Time-reversal waves

The waves that satisfy time-reversal Maxwell equations are time-reversal waves. There are two kinds of time-reversal waves. One is the time-reversal wave corresponding to the retarded wave, and another is the time-reversal wave corresponding to the advanced wave.

Here, the time-reversal wave is a possible physical wave that exists in nature. It satisfies time-reversal Maxwell equations.

Time-reversal waves

We know that Maxwell equations have two kinds of solution: retarded solution and advanced solution. Corresponding to the retarded solution, there is the retarded wave. Corresponding to the advanced solution, there is advanced wave. Advanced wave does not satisfy our traditional causality. Many people think the advanced wave is not a real wave, but there are a few very famous physicists who believe the advanced wave exists. Assume the retarded wave and advanced wave all exist. The question is, is the advanced wave the time-reversal wave of the retarded wave? The answer is negative. Maxwell equations are not time-reversible. We can define the time-reversal waves for the retarded wave and define the time-reversal wave for the advanced wave. The question is, do the time-reversal waves exist or not?

Conjugate transform

It is not clear who first introduced the concept of the conjugate transform, but the detailed theory of the conjugate transform can be found in （Jin Au Kong）. The conjugate transform can be seen as following:

$\mathbb{ℂ}(\mathbf{𝐄}(t),\mathbf{𝐇}(t),\mathbf{𝐉}(t),\mathbf{𝐊}(t),\mathit{ϵ}(t),\mathit{\mu }(t))=(\mathbf{𝐄}(-t),-\mathbf{𝐇}(-t),-\mathbf{𝐉}(-t),\mathbf{𝐊}(-t),\mathit{ϵ}(-t),\mathit{\mu }(-t))$

or

$\mathbb{ℂ}(\mathbf{𝐄}(\omega ),\mathbf{𝐇}(\omega ),\mathbf{𝐉}(\omega ),\mathbf{𝐊}(\omega ),\mathit{ϵ}(\omega ),\mathit{\mu }(\omega ))=({\mathbf{𝐄}}^{*}(\omega ),-{\mathbf{𝐇}}^{\mathbf{*}}(\omega ),-{\mathbf{𝐉}}^{\mathbf{*}}(\omega ),{\mathbf{𝐊}}^{*}(\omega ),{\mathit{ϵ}}^{\mathit{*}}(\omega ),{\mathit{\mu }}^{\mathbf{*}}(\omega ))$

Where $\mathbb{ℂ}$ is the conjugate transform. $\mathbf{𝐄}$ is electric field. $\mathbf{𝐇}$ Magnetic field. $\mathbf{𝐉}$ current intensity. $\mathbf{𝐊}$ magnetic current intensity. $\mathit{ϵ}$ is permittivity, $\mathit{\mu }$ is permeability, $t$ is time, $\omega$ is frequency.

It is important that if a field satisfies the Maxwell equations, after the conjugate transform, it still satisfies the Maxwell equations. If the original field is a retarded wave, after the transform it becomes an advanced wave. Vice versa, if the original field is an advanced wave, after the transform it becomes the retarded wave.

Time reversal transform

Time reversal transform is different from conjugate transform. It can be defined as following: $\mathbb{ℝ}(\mathbf{𝐄}(t),\mathbf{𝐇}(t),\mathbf{𝐉}(t),\mathbf{𝐊}(t),\mathit{ϵ}(t),\mathit{\mu }(t))=(\mathbf{𝐞}(-t),\mathbf{𝐡}(-t),-\mathbf{𝐣}(-t),-\mathbf{𝐤}(-t),\mathit{ϵ}(-t),\mathit{\mu }(-t),)$

or

$\mathbb{ℝ}(\mathbf{𝐄}(\omega ),\mathbf{𝐇}(\omega ),\mathbf{𝐉}(\omega ),\mathbf{𝐊}(\omega ),\mathit{ϵ}(\omega ),\mathit{\mu }(\omega ))=({\mathbf{𝐞}}^{*}(\omega ),{\mathbf{𝐡}}^{\mathbf{*}}(\omega ),-{\mathbf{𝐣}}^{\mathbf{*}}(\omega ),-{\mathbf{𝐤}}^{*}(\omega ),{\mathit{ϵ}}^{\mathit{*}}(\omega ),{\mathit{\mu }}^{\mathbf{*}}(\omega ))$

Where $\mathbb{ℝ}$ is time reversal transform. $[\mathbf{𝐞},\mathbf{𝐡}]$ is the time-reversal electromagnetic field. $[\mathbf{𝐣},\mathbf{𝐤}]$ is time-reversal electric current intensity and time-reversal magnetic current intensity. After the time-reversal transform $\mathbb{ℝ}$, a normal electromagnetic field becomes time-reversal magnetic fields. The time-reversal electromagnetic fields do not satisfy Maxwell equations, but they satisfy time-reversal Maxwell equations.

