cooling to

room temperature. We can explain why the reversed processes rarely occur by introducing a

probability distribution over initial conditions that deems improbable the kind of fine-tuning that

would be necessary for such reversed processes to be common. Albert (2000) has given a clear and

influential presentation of this kind of solution, calling the two posits that specify the probability

distribution over initial conditions “the past hypothesis” and “the statistical postulate.” Wallace

(2011) has written: “There are no consensus positions in philosophy of statistical mechanics, but

the position that David Albert eloquently defends in

Time and Chance

. . . is about as close as

we can get.”

In stark contrast (and despite considerable work on the subject), there has emerged no

consensus as to how we ought to solve the puzzle of temporal bias for wave phenomena in

general or for electromagnetic waves in particular. There is no generally accepted explanation

for the observed arrow of electromagnetic radiation. One popular strategy

is to give a statistical

explanation of the arrow of radiation.

However, supporters of such an explanation vary

For a list of authors in addition to North (2003) and Atkinson (2006) who take this option, see footnote 23.

considerably in the details and there are opponents defending quite different approaches. In

an effort to move the community towards consensus, here we are throwing our support behind

North’s (2003) statistical strategy for explaining the arrow of radiation, identifying the work that

remains to be done in developing the strategy, and comparing this strategy to its main rivals

(focusing on the comparison to distant rivals over close ones).

North (2003) builds on the broad agreement as to how we ought to explain thermodynamic

asymmetries and argues that we can explain wave asymmetries using the same tools. Converging

waves (of sound, water, light, etc.) are rare because it would require extreme fine-tuning in the

initial conditions for such waves to be common. We can introduce a probability distribution over

initial conditions for matter and field that makes such fine-tuning incredibly unlikely. Because

we believe that a single probability distribution over initial conditions will suffice to explain

deflating tires, cooling beverages, diverging waves in general, and diverging electromagnetic

waves in particular, we are seeking to “absorb” the arrow of electromagnetic radiation into a

broader explanatory schema—unifying the arrow of radiation with other arrows of time.

This article begins with a section on technical background followed by a section presenting

and defending the above-described statistical explanation of the arrow of radiation. Then,

we spend a section each on three competing strategies for explaining the arrow: First, one

can impose an additional time-asymmetric law (or postulate) that goes beyond Maxwell’s

equations in constraining the behavior of the electromagnetic field: the Sommerfeld Radiation

Condition. This condition requires that the electromagnetic field at any point in space and

time be attributable to past sources. Second, one can eliminate the electromagnetic field and

have charges interact with one another directly over spatial and temporal gaps in a retarded

action-at-a-distance theory, where the electromagnetic force on a given charge is determined by

the past behavior of other charges (a move that was advocated by Walther Ritz in his 1909 debate

with Albert Einstein). Third, one can adopt the Wheeler-Feynman half-retarded half-advanced

action-at-a-distance theory where the electromagnetic field is eliminated and the force on a

charge is determined by both the past and the future behavior of other charges.

We include subsections evaluating, in detail, the strengths and weaknesses of each strategy

for explaining the arrow of radiation. To briefly summarize, here are a few advantages of

the statistical strategy for explaining the arrow of radiation: The statistical strategy does not

require complicating the laws of electromagnetism through any addition or revision. Unlike the

three approaches just described, the statistical strategy gives a unified account of all wave and

thermodynamic asymmetries. The three alternative approaches draw a sharp distinction between

The arrow of electromagnetic radiation is distinct from the arrow of time itself, if there is such a thing.

The arrow of radiation is about the time-directed nature of certain wave phenomena, similar to the arrow of

entropy increase describing thermodynamic phenomena in our universe. For Maudlin (2007), time is intrinsically

directed. Nevertheless, one can still investigate the arrows of radiation and entropy increase with respect to

this fundamental arrow of time. Taking a different view, Albert (2000, 2015) and Loewer (2012a,b, 2020) deny

the existence of a fundamental arrow of time and argue that the arrows of time we observe (like the arrow of

entropy increase) can be explained without time itself being directed. Of the four strategies for explaining the

arrow of radiation explored here, two involve time-directed laws that seem to require a fundamental arrow of time

(the Sommerfeld Radiation Condition approach and the retarded action-at-a-distance approach) and two do not

require a fundamental arrow of time (the statistical approach that we favor and the Wheeler-Feynman approach).

In what follows, our focus will be on the arrow of radiation and not the arrow of time itself.

matter and field that we believe to be unwarranted—treating the electromagnetic field as either

unreal or as merely an emanation from charged matter. (Debates over the right explanation

of the arrow of radiation are thus tightly linked to debates over the ontological status of the

electromagnetic field—debates that are important for the sake of better understanding both

classical electromagnetism and quantum field theory.) The three alternative approaches do not

entirely avoid statistical reasoning and, given that such reasoning will feature in any explanation

of the arrow of radiation, we find a fully statistical explanation to be appealing. Having noted

some advantages of the statistical strategy, we should also be clear that there is much work to

be done in completing the story. In particular, the account relies on a Boltzmannian approach

to the statistical mechanics of fields. This is a subject that has not (to our knowledge) received

much attention in physics.

In addition to the observed asymmetry between diverging and converging electromagnetic

waves, there is a related asymmetry of radiation reaction. When a charged body is accelerated

from rest it emits electromagnetic radiation and feels a radiation reaction force opposing the

acceleration. If we take charged matter to be composed of extended charge distributions, then the

asymmetry of radiation reaction can be viewed as a consequence of the asymmetry of radiation.

As electromagnetic waves pass through an extended charged body on their way out they exert

a force on that body. Radiation reaction is the result of self-interaction. If we take charged

matter to be composed of point charges, then in most versions of electromagnetism one will

need to modify the Lorentz force law to account for radiation reaction (viewing the source of

the radiation reaction asymmetry as distinct from the source of the radiation asymmetry). The

exception is Wheeler-Feynman electrodynamics, where for point charges the arrows of radiation

reaction and radiation emission are both explained by the dynamical equations together with

assumptions about an absorbing medium that surrounds all of the charges.

Throughout the article, we focus on classical electrodynamics. This is standard practice in

the literature on the arrow of radiation, though there are exceptions (Arntzenius, 1993; Atkinson,

2006). One may wonder why we should try to solve the arrow of radiation with classical physics

when we know that classical physics has been superseded by quantum physics and classical

electromagnetism has been replaced by quantum electrodynamics. This question is especially

pressing given our discussions of the early universe and the ultimate composition of charged

bodies. We think it is important to see whether and how the arrows of radiation and radiation

reaction can be explained within classical electromagnetism. Our methodology is to push the

classical theory to its limits and see what it can do. Figuring out how the arrows of radiation

and radiation reaction are best explained in this context provides insight into the laws and

ontology of classical electromagnetism. Such work may also help us solve foundational problems

in quantum electrodynamics. In particular, studying self-interaction in classical electrodynamics

can provide clues as to how self-interaction should be handled in quantum electrodynamics (an

important, and notoriously difficult, subject).

This article is intended as an accessible entry point to debates about the arrow of radiation

See Sebens (2022a).

in classical electromagnetism and also as a comparative case for a particular explanation of this

arrow. Readers who are well-versed in the relevant literature my be particularly interested in the

following highlights: In section 2, we differentiate our understanding of the arrow of radiation

from characterizations of the arrow by Frisch and North. That section closes with a discussion

of the relation between the arrow of radiation and the arrow of radiation reaction for both

extended charges and point charges, noting that the point charge Lorentz-Dirac equation breaks

down if converging waves are present. In section 3, we discuss cosmic microwave background

radiation and, unlike North, do not take it to be evidence for the existence of free (unsourced)

electromagnetic fields. We also depart from North by showing that backwards causation can

be avoided in a statistical explanation of the arrow of radiation. We explain the significant

work that remains to be done in developing a complete statistical explanation of the arrow

(developing a Boltzmannian statistical mechanics for the electromagnetic field). In section 4,

we present serious problems for formulating the Sommerfeld Radiation Condition (prohibiting

unsourced electromagnetic fields) if there was a first moment and instead assume an infinite past.

