Can We Make Sense of Dissipation without Causality?

Relativity opens the door to a counterintuitive fact: A state can be stable to perturbations in one frame of reference and unstable in another one. For this reason, the job of testing the stability of states that are not Lorentz invariant can be very cumbersome. We show that two observers can disagree on whether a state is stable or unstable only if the perturbations can exit the light cone. Furthermore, we show that, if a perturbation exits the light cone and its intensity changes with time due to dissipation, then there are always two observers that disagree on the stability of the state. Hence, “stability” is a Lorentz-invariant property of dissipative theories if and only if the principle of causality is respected. We present 14 applications to physical problems from all areas of relativistic physics ranging from theory to simulation.

Popular Summary

Many dissipative processes in physics are currently modeled using equations that allow for faster-than-light communication. The most cited example is the heat equation, which permits energy transport at infinite speed. Equations of this kind are very popular also in relativistic settings (especially in relativistic astrophysics) and are employed in theoretical models and numerical simulations. Should we be worried? In this work, we find that the answer is probably yes.

We show that, when a system of partial differential equations violates causality, different observers disagree on whether the system tends to evolve toward thermodynamic equilibrium or away from it. In other words, acausal dissipative systems that are stable in one reference frame always end up being unstable in another. In causal theories, on the other hand, the equilibrium state is either stable or unstable in all reference frames at the same time.

One implication of this paper is that in relativity, we can never rely on acausal dissipative field equations, even when our intuition may suggest that faster-than-light signals may be “negligible.” They are not. There is always a reference frame in which causality violations produce infinitely fast instabilities that destroy the system in a fraction of a nanosecond. However, there is also some good news: If causality holds, stability (or instability) is a Lorentz-invariant property of the field equations. Hence, there is no need to assess the stability of a theory in reference frames in which the background is moving. One can always work in the rest frame of the medium.

Figure 1

Geometric visualization of the principle of causality. Take an arbitrary spacelike Cauchy 3D surface $\mathrm{\Sigma }$. The simplest example of such a surface is the hyperplane $\{t=0\}$. Divide $\mathrm{\Sigma }$ into two regions: $\mathcal{R}$ (dark blue) and ${\mathcal{R}}^c$ (light blue). “Paint in red” all the timelike and lightlike curves that originate from $\mathcal{R}$ and propagate toward the future. The red paint will cover a set $J^{+}(\mathcal{R})$ called the “domain of influence of $\mathcal{R}$” or “causal future of $\mathcal{R}$” or “future light cone of $\mathcal{R}$.” Causality demands the following: If we compare two arbitrary solutions (of the field equations) whose initial data differ on $\mathcal{R}$ but coincide on ${\mathcal{R}}^c$, such solutions can differ only inside $J^{+}(\mathcal{R})$. This is equivalent to saying that information coming from $\mathcal{R}$ can never exit $J^{+}(\mathcal{R})$.

Figure 2

Minkowski diagrams of the argument outlined in Sec. 3a. Reference frame of Alice (left panel): the perturbation moves superluminally from the left to the right, and its intensity decreases with time as a result of dissipation. Reference frame of Bob (right panel): the perturbation moves from the right to the left, and its intensity grows with time. The two points of view are connected by a Lorentz boost. The shades of red are a color map of the intensity of the perturbation (red large, white small); the arrows have the orientation induced by $\phi$ (see the main text); the blue dashed lines are the light cone.

Figure 3

Left panel: Minkowski diagram of a subluminal perturbation (in Alice’s coordinates). The blue segment is $\mathcal{R}$, where the perturbation is initially located; the black line is ${\mathcal{R}}^c$, where the perturbation is absent. Together, $\mathcal{R}$ and ${\mathcal{R}}^c$ constitute the initial-data hypersurface $\mathrm{\Sigma }$. The shaded red region is $J^{+}(\mathcal{R})$, where $\phi$ can propagate. The white region above $\mathrm{\Sigma }$ is ${\mathcal{D}}^{+}({\mathcal{R}}^c)$, where $\phi =0$. The shades of red are a color map of $\phi$ (red large, white small). The hyperplanes at constants $t_A$ and $t_B$ are, respectively, the horizontal and oblique lines. Right panel: visualization of the sets $\mathcal{C}$ and $\tilde{\mathcal{C}}$ constructed in the proof of Theorem 1.