Assume there is a retarded wave, the time-reversal wave corresponding to this retarded wave is not the advanced wave. The advanced wave is from current time going to the past time. The time reversal wave is from a future time going to the current time. Advanced wave still satisfies Maxwell equations; it is a normal electric field. Time-reversal wave does not satisfy Maxwell equations; it is not a normal electromagnetic field.

Time reversal Maxwell equations

Name	Integral equations	Differential equations	Meaning
time-reversal Gauss's law	$\partial \mathrm{\Omega }$ $\mathbf{𝐞}\cdot \mathrm{d}\mathbf{𝐒}=\frac{1}{{\epsilon }_0}\underset{\mathrm{\Omega }}{\iiint }\rho \mathrm{d}V$	$\mathrm{\nabla }\cdot \mathbf{𝐞}=\frac{\rho }{{\epsilon }_0}$	The electric flux through a closed surface is proportional to the charge inside an enclosed volume.
time-reversal Gauss's law for magnetism	$\partial \mathrm{\Omega }$ $\mathbf{𝐛}\cdot \mathrm{d}\mathbf{𝐒}=0$	$\mathrm{\nabla }\cdot \mathbf{𝐛}=0$	The magnetic flux through a closed surface is zero (i.e. there are no magnetic monopoles)
time-reversal Maxwell–Faraday equation (Faraday's law of induction)	${\oint }_{\partial \mathrm{\Sigma }}\mathbf{𝐞}\cdot \mathrm{d}\mathit{l}=\frac{\mathrm{d}}{\mathrm{d}t}\underset{\mathrm{\Sigma }}{\iint }\mathbf{𝐛}\cdot \mathrm{d}\mathbf{𝐒}$	$\mathrm{\nabla }\times \mathbf{𝐞}=\frac{\partial \mathbf{𝐛}}{\partial t}$	The work per unit charge required to move a charge around a closed loop equals the rate of decrease of the magnetic flux through an enclosed surface.
time-reversal Ampère's circuital law (with Maxwell's addition)	${\oint }_{\partial \mathrm{\Sigma }}\mathbf{𝐛}\cdot \mathrm{d}\mathit{l}={\mu }_0(\underset{\mathrm{\Sigma }}{\iint }-\mathbf{𝐣}\cdot \mathrm{d}\mathbf{𝐒}-{\epsilon }_0\frac{\mathrm{d}}{\mathrm{d}t}\underset{\mathrm{\Sigma }}{\iint }\mathbf{𝐞}\cdot \mathrm{d}\mathbf{𝐒})$	$\mathrm{\nabla }\times \mathbf{𝐛}={\mu }_0(-\mathbf{𝐣}-{\epsilon }_0\frac{\partial \mathbf{𝐞}}{\partial t})$	The magnetic field induced around a closed loop is proportional to the electric current plus displacement current (proportional to the rate of change of electric flux) through an enclosed surface.

Time reversal signal processing

It is worth saying that there is also a kind of technology referred to as the time-reversal focus method, which applies a normal electromagnetic field to simulate some property of time-reversal waves; please see Time reversal signal processing for details. Here, the time-reversal wave doesn't mean that kind of technology.

Background of time-reversal wave

Here, some things related to the time-reversal wave will be discussed. Perhaps the time-reversal wave doesn't involve the Alberta.

Action at a distance

The theory of action-at-a-distance is introduced by （K. Schwarzschild） （H. Tetrode） （A.D. Fokker）. According to this theory, an electric current will produce two electromagnetic potentials or two electromagnetic waves: one is the retarded wave, and another is the advanced wave. The emitter can send the retarded wave, but at the same time it also sends an advanced wave. The absorber can send the advanced wave, but at the same time it also sends a retarded wave. According to this theory, the sun cannot send the radiation wave out if it stayed alone in empty space. Infinite absorbers are the reason that the sun can radiate its light. The action can be written as following:

$S=-\sum _i{m}_{i}c\int (\frac{d{x}_{i\mu }}{d{\tau }_i}\frac{d{x}_i^{\mu }}{d{\tau }_i}{)}^{\frac{1}{2}}d{\tau }_i-\sum _i\sum _{j<i}\frac{e_i{e}_j}{c}\int \int \delta (s_{ij}^2)\frac{d{x}_{i\mu }}{d{\tau }_i}\frac{d{x}_j^{\mu }}{d{\tau }_j}d{\tau }_{i}d{\tau }_j=\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{m}\mathrm{u}\mathrm{m}$

where

$s_{ij}^2=(x_{i\mu }-x_{j\mu })(x_i^{\mu }-x_j^{\mu })$

$ds=c^{2}d{t}^2-d{x}_1^2-d{x}_2^2-d{x}_3^2$

Absorber theory

The absorber theory is introduced by Wheeler and Feynman （J. A. Wheeler） （J. A. Wheeler）. The absorber theory is built on the top of the above theory of the action-at-a-distance （A.D. Fokker） （K. Schwarzschild） （H. Tetrode） . According to this theory, the electromagnetic field has no freedom of its own. The electromagnetic field is an adjective field. It is only a bookkeeper for the action or reaction between at least two charges. That means without a test charge or absorber, only the emitter alone cannot produce the radiation. Absorber theory tries to offer a better explanation for the recoil force of an accelerated or decelerated charge in empty space. The recoil force has been introduced by Dirac （P. A. M. Dirac）. But Wheeler and Feynman do not accept that Dirac did not offer a reasonable reason for that formula. Wheeler and Feynman try to use the absorbers stayed on the infinite big sphere to explain the formula given by Dirac. The absorber theory also emphasizes the importance of the absorber in the radiation process.

Transactional interpretation for quantum mechanics

The transactional interpretation of quantum mechanics introduced by John Cramer (John Cramer). The transactional interpretation is built on the top of Wheeler–Feynman absorber theory. In this theory, the emitter can send an offer wave to the absorber. When the absorber receives the offering wave, it can send a confirmation wave to the emitter. These two waves can have a handshake. This handshake process is the transactional process. In this process the photon or other particle is produced. The confirmation wave is an advanced wave.

Welch's reciprocity theorem

This theory was introduced by W. J. Welch proposed in 1960 （W. J. Welch）. The theorem can be seen in the following:

$-\underset{t=-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\underset{V1}{\int }{\mathbf{𝐄}}_2(t)\cdot {\mathbf{𝐉}}_1(t)dVdt=\underset{t=-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\underset{V2}{\int }{\mathbf{𝐄}}_1(t)\cdot {\mathbf{𝐉}}_2(t)dVdt$

In order to prove the above formula, it is required to prove a surface integral vanishes. The surface is on the infinite big sphere. The proof of the vanish of the surface integral on infinite big sphere need the two waves, one is the retarded wave and another is the advanced wave.

Rumsey's reciprocity theorem

V.H. Rumsey has introduced his summary of the Lorentz reciprocity theorem as "action and reaction". He has applied the complex conjugate transform to his "action and reaction" theorem and obtained a new reciprocity theorem （V.H. Rumsey）,

$-\underset{V1}{\int }{\mathbf{𝐄}}_2^{*}(\omega )\cdot {\mathbf{𝐉}}_1(\omega )dVdt=\underset{V2}{\int }{\mathbf{𝐄}}_1(\omega )\cdot {\mathbf{𝐉}}_2^{*}(\omega )dV$

Inner product space for the electromagnetic fields on a closed Surface

Shuang-ren Zhao has defined the inner product for any two electromagnetic fields which are (Shuang-ren Zhao),

$({\xi }_1,{\xi }_2{)}_{\mathrm{\Gamma }}={\oint }_{\mathrm{\Gamma }}({\mathbf{𝐄}}_1(\omega )\times {\mathbf{𝐇}}_2^{*}(\omega )+{\mathbf{𝐄}}_2^{*}(\omega )\times {\mathbf{𝐇}}_1(\omega ))\cdot \hat{n}d\mathrm{\Gamma }$

where $\xi =[\mathit{E},\mathit{H}]$, $\tau =[\mathit{J},\mathit{K}]$, Shuang-ren Zhao has proved that the above inner products satisfy the Inner product space 3 definitions. If ${\tau }_2$ is taken as a unit vector of ether current ${\mathit{J}}_2$ or ${\mathit{K}}_2$, the field ${\tau }_1={\mathit{J}}_1,{\mathit{K}}_1$. ${\xi }_1$ can be calculated either on the original source ${\mathit{J}}_1$ or on the surface $\mathrm{\Gamma }$. $\mathrm{\Gamma }$ is any surface outside the two volumes $V_1$ and $V_2$.