In section 6 we separate out two absorber conditions in Wheeler-Feynman electromagnetism,

noting that it is the second absorber condition that yields time-asymmetry.

2 Waves in Classical Electromagnetism

As background for the upcoming discussion of the arrow of electromagnetic radiation, let us

briefly review some important features of classical electromagnetism. At every moment, the

magnetic field must be divergenceless,

∇·

= 0

(1)

and the divergence of the electric field must be proportional to the density of charged matter,

∇·

= 4

πρ .

(2)

These are Gauss’s laws for electricity and magnetism, two of Maxwell’s equations. The time

evolution of the electric and magnetic fields (the electromagnetic field) is determined by the

remaining two of Maxwell’s equations,

∇×

−

∂

∂t

(3)

∇×

∂

∂t

(4)

where

is the current density. The time evolution of matter is given by a force law (such as

the Lorentz force law or the Lorentz-Dirac force law) plus further equations governing other

interactions (that lie outside of classical electromagnetism). Solving all of these equations

together is difficult. In this section we will focus on the task of finding electric and magnetic

fields that obey Maxwell’s equations given a stipulated history for the charged matter. We will

model matter here as a continuous charge distribution, but one could derive equations for point

charges as a special case.

The electric and magnetic fields can be expressed in terms of the scalar potential

and the

vector potential

−

∇

−

∂

∂t

∇×

A .

(5)

Working with such potentials ensures that two of Maxwell’s equations, (1) and (3), will be

automatically satisfied. These potentials have a gauge freedom that can be partially fixed by

adopting the Lorenz gauge condition,

∇·

−

∂φ

∂t

(6)

With this condition in place, the remaining two Maxwell equations, (2) and (4), become

∇·

= 4

πρ

⇒

(

∇

−

∂

∂t

)

−

πρ

(7)

∇×

∂

∂t

⇒

(

∇

−

∂

∂t

)

−

J ,

(8)

These are wave equations for each potential.

The following expressions for

and

satisfy both (7) and (8),

(

~x,t

) =

ˆ

′

(

′

)

(

~x,t

) =

ˆ

′

(

′

)

(9)

where

is a vector that points from

′

−

′

, and

is the retarded time,

−

(the time that a signal traveling at the speed of light from

′

would have to have been emitted

for it to arrive at

). These are called the

retarded

solutions of (7) and (8). The potentials at

a point can be calculated by combining contributions to the field associated with bits of charged

matter at distant points at past (retarded) times. That is, one can find the values for

and

at a given point by integrating contributions to these potentials from the charge and current

densities at each point

′

at the appropriate moment in the past,

. You might interpret (9) as

telling us how past charged matter acts as source for the current electromagnetic field. For the

simple case of a charge that is briefly shaken back and forth, the electromagnetic field calculated

from the retarded potentials will describe diverging electromagnetic waves propagating outwards

after the charge is shaken, carrying away energy (figure 2.a).

The discussion in this section most closely follows that of Griffiths (2013, ch. 10), though the equations here

are written in Gaussian cgs units.

The solutions in (9) are not the only solutions to (7) and (8). There are also

advanced

solutions,

(

~x,t

) =

ˆ

′

(

′

)

(

~x,t

) =

ˆ

′

(

′

)

(10)

where the only difference from (9) is that the retarded time,

, is replaced by the advanced

time,

(the time that a signal traveling at the speed of light from

would arrive

′

). Using (10), the potentials at a point can be calculated by combining contributions to

the field associated with bits of charged matter at distant points at future (advanced) times.

You might interpret (10) as telling us how future charged matter acts as sink for the current

electromagnetic field. For a charge that is briefly shaken back and forth, the electromagnetic

field calculated from the advanced potentials will describe converging electromagnetic waves

propagating inwards, arriving at the charge as it is being shaken and depositing energy in the

charge (figure 2.b). For such an advanced solution, one might be tempted to say that the

presence of converging electromagnetic waves at some time before the shaking is caused by

the future shaking of the charge (that there is retrocausation). In this article, we will avoid

such language and generally work under the assumption that causes come before their effects.

One can retain the ordinary picture of causes preceding their effects in this case if one views the

converging waves at a particular moment as caused by the earlier presence of more widely spread

out converging waves and as causing the future motion of the charge (alongside other forces). To

avoid such waves that converge in from the infinite past and are not produced by earlier motions

of charged bodies, one might stipulate that it is the retarded and not the advanced solutions

that are to be used. We will consider the merits of such a proposal in section 4.

In addition to the retarded and advanced solutions in (9) and (10), there are also

free

solutions

which describe the propagation of electromagnetic waves in the absence of charges (when the

right-hand sides of (7) and (8) are zero). By the Kirchhoff representation theorem,

an arbitrary

solution to (7) and (8) can be written as the sum of the retarded solution (

ret

) and a

free solution (

) or, alternatively, as the sum of the advanced solution (

adv

) and a

different free solution (

out

). The solution can be written either as

tot

(

~x,t

) =

ret

(

~x,t

) +

(

~x,t

)

tot

(

~x,t

) =

ret

(

~x,t

) +

(

~x,t

)

(11)

tot

(

~x,t

) =

adv

(

~x,t

) +

out

(

~x,t

)

tot

(

~x,t

) =

adv

(

~x,t

) +

out

(

~x,t

)

(12)

See Earman (2011, sec. 2.3).

Figure 2: On the left, a charge is shaken and sends out diverging waves (the retarded solution).

On the right, the shaken charge functions as a sink for converging waves (the advanced solution).

The motion of charged matter is the same in both figures, but the non-electromagnetic forces

needed to account for that motion would be different (as energy is transferred from matter to

field in figure a and from field to matter in figure b). To simplify the depiction of electromagnetic

waves, the dark rings only show the distribution of field energy.

The “in” in (

) stands for “incoming,” as this contribution to the potentials cannot be

traced back to past charged matter sources and is thus thought of as coming in from the infinite

past. The “out” in (

out

) stands for “outgoing,” as this contribution to the potentials

cannot be traced forward to future charged matter sinks and is thus thought of as going out to

the infinite future.

Generalizing from these two specific ways of decomposing the field, one can write arbitrary

potentials satisfying (7) and (8) as the sum of some constant

times the retarded solution plus

−

times the advanced solution plus a free solution (that is

times the incoming solution plus

−

times the outgoing solution):

tot

(

~x,t

) =

αφ

ret

(

~x,t

) + (1

−

)

adv

(

~x,t

) +

αφ

(

~x,t

) + (1

−

)

out

(

~x,t

)

tot

(

~x,t

) =

ret

(

~x,t

) + (1

−

)

adv

(

~x,t

) +

(

~x,t

) + (1

−

)

out

(

~x,t

)

(13)

In section 6, we will discuss taking

to be one-half and eliminating the free field solution (the

Wheeler-Feynman approach).

Corresponding to the retarded, advanced, incoming, and outgoing potentials, we can speak

of the retarded, advanced, incoming, and outgoing electric and magnetic fields, using (5) to pass

from potentials to fields. Or, we can combine the electric and magnetic fields into the Faraday

tensor

μν

and speak of retarded, advanced, incoming, and outgoing electromagnetic fields. In

discussions of the arrow of radiation, the tensor indices are often dropped and these fields are

written as

ret

adv

and

out

. We will adopt this terse notation in future sections.

Although we used the scalar and vector potentials in the Lorenz gauge to pick out the retarded, advanced,

incoming, and outgoing electromagnetic fields, these separate fields can be written using the potentials in any

gauge or using the gauge-independent electric and magnetic fields (or the gauge-independent Faraday tensor).

To better understand the Kirchhoff representation theorem, let us again consider the earlier

example of a charge that is shaken and emits electromagnetic waves (figure 2.a). In the retarded

representation (11), we have a retarded electromagnetic field describing diverging electromagnetic

waves leaving the charge after it is shaken (and also the Coulomb field around the charge).

