Rindler transformations are highly non-linear and thus do not belong to the certain class of non-linear transformations discussed in [13]. However, we will see below that the embedding techniques developed in [13] can be extended to include this case. In order to see this, we consider now a basic dynamics governed by the equation \(i\partial_{t} \psi = -i\partial_{x}\psi \). This is a \(1+1\) Dirac equation for a massless particle where, for simplicity, we have traced out the internal degrees of freedom. Let us split the wave function *ψ* and an arbitrary operator *θ* as:

$$\begin{aligned} \begin{aligned} &\psi (x,t)= \frac{1}{2} \bigl\{ \bigl[ \psi (x,t) + \psi (\chi , \tau ) \bigr] + \bigl[ \psi (x,t) - \psi (\chi , \tau ) \bigr] \bigr\} , \\ &\theta (x,t)= \frac{1}{2} \bigl\{ \bigl[ \theta (x,t) + \theta ( \chi , \tau ) \bigr] + \bigl[ \theta (x,t) - \theta (\chi , \tau ) \bigr] \bigr\} . \end{aligned} \end{aligned}$$

(4)

Correspondingly, for the particular case \(\theta = \partial_{t,x}\), the time and spatial derivative operators are \(\partial_{t,x} = \frac{1}{2}[\partial_{t,x} + \partial_{\tau ,\chi }] + \frac{1}{2}[ \partial_{t,x}-\partial_{\tau ,\chi }]\). With these mappings, we can write the dynamical equation, \(i\partial_{t}\psi =-i\partial_{x} \psi \), in terms of its even \((e)\) and odd \((o)\) components as follows,

$$ i\bigl(\partial^{e}_{t} + \partial^{o}_{t}\bigr) \bigl(\psi^{e} + \psi^{o}\bigr) = -i\bigl( \partial^{e}_{x} + \partial^{o}_{x}\bigr) \bigl(\psi^{e} + \psi^{o}\bigr), $$

(5)

where \(\psi^{e, o} = \frac{1}{2} [ \psi (x,t) \pm \psi (\chi , \tau ) ]\), \(\partial_{t,x}^{e, o} = \frac{1}{2}[\partial_{t,x}\pm \partial_{\tau ,\chi }]\).

Now, we define a spinor \(\Psi (x, t)\) in the *enlarged space*, according to \(\Psi = (\psi^{e}, \psi^{o})^{T}\), where *T* is the transpose operation. The spinor Ψ is related to *ψ* through the expression \(\psi (x, t) = (1, 1)\Psi \). Moreover, since \(\psi (\chi ,\tau )=\psi^{e}-\psi^{o}\), then the spinor \(\Psi '\) in the enlarged space corresponding to \(\psi (\chi , \tau )\), is just \(\sigma_{z} \Psi \), i.e., \(\psi (\chi , \tau ) = (1, 1)\sigma_{z} \Psi (x, t)\). This means that a physical action like \(\sigma_{z}\), acting on the enlarged space, gives rise to a physically-forbidden action—an instantaneous Rindler transformation—on the wave function in the simulated space.

The dynamical equation for \(\Psi (x, t)\) can be obtained from Eq. (5) separating its even and odd components, giving rise to

$$ i\left ( \textstyle\begin{array}{@{}c@{\quad}c@{}} \partial_{t}^{e}& \partial_{t}^{o} \\ \partial_{t}^{o}& \partial_{t}^{e} \end{array}\displaystyle \right ) \left ( \textstyle\begin{array}{@{}c@{}} \psi^{e} \\ \psi^{o} \end{array}\displaystyle \right ) = -i \left ( \textstyle\begin{array}{@{}c@{\quad}c@{}} \partial_{x}^{e}& \partial_{x}^{o} \\ \partial_{x}^{o}& \partial_{x}^{e} \end{array}\displaystyle \right ) \left ( \textstyle\begin{array}{@{}c@{}} \psi^{e} \\ \psi^{o} \end{array}\displaystyle \right ) . $$

(6)

We can write \(\partial_{t, x}^{e, o}\) in terms of \(\partial_{t}\) and \(\partial_{x}\) as follows,

$$\begin{aligned} \partial^{e, o}_{t} =& \frac{1}{2} [ \partial_{t} \pm \partial _{\tau } ] \\ =& \frac{1}{2} \biggl[ \partial_{t} \pm \biggl(\sinh {\frac{ \tau }{\chi }}\partial_{x}+ \cosh {\frac{\tau }{\chi }}\partial_{t}\biggr) \biggr] \\ =& \frac{1}{2} \bigl[\partial_{t} \pm \bigl( \sqrt{a^{2}x^{2}-1}\,\partial _{x}+ax \partial_{t}\bigr) \bigr] , \end{aligned}$$

