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My question

Why can't effects propagate backwards in time, within the backwards light cone of a cause? For example, when I turn on a flashlight, why doesn't the light travel backwards in time just like it does forwards in time? I don't see why this is prohibited by the laws of physics.

I have a feeling this question actually has a very simple answer, and I'm just overthinking it.


Background

Causality, in one specific form which I will detail below, seems to appear in every theory as an assumption.

  • In classical Newtonian physics, an effect cannot occur at a time earlier than its cause, e.g. particles propagate forward in time, fields satisfying equation of motion propagate forward in time, etc. Solutions which propagate backwards in time are artificially "thrown out" because they violate causality.

  • In classical relativity, we must make the distinction between causality within and outside the light-cone. Absent tachyonic degrees of freedom, it is impossible for an effect and cause to be space-like separated, which would otherwise imply a possible violation of causality. However, nothing a piori says that an effect cannot propagate backwards in time within the backwards light cone. Just as in classical physics, we make an assumption: an effect must reside on or within the future light-cone of its cause, e.g. the tangent vector for a particle trajectory is always within the future light-cone, field configurations propagate forward in time from (spacelike) Cauchy surface, etc. So once again, we simply "throw out" solutions which travel backwards in time.

  • In quantum field theory (in a flat background), the "first" type of causality is encoded in the fourth Wightman axiom - namely that $\langle\left[\phi_i(x)\phi_j(y)\right]_{\pm}\rangle=0$ if $x-y$ is a space-like separation, where $[,]_{\pm}$ indicates the (anti-commutator)commutator for (fermion)boson fields. It is also convenient to know that imaginary-mass tachyonic particles (as they might exist in classical relativity) cannot exist due to tachyonic condensation. However, what about if $x-y$ is not space-like separated? Can an effect propagate backwards in time, within the backwards light cone? ("second" type of causality) This seems to be an assumption going into the analyticity of the S-matrix — see answers to S-matrix analyticity and causality.

  • I don't know string theory well, but it seems like it will not provide anything new for causality compared to relativity and QFT, especially if the target space has a unique timelike direction (Killing vector), as in flat Minkowski space (in $d=10+1$). I have read the following PO thread: How is causality encoded in string theory?. It seems to me that no satisfactory answer was given for why effects cannot propagate backwards in time within the light-cone.

Maybe the reason is because such violations of causality would be self-inconsistent and hence couldn't possibly exist, e.g. via something like the Novikov self-consistency principle.

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Why can't effects propagate backwards in time, within the backwards light cone of a cause? For example, when I turn on a flashlight, why doesn't the light travel backwards in time just like it does forwards in time? I don't see why this is prohibited by the laws of physics.

The raw differential equations describing, e.g. the propagation of light are time reversal symmetric. If you provide sufficient data about the fields at $t = 0$, then you can compute what the fields are at $t > 0$, but you can also compute what the fields are at $t < 0$. Whether the data at $t = 0$ "causes" the field values at $t > 0$, or at $t < 0$, is not embedded in the equations, but rather is a verbal description we use to help us understand what the equations say. It's like asking if $F=ma$ means $F$ "causes" $a$, or $a$ "causes" $F$, or if Gauss's law means charge "causes" flux or flux "causes" charge. The equations are just relations.

In the case of a flashlight, you have something that drives the electromagnetic field. So if you turn it on for a moment at $t = 0$, you effectively say something about how the fields change at $t = 0$. If you fix the fields at $t < 0$ (say, demanding that they're zero), then you can use this information to calculate the fields at $t > 0$. But it works in reverse too: if you fix the fields at $t > 0$ (say, demanding that they're zero), then you can compute the fields at $t < 0$.

The first situation looks like beginning with darkness, and having the flashlight emit light. The second situation looks like beginning with light, aimed in such a way so that it all lands inside the flashlight at $t = 0$ and gets absorbed, with darkness thereafter. Both are valid solutions to the equations. The asymmetry is that it's easy to set up the first situation but it's almost impossible to set up the second, because of the second law of thermodynamics.

It's not really any more complicated for your other examples. For example, the S-matrix maps initial states to final states, but you could just invert it to get a map from final states to initial states. We talk about the former because the initial state is what is under our control in reality. There's no easy way to fix the final state.

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  • $\begingroup$ Thanks, this is simple and I understand my confusion now. If we provide enough information about a system at $t=0$, the state of that system is determined for all $t\neq 0$, both for $t>0$ and $t<0$. If you ignore certain parts of that system (e.g. electrical circuitry in the flashlight, the complex dynamics of the human that turned it on, etc.), then in order to get an accurate description of what you didn't ignore (e.g. the EM field) you will need to impose "ad-hoc" constraints (e.g. $A^{\mu}(\vec x,t<0)=0$ ) that account for the contributions from the parts that you did ignore. $\endgroup$ – Arturo don Juan Oct 12 '20 at 6:53
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From a classical relativity point of view, there are at least a couple different ways of looking at time. See What is time, does it flow, and if so what defines its direction?

For example, this answer says that time does not flow. The universe is a static block of events that just exist.

On the other hand, my answer says that while the Block Universe works, so does the flow of time. We don't know why time flows forward. It just does.

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Because of the second law of thermodynamics. The second law of thermodynamics says that the entropy of the Universe must keep increasing, and if we reverse the direction of time we would be travelling in the direction where the entropy of the Universe would decrease rather than increase. I believe this is the fundamental reason for the direction of the flow of time.

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  • $\begingroup$ Yes, much of what people think of as the "forward arrow of time" is explained by the second law of thermodynamics, but the second law of thermodynamics is an emergent statistical and/or macroscopic phenomenon which doesn't actually have anything to do with the question I am asking (about the microscopic nature of nature/physics). $\endgroup$ – Arturo don Juan Oct 11 '20 at 21:36
  • $\begingroup$ @ArturodonJuan Thermodynamics is valid at all scales, and it is what defines the arrow of time. $\endgroup$ – SK Dash Oct 12 '20 at 0:23
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    $\begingroup$ I understand my confusion now, I was under the impression that there was a microscopically preferred direction of time, but now I see that that isn't true. However to respond directly to your comment, yes and no. Yes, I will now agree that the arrow of time is an anthropic concept emerging from increasing entropy (i.e. thermodynamics). But no, thermodynamics is an emergent phenomenon from the collective behavior of many constituents. At the root of thermodynamics is statistical mechanics. Even more, entropy is a bit of an anthropic concept. $\endgroup$ – Arturo don Juan Oct 12 '20 at 7:02

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