There is no incoming (free) field. In the advanced representation (12) of the very same history,

we have an advanced electromagnetic field describing electromagnetic waves converging on the

charge—reaching it when it shakes (and also the Coulomb field around the charge). In addition

to this advanced field, there is an outgoing (free) field. Before the shaking, the outgoing field

cancels the waves in the advanced field (destructive interference). After the shaking, the outgoing

field describes the very same diverging electromagnetic waves that the retarded field described

in the retarded representation.

Thus, diverging waves can be expressed using either retarded or

advanced fields (with the appropriate free fields). In this case, energy is emitted from charged

matter into the electromagnetic field and we see that this energy emission can be described

using either the retarded or advanced representation. Similarly, cases of energy absorption can

be described using either the retarded or advanced representation.

The asymmetry we seek to explain is the observed asymmetry between converging and

diverging electromagnetic waves in the total electromagnetic field. Why is it that symmetric

converging waves are rare? Of course, waves that increase in strength over time are not so rare.

Consider, for example, the way in which ocean waves converge at an uneven shoreline or the way

in which ear trumpets concentrate sound waves to aid hearing. In this article, when we speak

of “converging” waves we mean waves that approach a central point or region in a symmetric,

coordinated manner. If you oscillate a charge up and down in the z direction, it will produce

a diverging electromagnetic wave that is symmetric about the z axis. The time-reverse of this

process is a converging wave that is symmetric about the z axis. We seek to explain why this kind

of converging wave is so rare. Choosing a particular representation does not give an explanation

as to why converging waves are rare. Choosing the retarded representation and stipulating that

there are no incoming free fields does yield such an asymmetry (section 4). However, we think

there is a better way to explain the absence of converging waves: they are improbable.

There is disagreement in the literature as to the arrow of electromagnetic radiation that needs

There is no need to ontologically privilege the Lorenz gauge, though one may choose to do so (Maudlin, 2018

discusses the pros and cons).

North (2003, pg. 1089) describes a similar case, considering the turning on of a light bulb in the advanced

representation.

On the point that both representations are fully capable of describing energy emission and absorption, see

North (2003, pg. 1088); Frisch (2005, pg. 141); Zeh (2007, pg. 18).

to be explained.

Frisch (2000, pg. 384) initially sought to explain why the incoming free field

is zero in the retarded representation, a formulation of the problem that we would resist because

some proposed explanations of the arrow of radiation involve non-zero incoming free fields. In

his later book, Frisch (2005, pg. 108) describes the asymmetry-to-be-explained as follows:

“There are many situations in which the total field can be represented as being

approximately equal to the sum of the retarded fields associated with a small number

of charges (but not as the sum of the advanced fields associated with these charges),

and there are almost no situations in which the total field can be represented as

being approximately equal to the sum of the advanced fields associated with a small

number of charges.”

This is a better formulation of the arrow, but it is a step removed from what we actually observe:

diverging waves in the total electromagnetic field (Price, 2006, sec. 2.4). North (2003, sec. 2)

seeks to explain why “accelerated charges produce retarded and not advanced radiation” or, put

more precisely, why “the retarded solution requires a much more natural free-field component

than the advanced solution.” In particular, North takes the weak and relatively uniform cosmic

microwave background (CMB) to be the incoming free field in the retarded representation. As

will be discussed in section 3.2, we do not want to assume that the CMB is a truly free field

or that the true incoming free field is simple in the retarded representation. Such assumptions

could be part of an explanation of the arrow of radiation, but they are not part of the arrow

itself.

Before embarking on the project of explaining the arrow of radiation, we should pause to

discuss both the time-reversal invariance of electromagnetism and the asymmetry of radiation

reaction. First, let us briefly consider the question as to whether the laws of electromagnetism

are time-symmetric (or, put another way, whether they are time-reversal invariant). Recall that

the puzzle of temporal bias for electromagnetic waves (posed in section 1) asks why we observe

waves behaving in a time-directed way (diverging but not converging) when the underlying laws

are time-symmetric. In fact, there is debate as to whether the laws of electromagnetism are

truly time-symmetric.

If you simply reverse the order of instantaneous states for matter and

field without altering the electric or magnetic fields at each moment, then the time-reverse of a

history obeying Maxwell’s equations will, in general, not obey Maxwell’s equations. If, instead,

you reverse the order of states and flip the orientation of the magnetic field, then the time-reverse

Price (1996, 2006) takes the arrow of radiation to be a macroscopic effect:

“According to this view the radiative asymmetry in the real world simply involves an imbalance

between transmitters and receivers: large-scale sources of coherent radiation are common, but large

receivers, or ‘sinks,’ of coherent radiation are unknown. ... At the microscopic level things are

symmetric, and we have both coherent sources and coherent sinks. At the macroscopic level we

only notice sources, however, because only they combine in an organized way in sufficiently large

numbers.” (Price, 1996, pg. 71)

We find it odd to draw such a sharp distinction between transmitters and receivers. Really, there are just charges

and charges sometimes emit and sometimes absorb energy. Also, radiation reaction illustrates that there is time

asymmetry at both the macro and micro level. We thus think it is wrong to say that “at the microscopic level

things are symmetric” (see Frisch, 2005, pg. 139–142, especially the point about synchrotron radiation).

See Albert (2000, ch. 1); Malament (2004); Arntzenius & Greaves (2009); Allori (2015); Struyve (2020);

Roberts (2021).

of a history obeying Maxwell’s equations will obey Maxwell’s equations. Applying this second

form of time reversal, the time-reverse of a history where the electromagnetic field is purely

retarded (with no incoming field) is a history where the electromagnetic field is purely advanced

(with no outgoing field). The time-reverse of a series of diverging waves is a series of converging

waves. Whether or not we count this form of time-reversal as true time-reversal, it is sufficient to

get the puzzle off the ground: If for every history with diverging waves there is a corresponding

history with converging waves, why do we never see converging electromagnetic waves?

In addition to the asymmetry between diverging and converging electromagnetic waves, there

is a related asymmetry in electromagnetic phenomena that needs to be explained: radiation

reaction. This asymmetry will not be our main focus, but we will discuss whether different

proposals for explaining the wave asymmetry can also explain the asymmetry of radiation

reaction. Here is the asymmetry: When a charged body accelerates, it emits radiation. That

radiation carries energy and momentum. Because momentum is conserved, the change in

field momentum is balanced by a change in momentum of the charged body (or whatever is

accelerating it). This radiation reaction is a time-asymmetric phenomenon somewhat similar to

friction. An accelerating charge will feel a reactive damping force that it would not feel if it were

uncharged. The time reverse of this effect would be an anti-damping force where the field gives

energy to the charge and helps it accelerate. That is not observed in nature.

For extended charged bodies, we can explain radiation reaction by analyzing the way that

electromagnetic waves propagate through bodies on their way out.

If we can explain why

electromagnetic waves diverge, we can explain radiation reaction. For point charges, the situation

is more complicated. Calculating the force on a point charge in classical electromagnetism is

problematic because the electromagnetic field becomes infinitely strong as one approaches the

charge (and the value of the field is undefined at the location of the charge). One could try

stipulating that point charges do not notice their own fields and only experience the standard

forces from the fields of other particles (via the Lorentz force law,

), but then one

would miss radiation reaction and have violations of both energy and momentum conservation.

One way out of these troubles is to argue that there are no point charges in nature.

Should

one desire to work with point charges, there are a variety of strategies available for patching

up classical electromagnetism. For example, one might replace the Lorentz force law with the

Lorentz-Dirac force law,

which can be written in outline as

ext

rad

inc

(14)

The first term,

ext

, is the Lorentz force from the retarded electromagnetic fields associated with

each of the other charges. The second term,

rad

, gives the (time-asymmetric) radiation reaction

force on the charge. This radiation reaction force can be expressed in terms of time derivatives

See Rohrlich (1999, 2000); Sebens (2022b, sec. 2.2).

For discussion of this idea, see Arntzenius (1993, sec. 3); Jackson (1999, ch. 16); Frisch (2005, pg. 55–58,

117–118); Pietsch (2012, pg. 145); Sebens (2020, sec. 8); Wald (2020, sec. 1.4); Sebens (2022a).