(7)

where in the last step we have used that \(\chi =1/a\), and also

$$\begin{aligned} \partial^{e, o}_{x}&=\frac{1}{2} [\partial_{x} \pm \partial _{\chi } ] \\ &= \frac{1}{2} \biggl[\partial_{x} \pm \biggl(\cosh {\frac{\tau }{\chi }}-\frac{\tau }{\chi }\sinh {\frac{\tau }{\chi }} \biggr)\partial _{x}+\biggl(\sinh {\frac{\tau }{\chi }}-\frac{\tau }{\chi }\cosh {\frac{\tau }{\chi }}\biggr)\partial_{t} \biggr] \\ &= \frac{1}{2} \biggl[\partial_{x} \pm \biggl(\biggl(ax- \sqrt{a^{2}x^{2}-1} \operatorname{arctanh}\biggl( \frac{\sqrt{a^{2}x^{2}-1}}{ax}\biggr)\biggr)\partial_{x} \\ &\quad {}+\biggl(\sqrt{a^{2}x^{2}-1}-a x \operatorname{arctanh}\biggl(\frac{\sqrt{a ^{2}x^{2}-1}}{ax}\biggr)\biggr)\partial_{t}\biggr) \biggr] . \end{aligned}$$

(8)

We can substitute these expressions in Eq. (6) in order to obtain a Schrödinger equation for Ψ. After some algebra, we can write it in this way:

$$ i\partial_{t} \Psi = -i \bigl[f(x) I + g(x) \sigma_{x} \bigr]\partial _{x}\Psi , $$

(9)

with

$$\begin{aligned} \begin{aligned} &f(x) =\frac{(ax+\sqrt{a^{2}x^{2}-1})(1-\frac{ \operatorname{arctanh}(\frac{\sqrt{a^{2}x^{2}-1}}{ax})}{2})}{ax+\sqrt{a ^{2}x^{2}-1}-ax\,\operatorname{arctanh}(\frac{\sqrt{a^{2}x^{2}-1}}{ax})}, \\ &g(x)=\frac{(\sqrt{a^{2}x^{2}-1}-ax)\operatorname{arctanh}(\frac{\sqrt{a ^{2}x^{2}-1}}{ax})}{2 (ax(1-\operatorname{arctanh}(\frac{\sqrt{a ^{2}x^{2}-1}}{ax}))+\sqrt{a^{2}x^{2}-1})}. \end{aligned} \end{aligned}$$

(10)

These equations are valid as long as the denominator is non-zero, that is, \(ax+\sqrt{a^{2}x^{2}-1}\neq ax\,\operatorname{arctanh}(\frac{\sqrt{a ^{2}x^{2}-1}}{ax})\). This is due to the fact that, in order to obtain Eq. (9), we need to manipulate a equation of the form \(iA\partial_{t}=-iB\partial_{x}\), where *A* and *B* are \(2\times 2\) matrices. Therefore, we need to invert *A*, which is only invertible if the above condition is met. Otherwise, the entries of *A* are all 1, and the dynamics cannot be described by a Dirac-like dynamics.

Using that \(ax=\cosh [\operatorname{arctanh}(v)]\) we can rewrite Eq. (10) in terms of the velocity boost only. In Fig. 1, we see the behavior of \(f(x)\) and \(g(x)\) ranging from \(a x=1\) (\(v=0\)) and \(a x=20\), (\(v=0.998749\simeq c\)).

We are now able to analyse two interesting regimes. We consider first the non-relativistic regime where \(v\simeq \phi \ll1\), and then \(ax\simeq 1+v^{2}/2\). By considering this limit in Eq. (10), we recover, as expected the embedded dynamics of a Galileo boost [13]:

$$\begin{aligned} i\partial_{t} \Psi = -i \biggl[\biggl(1+ \frac{v}{2}\biggr) I -\frac{v}{2} \sigma _{x} \biggr] \partial_{x}\Psi . \end{aligned}$$

(11)

Notice however that in this case *v* is spacetime dependent, unlike in standard Galileo boosts.

Now we consider the opposite regime, that is, an ultra relativistic observer \(v=1-\delta \), where \(0<\delta \ll 1\). In this case, \(ax=\cosh [\operatorname{arctanh}(1-\delta )]\). Expanding in *δ*, we obtain:

$$\begin{aligned} i\partial_{t} \Psi = -i \bigl[ \bigl(1+f(\delta ) \bigr) I -f(\delta ) \sigma_{x} \bigr]\partial_{x}\Psi, \end{aligned}$$

(12)

where

$$ f(\delta )=-\frac{\delta }{2(1+\frac{4}{\log [\frac{\delta }{2}]})}. $$

(13)

Note that is restricted to values of *δ* that are sufficiently far from \(\log [\frac{\delta }{2}]=-4\), which corresponds to \(ax+\sqrt{a ^{2}x^{2}-1}= ax\,\operatorname{arctanh}(\frac{ \sqrt{a^{2}x^{2}-1}}{ax})\), which is the singular point described above. Under this additional condition, we can write:

$$ f(\delta )\simeq -\frac{\delta }{2}\simeq -\frac{1}{4\,a^{2}\,t^{2}}. $$

(14)

Notice that in the limit \(\delta =0\) (\(v=c=1\)) we obtain the same trivial dynamics as in \(v=0\). This is because in the case \(v=c\), the Rindler transformation becomes a standard Lorentz time-independent boost, that is, the transformed reference frame is inertial. Accordingly, the coordinate transformation does not change the Lorentz-invariant Dirac dynamics in the simulated space.