$\hat{n}$ is a unit surface normal vector. Shuang-ren Zhao has proved that this kind of inner product satisfies inner product space 3 conditions.

Zhao's mutual energy theorem

Shuang-ren Zhao has introduced the mutual energy theorem (Shuang-ren Zhao) in early 1987. Shuang-ren Zhao emphasizes that the mutual energy theorem is an energy theorem instead of some kind of reciprocity theorem. The theorem described an energy in space.

$-({\tau }_1,{\xi }_2{)}_{V_1}=({\xi }_1,{\tau }_2{)}_{V_2}$

where

$\mathrm{\Gamma }$ is any close surface or infinite big surface separating $V_1$ and $V_2$. We take the direction of $\hat{n}$ is from $V_1$ to $V_2$.

$({\tau }_1,{\xi }_2{)}_{V_1}=\underset{V1}{\int }({\mathit{J}}_1(\omega )\cdot {\mathit{E}}_2^{*}(\omega )+{\mathit{K}}_1(\omega )\cdot {\mathit{H}}_2^{*}(\omega ))dV$

$({\xi }_1,{\tau }_2{)}_{V_2}=\underset{V2}{\int }({\mathit{E}}_1(\omega )\cdot {\mathit{J}}_2^{*}(\omega )+{\mathit{H}}_1(\omega )\cdot {\mathit{K}}_2^{*}(\omega ))dV$

de Hoop's reciprocity theorem

Adrianus T. de Hoop published the time-domain cross-correlation reciprocity theorem at the end of 1987 （Adrianus T. de Hoop） which can be seen as following:

$-\underset{t=-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\underset{V2}{\int }{\mathbf{𝐄}}_2(t+\tau )\cdot {\mathbf{𝐉}}_1(t)dVdt=\underset{t=-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\underset{V1}{\int }{\mathbf{𝐄}}_1(t)\cdot {\mathbf{𝐉}}_2(t+\tau )dVdt$

The forgot second Lorentz reciprocity theorem

I. V. Petrusenko introduced the forgot second Lorentz reciprocity theorem in 2009 （I. V. Petrusenko）.

It is similar to the Rumsey's reciprocity theorem or Zhao's mutual energy theorem.

Some property of time-reversal wave

Poynting theorem of the time-reversal wave

Apply time-reversal transform to the Poynting theorem,

$-\mathrm{\nabla }\cdot (\mathbf{𝐄}\times \mathbf{𝐇})=\frac{\partial U}{\partial t}+\mathbf{𝐉}\cdot \mathbf{𝐄}$

where

$U=\frac{1}{2}({ϵ}_0\mathbf{𝐄}\cdot \mathbf{𝐄}+\frac{1}{{\mu }_0}\mathbf{𝐁}\cdot \mathbf{𝐁})$

we can obtain the Poynting theorem for time-reversal wave, which is in the following:

$\mathrm{\nabla }\cdot (\mathbf{𝐞}\times \mathbf{𝐡})=+\frac{\partial u}{\partial t}+\mathbf{𝐣}\cdot \mathbf{𝐞}$

$u=\frac{1}{2}({ϵ}_0\mathbf{𝐞}\cdot \mathbf{𝐞}+\frac{1}{{\mu }_0}\mathbf{𝐛}\cdot \mathbf{𝐛})$

Time-reversal wave can balance out or cancel the energy of the normal wave

Here, the normal waves are the retarded wave and the advanced wave which satisfy Maxwell equations. The normal wave, for example, the retarded wave, can be canceled or balanced out by the time-reversal wave corresponding to the retarded wave. Similarly, the advanced wave can be canceled or balanced out by the time-reversal wave corresponding to the advanced wave.

If the time-reversal wave exists, it can balance out the normal electromagnetic fields, which means it is possible that

$-\mathrm{\nabla }\cdot (\mathbf{𝐄}\times \mathbf{𝐇})+\mathrm{\nabla }\cdot (\mathbf{𝐞}\times \mathbf{𝐡})=0$

$\frac{\partial U}{\partial t}+\frac{\partial u}{\partial t}=0$

$\mathbf{𝐉}\cdot \mathbf{𝐄}+\mathbf{𝐣}\cdot \mathbf{𝐞}=0$

Hence, if a time-reversal wave exists, and if they exist together with the normal waves, that will make the normal wave not carry energy.

In quantum physics, the waves are probability waves. A probability wave cannot carry energy. In order to make the wave not carry energy, the time-reversal wave is required.

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