See Dirac (1938); Wheeler & Feynman (1945); Frisch (2005, sec. 3.3); Earman (2011, sec. 3); Kiessling (2011);

Lazarovici (2018, sec. 3.1).

of the charge’s position without referencing any electromagnetic fields. The final term,

inc

, is

the Lorentz force from the free incoming electromagnetic field:

inc

. For

some free incoming fields, this force will be well-defined. However, if the incoming field contains

electromagnetic waves that converge on the charge, then this force will not be well-defined. For

example, consider the purely advanced solution in figure 2.b, where the total electromagnetic

field is a wave that converges on a charge (which we will assume here to be a point charge).

In the retarded representation (11), the incoming free field would be ill-defined at the location

of the point charge when the wave converges. Thus, the Lorentz-Dirac equation can break

down. In general, the time-reverse of a history of charged particles and fields that obeys the

Lorentz-Dirac equation will be a history where the Lorentz-Dirac equation breaks down because

inc

is not always well-defined at the particle locations. For the law to function properly, we

must explain why the problematic incoming fields do not occur in nature. An explanation as

to why electromagnetic waves diverge might also explain this. We will come back to this point

in future sections. To review, the Lorentz-Dirac equation for point charges is time-asymmetric

but should not be viewed as the sole source of time-asymmetry in electromagnetism because

its applicability presupposes time-asymmetric radiation. There is more to be said about the

strengths and weaknesses of the Lorentz-Dirac equation, but let us not go too deep into the

problems of point charges.

3 Strategy 1: A Statistical Explanation

The motivating idea behind the statistical strategy for explaining the arrow of electromagnetic

radiation is stated clearly by Frisch (2015): converging waves are rare because “a converging

wave would require the co-ordinated behaviour of ‘wavelets’ coming in from multiple different

directions of space—delicately co-ordinated behaviour so improbable that it would strike us as

nearly miraculous.”

As an example, consider the advanced solution in the case of a single

charge briefly shaken (figure 2.b). There we have electromagnetic waves that approach the

charge from all sides in a coordinated manner that seems improbable. To justify the claim that

such histories are improbable, we need to say more about the probabilities for different histories.

Zeh (2007, pg. 18) criticizes this kind of statistical explanation, writing:

“The popular argument that advanced fields are not found in Nature because they

would require improbable initial correlations is known from statistical mechanics, but

totally insufficient . . . The observed retarded phenomena are precisely as improbable

among

all possible

ones, since they describe equally improbable

final

correlations.”

Note that (by the Kirchhoff representation theorem) we cannot actually say that advanced

In a similar complaint about the miraculous nature of converging waves, Popper (1958) writes: “If not steered

by an expanding wave, the contracting wave, though not in itself physically impossible, would nevertheless have

the character of a physical miracle: it would be like a conspiracy, undertaken by many people, each carefully

acting so as to support all the others, but without any previous arrangement, or anything like a prepared plan.”

Of course, converging waves are possible when you have a prepared plan. With the right electromagnetic wave

emitters arranged in a ring and set on a timer, you could get electromagnetic waves propagating inwards toward

a central point. (A similar example with water waves was given in the introduction.)

fields are absent in nature because the electromagnetic field can always be decomposed into

an advanced field and an outgoing (free) field (12). What must be explained is the fact that

converging electromagnetic waves are rarely found in nature. We can break the symmetry that

Zeh identifies by adding a statistical postulate assigning probabilities over initial conditions, not

final conditions. This is exactly the same move that is standardly made when one uses statistical

mechanics to explain the observed asymmetries of thermodynamics. Let us take a moment to

review the philosophical foundations of statistical mechanics before returning to the arrow of

electromagnetic radiation.

3.1 The Past Hypothesis and the Statistical Postulate

We will follow the current trend among philosophers of adopting a “Boltzmannian” or

“neo-Boltzmannian” (as opposed to a “Gibbsian”) approach to statistical mechanics.

On this

approach, we can take the entropy

of a system in a particular microstate to be proportional

to the logarithm of the volume

of the system’s macrostate in the space of all accessible

microstates,

log

W ,

(15)

where

is Boltzmann’s constant. The previous sentence requires some unpacking. For a

monatomic gas in a box, the microstate is specified precisely by microscopic variables: the

positions and velocities of all the atoms in the gas. The macrostate is specified imprecisely by

macroscopic variables (macrovariables) like the pressure, temperature, and volume of the gas.

The space of all accessible microstates for a closed system is an energy hypersurface within phase

space. Phase space is a 6

-dimensional space with dimensions for the

, and

components

of the position and velocity of each of

atoms. By imposing constraints like the boundaries of

the box, the total energy of the gas, and that the walls of the box do not transfer energy to the

environment, we arrive at a 6

−

1-dimensional constant-energy hypersurface that is the accessible

subspace within phase space. The macrostate picks out a region of this energy hypersurface and

the microstate singles out a particular point in that region (see figure 3).

The motion of the particular point representing the gas is determined by the laws governing

the collisions of atoms, which we might model as repulsive Newtonian forces that depend

on the distances between the atoms.

The laws for collisions are ordinarily taken to be

time-reversal-invariant, such that any sequence of events allowed by the laws is also allowed

to occur in the opposite order. However, the behaviors we observe are time-directed. (This was

the puzzle of temporal bias mentioned in the introduction.) For example, a gas confined to the

left half of a box will expand to fill the entire box if the barrier is removed. The reverse process

of a gas contracting to fill only half of a box never occurs. This is just one instance of the Second

Law of Thermodynamics. In one of its formulations, it says the following:

For an introduction to this Boltzmannian approach and a comparison to the Gibbsian alternative, see

Callender (1999); Albert (2000); Goldstein (2001); Uffink (2007, sec. 4 and 5); Frigg (2008); Carroll (2010,

ch. 8); North (2011); Wallace (2015); Frigg & Werndl (2019); Goldstein

et al.

(2020); Myrvold (2021, ch. 7).

Tim Maudlin has emphasized this point in his talk, “Boltzmann Entropy, the Second Law, and the

Architecture of Hell.”

Figure 3: This figure gives a depiction of the space of all accessible microstates, with small

low-entropy microstates in the corners and a large maximum-entropy equilibrium macrostate in

the center. The point representing the microstate begins in the medium-sized gray macrostate

and its evolution illustrates the second law of thermodynamics, moving through regions of

increasing entropy until the system reaches thermal equilibrium. This is exactly the evolution

one would expect from the structure of this space. Although it is not apparent in this simplified

image, in reality the space is high-dimensional, and the vast majority of the many ways out of

a small macrostate take you to a larger macrostate.

The Second Law of Thermodynamics:

The entropy of a closed system will

almost always either increase or remain the same.

Applying the definition of entropy in (15), this means that the point in phase space representing

a gas will move into macrostates of equal or greater volume until it reaches the equilibrium

macrostate (figure 3).

To explain the gas’s time-directed behavior, we can apply a statistical postulate over

the initial conditions at the moment the barrier is removed—assigning a uniform probability

distribution over the region of the energy hypersurface compatible with the known values for the

(appropriate) macrovariables. According to this probability distribution, it is overwhelmingly

likely that the gas’s microstate will move through ever larger macrostates until it reaches

equilibrium (and the gas is spread evenly throughout the entire box). Motion into a smaller

macrostate is physically possible but very unlikely.

Imposing this kind of statistical postulate at the beginning explains why we get time-directed

behavior (obeying the Second Law of Thermodynamics) afterwards. But, such a postulate makes

poor predictions about the past. To see why, consider applying this kind of postulate at a

time after the barrier is removed but before equilibration. At this moment, one might include

macrovariables that describe the unequal pressures (or densities) in the left and right halves

of the box. When we consider the positions and velocities for atoms that are consistent with

this macrostate and assign a uniform probability distribution over the microstates compatible

The Boltzmann equation mathematically describes how the density of the gas changes in the box. Boltzmann

derived this equation by making a statistical assumption about the collisions of the gas molecules, dubbed the

Stoßzahlansatz

, which postulates that the gas molecules are typically uncorrelated when the barrier is removed.