In order to determine the dynamics associated with Eq. (9), one just has to define the initial condition for Ψ, i.e., \(\Psi (x,0) = \frac{1}{2} [ \psi (x,0) + \psi (\chi (x,0), \tau (x,0) ) , \psi (x,0) - \psi (\chi (x,0), \tau (x,0) ) ]^{T}\). Notice that in this case \(\psi (\chi (x,0), \tau (x,0) )=\psi (x,0)\) and therefore the dynamics is determined by the knowledge of the initial wave function only.

The techniques of [13] for relating observables in the enlarged and simulated spaces are also applicable here. We can obtain any expectation value of either the inertial or Rindler wave functions through observables in the enlarged space as follows:

$$\begin{aligned}& \langle O \rangle_{\psi (x, t)} =\langle \psi \vert O \vert \psi\rangle =\Biggl\langle \Psi \left\vert \left ( \textstyle\begin{array}{@{}c@{}} 1 \\ 1 \end{array}\displaystyle \right ) O( 1 \quad 1) \right\vert \Psi \Biggr\rangle \\& \hphantom{\langle O \rangle_{\psi (x, t)}}{}= \bigl\langle \Psi \bigl\vert (I + \sigma_{x})\otimes O \bigr\vert \Psi \bigr\rangle , \end{aligned}$$

(15)

$$\begin{aligned}& \langle O \rangle_{\psi (\chi , \tau )} = \bigl\langle \psi '\vert O \vert \psi ' \bigr\rangle = \Biggl\langle \Psi \left\vert \sigma_{z}\left ( \textstyle\begin{array}{@{}c@{}} 1 \\ 1 \end{array}\displaystyle \right ) O( 1 \quad 1) \sigma_{z}\right\vert \Psi \Biggr\rangle \\& \hphantom{\langle O \rangle_{\psi (x, t)}}= \bigl\langle \Psi \bigl\vert (I - \sigma_{x})\otimes O \bigr\vert \Psi \bigr\rangle , \end{aligned}$$

(16)

where we use \(\langle x|\psi \rangle = \psi (x, t)\), \(\langle \chi | \psi '\rangle = \psi (\chi , \tau )\), and \(\langle x|\Psi \rangle = \Psi (x, t)\). We are also able to analyse correlations between \(\psi (x, t)\) and \(\psi (\chi , \tau )\) out of the dynamics in the enlarged space only:

$$\begin{aligned} \langle O \rangle_{\psi (x, t), \psi (\chi , \tau )} = &\bigl\langle \psi \vert O \vert \psi '\bigr\rangle = \Biggl\langle \Psi \left\vert \left ( \textstyle\begin{array}{@{}c@{}} 1 \\ 1 \end{array}\displaystyle \right ) O( 1 \quad 1) \sigma_{z}\right\vert \Psi \Biggr\rangle \\ =& \bigl\langle \Psi \bigl\vert (\sigma_{z} - i \sigma_{y})\otimes O \bigr\vert \Psi \bigr\rangle . \end{aligned}$$

(17)

For instance, this would allow to reveal the existence of twin-paradox time dilation effects through quantum measurements [20] as well as the degradation of correlations between an inertial and an accelerated observer close to a black hole horizon [21].This is so because \(\psi (\chi ,\tau )\) would be the natural description of the spinor *ψ* under uniform acceleration. Thus, the experiments would be straightforward in a two-particle Dirac simulator. One particle would remain inertial—thus subject to the standard Dirac Hamiltonian- while the other would undergo a period of simulated uniform acceleration by means of the implementation of the Rindler transformation—in the black hole case—or a trajectory with several acceleration and deceleration steps—several Rindler transformations—and several inertial steps, in order to simulate a twin-paradox trajectory. Then, a comparison of the time coordinates of the two particles would allow to measure a simulated relativistic time dilation and the measurement of the two-particle correlations would allow to detect a degradation of correlations due to acceleration or gravity. This degradation might be linked with the black-hole information problem, since it suggests that quantum information cannot be a solution for the information loss. However, a deeper analysis of this problem would require the simulation of a more complete theory of quantum gravity. While quantum field theory phenomenology is out of reach in this single-particle experiment, one main advantage would be the possibility of simulating higher values of the acceleration.