Figure 4

Left panel: an observer inside the red region ${\mathcal{D}}^{+}(\mathcal{R})$ cannot know what the initial state of the system was for $x_A<0$ (at $t_A=0$). Right panel: therefore, both $\phi (t_A=0)$ and ${\phi }^{\star }(t_A=0)=\mathrm{\Theta }(x_A)\phi (t_A=0)$ are initial states which are consistent with the data available to such observer; no experiment performed inside ${\mathcal{D}}^{+}(\mathcal{R})$ can tell $\phi$ and ${\phi }^{\star }$ apart.

Figure 5

Minkowski diagram of the two solutions $\phi$ (left panel) and ${\phi }^{\star }$ (right panel) in Bob’s coordinates. The shades of red are a color map of the perturbation (the oscillatory behavior of the periodic part is averaged out). In the gray area, we do not know the actual intensity of ${\phi }^{\star }$. Left panel: $\phi$ is an unstable Fourier mode (i.e., a growing plane wave) in the $B$ frame; it is well defined across the whole spacetime; its oscillation amplitude is constant along hyperplanes $t_B=\text{const}$ (horizontal lines) and grows exponentially for growing $t_B$ ($\phi \propto e^{{\mathrm{\Gamma }}_B{t}_B}$). Right panel: ${\phi }^{\star }$ is constructed on the half spacetime $\{t_A\ge 0\}$ by “gluing” initial data at $t_A=0$. On the right (on $\mathcal{R}$), we take ${\phi }^{\star }(t_A=0)=\phi (t_A=0)$ so that (by causality) ${\phi }^{\star }=\phi$ on ${\mathcal{D}}^{+}(\mathcal{R})$. On the left, we set ${\phi }^{\star }(t_A=0)=0$ (hence, ${\phi }^{\star }=0$ on the respective Cauchy development). In this way, ${\phi }^{\star }$ has a well-defined Fourier transform on $\{t_A=0\}$, but it diverges on ${\mathcal{D}}^{+}(\mathcal{R})$ (in the upper right corner), signaling an instability in Alice’s frame.

Figure 6

Minkowski diagram of the “acoustic-cone argument.” An external source at $p$ (yellow star) generates a perturbation in the medium, which propagates within the outermost characteristic cone of the field equations (acoustic cone) and decays by dissipation as we move away from the source. If the field equations are acausal, the acoustic cone extends outside the light cone. Hence, there is an observer Bob in whose frame the perturbation exists before $p$ has occurred. On the region $\{t_B<t_B(p)\}$, Bob observes a solution of the sourceless field equations, which is at equilibrium for $t_B\ll t_B(p)$ but grows spontaneously as $t_B$ approaches $t_B(p)$ from below.

Figure 7

Boosted Green’s function of the heat equation for $t_B<0$. We set $D=30$ and $v=3/4$. Each curve represents a snapshot of $T(t_B,x_B)$, for different choices of $t_B$. If someone knows the entire history of the system, the interpretation of this figure is quite straightforward: Alice injects a spike of energy at $t_B=x_B=0$; because of acausality, a portion of such a spike propagates toward the past; as it travels backward in time, the spike diffuses and flattens. On the other hand, to Bob (who cannot predict the decisions of Alice), the situation looks very different. From his perspective, the material is initially in thermodynamic equilibrium (at $t_B=-\infty$). Then, a perturbation builds up spontaneously, developing a superluminal front on the characteristic line $x_B=-t_B/v$. As time goes ahead, the perturbation “antidiffuses,” becoming more and more peaked. Eventually, when $t_B\to 0$, the peak diverges at $x_B=0$. What we are observing is just an inversion of chronology (see Fig. 2).

Figure 8

The same plane wave viewed by two observers in motion with respect to each other. The full spacetime dependence of the temperature perturbation is assumed to be $\delta T(t_A,x_A)=e^{-t_A}\sin (10x_A)$. In the “($A$)” frame, we have a conventional Fourier mode (left panel), whose amplitude decays in time. In the “($B$)” frame, there is an exponential modulation also in space (right panel). This is a direct consequence of the relativity of simultaneity: If something decays only in time in one frame, it may decay both in time and space in another frame. We choose the boost velocity $v=3/4$, which corresponds to the Lorentz factor $\gamma \approx 1.5$.