According to the Boltzmann equation, it is overwhelmingly likely that the gas will fill the box once the barrier is

removed (Brown

et al.

, 2009).

with the macrostate, the forward evolution will be exceedingly likely to fill the box. So will the

backwards evolution. Knowing only the macrostate, one would not predict the gas to have been

further concentrated in the left half of the box at earlier times (though in reality it was).

In general, if you apply the above kind of statistical postulate to a particular system at a

particular time, you will predict that entropy will increase (or stay the same) both forwards

and backwards in time. To generate correct predictions for some time period of interest, you

should apply a statistical postulate at the beginning of the time period. If we are interested

in everything that has happened in the history of our universe, we can apply such a statistical

postulate sometime soon after the big bang:

The Statistical Postulate:

For the purposes of making predictions in the future

of some time

soon after the big bang, we should apply a uniform probability

distribution to microstates compatible with the macrostate at

—where that

macrostate is a region of the relevant energy hypersurface within phase space (or

some successor to it) specified using appropriate macrovariables.

This statement resembles the formulation in (Albert, 2000, pg. 96), though we are focusing here

only on a single early moment (as in Loewer, 2020) and not directly specifying the probability

distributions to be used for subsystems at later times. The parenthetical about phase space

allows for a revision of the degrees of freedom available to a system when we move to a

physical description that includes more than just the positions and velocities of bodies—such as

quantum theories and, as we will see shortly, theories with fields. The choice of “appropriate”

macrovariables is left unspecified.

The Statistical Postulate will generate different predictions depending on the macrostate that

is posited at

. If the universe were in equilibrium at

, we would expect it to stay in (or near)

equilibrium. To generate accurate predictions, we can posit a low-entropy state:

The Past Hypothesis:

At some time

soon after the big bang, the universe was

in a particular low-entropy macrostate “that the normal inferential procedures of

cosmology will eventually present to us” (Albert, 2000, pg. 96).

Putting the Past Hypothesis and Statistical Postulate together, we have “a probability map

of the universe” (Loewer, 2020) assigning probabilities to all possible initial states and thus to

all possible histories of the universe. Using this probability distribution, we can predict that

systems will behave in a time-directed manner (obeying the Second Law of Thermodynamics)

even if the underlying dynamical laws are time-symmetric. We thus have a resolution of the

puzzle of temporal bias, at least for certain phenomena. Soon, we will see that this explanatory

schema works for waves as well.

At this point, one might naturally wonder about the nature of the probabilities specified by

the Past Hypothesis and the Statistical Postulate. What exactly are we saying when we specify

certain probabilities over initial conditions for the universe? This is a reasonable point of concern,

but not one that we will address here (see Allori, 2020). The interpretation of the probabilities

will eventually need to be settled to complete the Boltzmannian approach outlined here, as well

as our soon-to-come application of this approach to explaining the arrow of radiation.

For the goal of making accurate predictions about thermodynamic processes, many different

probability distributions would work just as well as the uniform one employed by the Statistical

Postulate (Wallace, 2011). Thus, although we would like to defend a statistical explanation as to

why electromagnetic waves diverge, we are not committed to the specific probability distribution

given by the combination of the Past Hypothesis and the Statistical Postulate.

3.2 Incorporating the Electromagnetic Field

As formulated above, the Past Hypothesis and the Statistical Postulate do not explicitly mention

the state of the electromagnetic field. But, a full specification of the microstate of the universe

would require specifying the state of the electromagnetic field. An accurate description of

the state of the electromagnetic field at this time in the history of the universe would have to

be quantum field theoretic.

For our purposes here, we will stick to classical physics (following

the plan set out in the introduction).

We can use this simplified context to give a fictional account of the early universe that

illustrates the way in which the arrow of electromagnetic radiation can be explained statistically,

noting that the details of that explanation will change as one moves to more advanced physics:

Long ago (at

charged matter lived in a bath of electromagnetic radiation so intense and

chaotic that, by inspection, one would not be able to discern clear converging or diverging

waves. At this time, there was no arrow of radiation in the phenomena to be explained. It is

here that we can apply the Statistical Postulate, adopting a uniform probability distribution

over microstates compatible with the macrostate for matter and field (following North, 2003

and Atkinson, 2006

). As the universe expanded, we were left with charged matter in a

One idea for avoiding unjustified precision is to collect all of the admissible probability distributions, form an

equivalence class, and reformulate the Statistical Postulate with this equivalence class of probability distributions.

(It is debated whether equivalence classes of probability distributions capture the right degree of precision. Rinard,

2017 argues in a different context on imprecise probabilities that these equivalence classes are still too precise, see

also Chen, forthcoming-a, p. 5 for this argument.) Going this route, we would no longer be interested in the exact

probability distribution over initial conditions but rather in something more coarse-grained: what kinds of initial

conditions are overwhelmingly likely (or “typical”) and what kinds are overwhelmingly unlikely (or “atypical”).

One could show that typical initial conditions yield the usual thermodynamic asymmetries. This approach to the

statistical postulate has been dubbed the

typicality account

(see, for instance, Goldstein, 2001, 2012; Maudlin,

2020; Hubert, 2021, who defend this account). For our purposes, we will stick to the ordinary statistical postulate

given above. One could easily adapt our soon-to-be-given explanation of the arrow of electromagnetic radiation

by using a modified statistical postulate if one wished to fold our explanation into a typicality account.

See North (2003, pg. 1095–1096); Atkinson (2006, pg. 539).

You might think of this fictional time as around 400,000 years after the big bang (Hartle, 2005, appendix A),

when charged matter formed a plasma in thermal equilibrium with the electromagnetic field (though the entire

universe was not in equilibrium, as can be inferred from the potential for further expansion).

Other authors have defended broadly similar explanations of the arrow of radiation. O. Penrose & Percival

(1962) introduce a “law of conditional independence” saying that you could not have distant parts of the universe

coordinate to form a converging wave because those parts of the universe were never in causal contact (see also

O. Penrose, 2001). R. Penrose (1979) gives a cosmological explanation of the arrow of electromagnetic radiation,

viewing that arrow and the thermodynamic arrows as all explained by a low-entropy initial state (presenting a

version of the Past Hypothesis that is explicit about the low gravitational entropy in the early universe). Hartle

(2005, appendix A) similarly employs a version of the Past Hypothesis to explain the arrow of radiation and the

thermodynamic arrows of time, describing the radiation that was present in the early universe as lacking the kind

of correlations that would give rise to converging waves. Earman (2011, pg. 524) ends his article with a conjecture

that “any [electromagnetic] asymmetry that is clean and pervasive enough to merit promotion to an arrow of time

is enslaved to either the cosmological arrow or the same source that grounds [the] thermodynamic arrow (or a

combination of both).” Arntzenius (1993, pg. 30) seeks a unified explanation of the arrow of radiation and other

very weak bath of radiation (the cosmic microwave background, CMB). Although a weak

bath of radiation could conceivably contain converging waves that are destined to grow in

strength as they approach charges, the Statistical Postulate applied to the earlier strong bath

makes the presence of any such waves exceedingly unlikely. Converging waves would require

precise coordination of electromagnetic field values at distant locations. Such fine-tuned initial

conditions are implausible, and rightly ruled improbable by the Statistical Postulate.

As North (2003, pg. 1088, 1091) tells the story, the cosmic microwave background is composed

of free fields—incoming fields that would have to be added to the retarded fields to get the total

electromagnetic field at some point now. Although for practical purposes it is reasonable to

treat the cosmic microwave background radiation as free, the beginning of our universe was so

violent that it seems impossible to say whether any of the radiation was truly free or whether it

was all at some point emitted by charged matter.

That is, it seems impossible to tell whether

the retarded representation (11) of the electromagnetic field—with no bounds on the integrals

in (9)—includes an incoming free field. Given the difficulty of ascertaining such a thing, we will

not argue that there is an empirical case to be made for this statistical approach to explaining

the arrow of electromagnetic radiation over the alternative approaches to be discussed in the

following sections (that do not allow for the possibility of free fields). There are multiple ways

to explain the observed absence of converging electromagnetic waves. As we cannot settle the

matter with data, we must look to other considerations.

3.3 Evaluation

We find the above brief account as to the origin of the arrow of radiation to be attractive for

four main reasons (all mentioned in North, 2003). First, there is no need to modify the standard

laws of electromagnetism to explain the arrow of radiation. Some view versions of the Past

Hypothesis and the Statistical Postulate as together forming an additional law (or pair of laws)

(Chen, 2022, forthcoming-b; Loewer, 2020). If the initial probability distribution specified by the

Past Hypothesis and Statistical Postulate is a law, then it is a law we already need to account for

other asymmetries and not an additional law peculiar to this strategy for explaining the arrow

of radiation. One might consider the Lorentz-Dirac equation (14) to be a modification to the

“arrows of time” within quantum field theory: “For a simpleminded philosopher like me, it would seem most

satisfactory if a unified account could be given of all arrows of time. ... I have the hope of a unified statistical

account within quantum field theory of all arrows of time.” Zeh (2007, sec. 2.2) gives a cosmological explanation

of the arrow of radiation (criticized in Frisch, 2000, sec. 4) that is not statistical, describing the early universe

as an ideal absorber with properties that allow us to ignore any free (incoming) fields that might have preceded

it. Although we would like to be able to claim Einstein as an ally, he does not consistently defend a statistical

explanation of the arrow of radiation. Ritz & Einstein (1909) write “Einstein believes that irreversibility is

exclusively due to reasons of probability” (Ritz & Einstein, 1990), but elsewhere Einstein (1909) concludes that

“The elementary process of the emission of light is, thus, not reversible” (Frisch, 2005, pg. 112). (For more on

Einstein’s views, see Frisch, 2005, pg. 109–114; Frisch & Pietsch, 2016.) Our goal here is not to focus on the

subtle differences between the accounts of the authors just listed, but instead to present a strong version of the

statistical approach that they are all clustering around (so that we can compare it to the very different approaches

in section 4–6).

On this point, Lazarovici (2018, pg. 159) (who opposes free fields) writes “We will never be able to determine

that some observed radiation is truly source-free, coming in ‘from infinity’. In fact, good scientific practice is to

assume that it is

not

and look for—or simply infer—the existence of material sources.” (See also Zeh, 2007, ch.

2; Pietsch, 2012, sec. 7.1; Wald, 2020, pg. 9.)

standard laws of electromagnetism. This equation is not needed to explain the prevalence of

diverging waves (the arrow of radiation) or to explain radiation reaction for extended charges,

but, as will be discussed below, the Lorentz-Dirac force law can be adopted to explain radiation

reaction for point charges. Some such modification would be needed to explain the arrow of

radiation reaction for point charges within any of the approaches to explaining the arrow of

radiation presented here except for the Wheeler-Feynman approach (see section 6).

Second, the statistical explanation gives a unified account of all wave asymmetries. In general,

converging waves are improbable because the strange initial conditions needed for them to occur

are improbable. Davies (1977, pg. 119) explains this well for the case of waves in a pond,

“. . . waves on real ponds are usually damped away at the edges by frictional effects.

The reverse process, in which the spontaneous motion of the particles at the edges

combine favourably to bring about the generation of a disturbance is overwhelmingly

improbable, though not impossible, on thermodynamic grounds.”

The fact that such coordination is improbable follows from the Statistical Postulate. This

postulate also explains why converging electromagnetic waves are improbable.

Third, the statistical explanation unifies wave asymmetries with familiar thermodynamic

asymmetries—gases expand, ice cubes melt, etc. These are two kinds of asymmetries that one

might have expected would receive different explanations. In fact, all of these asymmetries follow

from the Past Hypothesis and the Statistical Postulate, provided we include the electromagnetic

field in our descriptions of microstates and macrostates. The same probability distribution

explains both why waves diverge and why entropy increases.

Fourth, the symmetric treatment of charged matter and electromagnetic field fits well with

quantum field theory where charged matter and the electromagnetic field are modeled by very

similar equations (suggesting that they are the same kind of thing—see Sebens, 2022a). In the

statistical approach advocated here, the field is just as real as matter and it has independent

degrees of freedom (its state is not fixed by the behavior of charged matter, as in section 4,

though it is constrained by it, as will be discussed shortly). The appeal of such a picture

has been expressed in a memorable way by Penrose (1979, pg. 590)

while criticizing the

Wheeler-Feynman approach (which we will come to in section 6),

“And I have to confess to being rather out of sympathy with the whole

[Wheeler-Feynman] programme, which strikes me as being unfairly biased against

the poor photon, not allowing it the degrees of freedom admitted to all massive

particles!”

Wald (2020, pg. 2, 9–10) makes a similar remark when he addresses the “pernicious myth” that

electromagnetic fields are produced by charged matter (as in section 4),

“. . . the view that electromagnetic fields are produced by charges is particularly

untenable in quantum field theory, since it is essential for the understanding of

See also Rohrlich (2007, pg. 196); Pietsch (2012, pg. 141–142).

such phenomena as the vacuum fluctuations of the electromagnetic field that the

electromagnetic field have its own dynamical degrees of freedom, independently of

the existence of charged matter.”

Wald presents this lesson from quantum field theory at the beginning of his book on classical

electromagnetism, presumably because he thinks that we can learn about the proper formulation

of classical electromagnetism by studying its successor, quantum electrodynamics. We also think

that debates within one theory can sometimes be informed by looking to deeper physics. Ideally,

these two theories should fit together neatly with classical electromagnetism arising as a classical

limit to quantum electrodynamics and quantum electrodynamics derivable by quantizing the

classical electromagnetic field.

Having noted some reasons in favor of the above statistical strategy for explaining the arrow

of electromagnetic radiation, let us now respond to three potential objections. The first objection

we will consider is the entirely reasonable request for more details. In particular, a request for

details on the correct probability distribution to apply—in the early universe, and in simpler

situations analogous to the gas in a box discussed earlier. The most similar case to the gas in a box

would be a box filled with radiation (Bohm, 1951, ch. 1). Following the Boltzmannian approach

from sections 3.1 and 3.2, one would need to first select an appropriate set of macrovariables

to describe the radiation in the box. Then, one could assign probabilities over microstates

for a given macrostate by adopting a uniform probability distribution over the region of the

constant energy hypersurface (within the space of all possible field states) picked out by the

values of the macrovariables.

The equilibrium macrostate for the electromagnetic field would

be the macrostate with the largest volume out of all the macrostates that partition the energy

hypersurface. Completing the account sketched here would require developing a Boltzmannian

statistical mechanics for the classical electromagnetic field, a project that we have not seen

carried out in detail.

Although it appears difficult, we do not see any principled reason why

this project would be impossible. One would not expect the predictions of this classical statistical

mechanics for the electromagnetic field to always be accurate as it is, after all, purely classical. It

may be possible to incorporate some quantum insights and arrive at an improved semi-classical

treatment that can be used for certain applications (e.g., by forbidding certain microstates for

quantum reasons).

Historically, attempting to use statistical mechanical tools and classical electromagnetism to

predict the equilibrium state of a box of radiation led Rayleigh and Jeans to the ultra-violet

catastrophe for black-body radiation: the prediction that the box should contain an infinite

The space of possible states for a classical electromagnetic field is infinite-dimensional, but that does not

rule out the possibility of a probability density over states in that space. Such probability densities over classical

electromagnetic field states appear in wave functional approaches to quantum field theory (Bohm, 1952, appendix

A; Hatfield, 1992, sec. 10.2; Kaloyerou, 1994; Struyve, 2010, sec. 4).

Max Planck was probably the first to study the problem of the arrow of radiation. Before he derived the

correct spectrum for black-body radiation, he wanted to solve the problem why certain electromagnetic phenomena

happen in one direction but not the other. He derived electromagnetic analogues of the Stoßzahlansatz and

the Boltzmann equation, which he called respectively the assumption of

natural radiation

and

the fundamental

equation

(see Kuhn, 1978; Darrigol, 1992, for the historical details). The neo-Boltzmannian approach we envision,

which relies on carving the space of microstates into macrostates and assigning a probability distribution, is

different from Planck’s approach, which is based on the Boltzmann equation.

amount of energy.

That particular danger would be avoided in the above Boltzmannian

approach because all of the states under consideration lie within a hypersurface of constant

(finite) energy.

When charged matter is present, there are some subtleties in specifying the allowed

microstates for the electromagnetic field in a Boltzmannian approach. Gauss’s law (2) requires

a certain coherence between states of matter and field. However, states of the electromagnetic

field that obey this constraint might still be unacceptable. As Hartenstein & Hubert (2021) have

shown, generic states of the electromagnetic field obeying the synchronic Maxwell equations, (1)

and (2), will give rise to pathological future behavior where shock fronts disrupt the dynamics

and cause the theory to break down (see also Lazarovici, 2018, sec. 8.1). Thus, we run into

challenges getting off the ground with a specification of allowed microstates for matter and

field.

The above general issues regarding macrostates, microstates, and probabilities in classical

electromagnetism are compounded when we focus on the early universe, where an adequate

description of the physics would require quantum field theory (in particular, quantum

electrodynamics).

Arntzenius (1993), North (2003, pg. 1096), and Atkinson (2006) have

discussed the importance of quantum electrodynamics for explaining the arrow of electromagnetic

radiation. We agree that the ultimate explanation of the arrow of radiation should appeal to

quantum electrodynamics. Still, we think the simplified classical parable told above is helpful

for getting a flavor for the kind of explanation that we expect quantum electrodynamics to yield.

It is correct in spirit, thought not in details.

A second objection to the above statistical explanation of the arrow of radiation is that it

allows for backwards causation—deeming it merely improbable and not impossible. We do not

think the statistical explanation requires allowing for the possibility of backwards causation,

though it has been paired with this view elsewhere. North (2003, pg. 1095) writes:

“The temporally symmetric laws say that both advanced and retarded radiation

could be emitted.

However, given the universe’s thermal disequilibrium, the

charges are overwhelmingly likely to radiate towards the future, as part of the

overwhelmingly likely progression towards equilibrium in that temporal direction.

They are overwhelmingly unlikely to radiate towards the past because the universe

According to the equipartition theorem in classical statistical mechanics, each independent degree of freedom

of a classical-mechanical system at equilibrium has average energy

kT/

2. Rayleigh and Jeans extended this idea

to the electromagnetic field, treating each mode of the field as an independent degree of freedom. Since there are

infinitely many such modes within the electromagnetic field, the energy of the electromagnetic field at equilibrium

is predicted to be infinite (the ultra-violet catastrophe). Despite this failure, Rayleigh and Jeans were able to

derive a law that approximates the spectrum of black-body radiation well for low frequencies (Kuhn, 1978). Wien

(1897) gave a different classical derivation of the black-body spectrum by modeling the way charges in the walls

of the black-body emit radiation using the Maxwell-Boltzmann distribution for their velocity distribution. Wien’s

law avoids ultra-violate catastrophe and gives an accurate approximation to the black-body radiation spectrum

at high frequencies, but not at low frequencies. Ultimately, accurately describing black-body radiation appears to

require some quantum physics. Planck arrived at the correct spectrum by treating the problem semi-classically,

assuming that certain frequencies will be absorbed and not reemitted by the walls and also allowing the energy

to depend on the frequency (contra the equipartition theorem).

One way to resolve the problems raised by Hartenstein & Hubert (2021) would be to adopt an approach

where the electromagnetic field does not have any independent degrees of freedom, as in sections 4–6.

was at thermal equilibrium in that direction. Note that on this view the retarded

nature of radiation is statistical: advanced radiation is not prohibited but given

extremely low probability.”

When North speaks of “advanced radiation” or “radiat[ing] towards the past,” she is talking

about situations where there is a converging wave in the total electromagnetic field approaching

a particular charge that resembles the advanced field of that charge (so that, locally, the advanced

representation seems more natural than the retarded representation). Such situations may be

described as involving backwards causation.

But, they do not need to be. Even when you

consider a converging electromagnetic wave that can be represented by a purely advanced field

(with no outgoing field), as in figure 2.b, you do not need to view the electromagnetic field at

any point in space and time as caused by future charges. You can instead view it as caused by

earlier states of the electromagnetic field (see section 2). The wave moves towards the charge

because it has been moving towards the charge. In general, whether the electromagnetic field

is purely retarded, purely advanced, or neither, it is possible to understand its time evolution

purely in terms of forward causation.

A third objection that might be raised to our account is that the Past Hypothesis and

Statistical Postulate, when spelled out precisely for electromagnetic field and matter, will be

complicated. The exact degree of complexity remains to be seen and the cost of that complexity

will depend on whether one views these principles as laws of nature or as something else. For now,

let us just note that if you would like to adopt versions of the Past Hypothesis and Statistical

Postulate to explain thermodynamic asymmetries, you cannot confine these principles to matter

and ignore the electromagnetic field (seeking simplicity). Specifying the initial positions and

velocities of charged bodies will fail to determine the future evolution of matter because there

are many states of the field compatible with any such state of charged matter (e.g., a given

electromagnetic wave could be present or absent). We need a way of selecting a particular state

of the electromagnetic field, or of assigning probabilities over different states, if we want to be

able to make predictions about the future motion of matter.

A fourth potential objection to our account is that we have not yet explained radiation

reaction. As was discussed in section 2, for extended charged bodies radiation reaction can be

explained by analyzing the way that electromagnetic waves propagate through such bodies on

their way out. The arrow of radiation reaction follows from the arrow of radiation. That kind

of explanation can go through even if the (retarded) waves emitted by the charged body are

not the only electromagnetic waves in existence. There may be other waves that were emitted

by other charged bodies in the past or waves that are part of the free incoming electromagnetic

field. So long as those waves do not conspire to converge on the accelerating charge, we will

see radiation reaction (as well as reaction to the forces from the other waves). If all charges

Although we will generally view causes as preceding their effects, we see the appeal of allowing for causes

that are in the future of their effects if there are periods of time (or regions of spacetime) where (relative to what

we call past and future) entropy decreases and waves converge. Boltzmann’s hypothesis that the low-entropy of

the early universe arose as a fluctuation from a high-entropy distant past would give rise to such periods of time

before the early universe reached its low-entropy state (Carroll, 2010, ch. 10).

are extended charges, we can stop there. For point charges, there are multiple ways of handling

radiation reaction. As was discussed in section 2, one idea is to replace the Lorentz force law with

the Lorentz-Dirac force law (14). If we treat the electromagnetic field at

as a free incoming

field, then the Lorentz-Dirac force law gives a well-defined equation of motion so long as

inc

well-defined at every point in spacetime that a charge passes through and no waves in the initial

field converge precisely on any of the point charges. Although our statistical explanation of the

arrow of radiation does not deem such precisely converging waves impossible, they are effectively

ruled out as they would only occur in a set of measure zero among the allowed initial conditions.

One should expect waves that converge on a region to be rare and waves that converge on a

point to be absent. Thus, the Lorentz-Dirac force law can be used to explain radiation reaction

once statistical moves have been made to tame the free field.

4 Strategy 2: The Sommerfeld Radiation Condition

An alternative strategy for explaining the arrow of radiation is to modify the laws of

electromagnetism.

The cleanest way of doing this is by restricting the space of physical

possibilities allowed by the theory to histories of matter and field where the electromagnetic

field has no free (incoming) component in the retarded representation,

ret

The Sommerfeld Radiation Condition:

The total electromagnetic field is purely

retarded. At every point in spacetime,

= 0 and

ret

This condition eliminates free fields from the retarded representation, though they would still

be present if one chose to use the advanced representation,

ret

adv

out

. If the

electromagnetic field is purely retarded (

= 0) at one time, it will be purely retarded at all

times. Thus it is equivalent to require that the field be purely retarded at one time, or to require,

as above, that it be purely retarded at all times.

Assuming that there was an infinite past, the Sommerfeld Radiation Condition states that

the electromagnetic field at a point in spacetime can be calculated by integrating contributions

from progressively further distances and earlier times out to spatial and past infinity along the

light-cone (9).

If there was a first moment, one might attempt to impose a version of the

Sommerfeld Radiation Condition by restricting the spatial integrals for the retarded potentials

(9) so that the retarded times being integrated over never precede the first moment (as a way of

positing that

= 0 at the initial moment and thus at all future moments). But, that will not

work. The recipe just described would have the consequence that, at the first moment, there is no

Sommerfeld wrote down the original formulation of the condition in 1912 in order to have unique solutions to

the Helmholtz equation (for the history, see Schot, 1992). This equation is time-independent, and the condition

is accordingly a restriction on the solutions at

spatial

infinity. This original boundary condition evolved into

the above Sommerfeld Radiation Condition, requiring that all radiation be attributable to past sources. In his

textbook, Sommerfeld (1949, p. 189) gives the following motivation for his boundary condition: “We call it the

condition of radiation

: the sources must be

sources

, not

sinks

, of energy. The energy which is radiated from the

sources must scatter to infinity;

no energy may be radiated from infinity into the prescribe singularities of the

field

[. . . ].”

See Frisch (2005, pg. 156–157).

See North (2003, pg. 1087); Price (2006); Earman (2011, sec. 2.8).

electromagnetic field at any point in space where there is no charged matter—even right next to

a charged body (in violation of Gauss’s law, one of Maxwell’s inviolable equations). Instead, one

might attempt to stipulate that the electromagnetic field at the first moment is just the field of

each bit of charge at the first moment. For example, a point charge at rest would be surrounded

by a Coulomb electric field. However, this strategy breaks down because the field generated by

a bit of charged matter via (9) depends on its imagined past, and multiple fictional pasts will be

compatible with the initial state of charged matter at the first moment (Hartenstein & Hubert,

2021). Thus, we do not see a precise way of stating the Sommerfeld Radiation Condition under

the assumption of a first moment.

To move forward with our assessment of this proposal, let

us assume that there was an infinite past.

4.1 Justification

In his excellent and widely-used textbook on classical electromagnetism, Griffiths (2013, pg.

446–447) gives the following justification for adopting the Sommerfeld Radiation Condition:

“Although the advanced potentials are entirely consistent with Maxwell’s equations,

they violate the most sacred tenet in all of physics: the principle of

causality

They suggest that the potentials

now

depend on what the charge and the current

distribution

will be

at some time in the future—the effect, in other words, precedes

the cause. Although the advanced potentials are of some theoretical interest, they

have no direct physical significance.

“... the theory itself is

time-reversal invariant

, and does not distinguish ‘past’

from ‘future.’ Time asymmetry is introduced when we select the retarded potentials

in preference to the advanced ones, reflecting the (not unreasonable!) belief that

electromagnetic influences propagate forward, not backward, in time.”

Similar reasoning appears in Schwinger

et al.

(1998, pg. 346); Jefimenko (2000);

Rohrlich

(2000, 2002, 2006, 2007). There are at least three points where one might criticize Griffiths’

argument. First, although it is true that, for the purely advanced potentials (10), the value

Frisch (2005, pg. 107) suggests that we might ignore the Coulomb fields when applying the Sommerfeld

Radiation Condition at a given moment such as the initial moment. One way to do this, for point charges, would

be to include, at the initial moment, only the generalized Coulomb field for each charge (Zeh, 2007, pg. 29). This

amounts to calculating the retarded fields that would have been generated if each particle had always been moving

before the initial moment with the same velocity that they have at the initial moment. Hartenstein & Hubert

(2021, sec. 3.3) show that there will be persistent and proliferating discontinuous jumps in the electromagnetic

field values if you only match the velocities (and not the accelerations) between the hypothetical past trajectories

and the actual future trajectories of charged particles. The fact that we do not observe such discontinuities speaks

strongly against this proposal.

Jefimenko incorporates the principle of causality into a broader vision regarding how fundamental laws in

physics should be formulated:

“Causal relations between phenomena are governed by the

principle of causality

. According to this

principle, all present phenomena are exclusively determined by past events. Therefore equations

depicting causal relations between physical phenomena must, in general, be equations where a

present-time quantity (the effect) relates to one or more quantities (causes) that existed at some

previous time.” (Jefimenko, 2000, pg. 4)

Jefimenko’s equations (16), giving the current state of the electromagnetic field in terms of the past behavior of

charges, fit this mold.

of the electromagnetic field at a given point in spacetime can be calculated mathematically by

examining charges in the future (along the future light cone), it does not automatically follow

that effects will precede their causes. As we discussed in section 2, advanced solutions can

be interpreted with an ordinary order of cause and effect. For example, the purely advanced

converging wave in figure 2.b can be seen as a cause of the charge’s motion (instead of as an effect

that precedes this cause). Second, to justify the use of purely retarded solutions Griffiths must

reject not only purely advanced solutions, but also solutions that are neither purely retarded nor

purely advanced and involve free fields whether they are expressed in the retarded or advanced

representation. Finally, one could of course contest the “principle of causality” requiring causes

to precede effects, though we will not explore that avenue here.

In a solution to Maxwell’s equations that violates the Sommerfeld Radiation Condition and

is not purely retarded, the value of the electromagnetic field at a given point in space cannot

be fully attributed to past sources. One could attempt to defend the Sommerfeld Radiation

Condition by arguing that the electromagnetic field is created by charges and must always be

fully attributable to past sources. But, why make this assumption? We see that defense as

begging the question as to whether the condition should be adopted.

Frisch (2000, sec. 5)

argues that the Sommerfeld Radiation Condition is justified not

by a deeper principle (like a principle of causality), but by the same kind of evidence that

justifies Maxwell’s equations.

We accept Maxwell’s equations because of their success in

explaining and predicting the behavior of charged matter and the electromagnetic field. With

the Sommerfeld Radiation Condition, the predictive and explanatory power of electromagnetic

theory arguably increases, as this condition may be used to explain why electromagnetic waves

generally diverge by ruling out certain solutions to Maxwell’s equations containing converging

waves. If imposing the Sommerfeld Radiation Condition were the only way to explain the

asymmetry of electromagnetic radiation, this would be a decisive argument for its inclusion

among the laws of classical electromagnetism. However, the presence of competing proposals

leaves room for debate as to whether the condition should be adopted.

In his later book, Frisch (2005, pg. 152) defends a variant of the Sommerfeld Radiation Condition that he

calls the “retardation condition”: “. . . each charged particle physically contributes a fully retarded component to

the total field.” This causal claim appears to leave open the possibility of there being a genuinely free incoming

field in addition to the retarded fields associated with charges. Thus, we do not see in this claim any restriction

on the space of physical histories allowed within classical electromagnetism. (For Frisch, it is a claim about

counterfactuals.) Without restricting the allowed histories or assigning a probability distribution over them, we

do not yet have the kind of resources that would be needed to explain the arrow of electromagnetic radiation. Frisch

(2005, pg. 152) combines his retardation condition with a time-asymmetric assumption about the distribution of

absorbing media that might be called an “absorber condition”: “. . . space-time regions in which we are interested

generally have media acting as absorbers in their past. . . . fields that are not associated with charges that are

relevant to a given phenomenon can generally be ignored, and it is easy to choose initial-value surfaces on which

the incoming fields are zero.” Setting the toothless retardation condition aside, this absorber condition could

be used as part of a statistical explanation of the arrow of radiation (as it would require statistical reasoning

to explain why certain media act as absorbers and not emitters—see Frisch, 2000, sec. 4; Price, 2006, sec. 3).

That being said, we do not think the condition is necessary to explain the asymmetry between converging and

diverging waves. A statistical explanation that assigns probabilities over states of the electromagnetic field in the

early universe will render converging waves automatically improbable (section 3). There is no need to rely on

assumptions about their eventual absorption.