# Can Time Unfold in the Wrong Direction?

The unfolding of time is typically described as a sequence of spatial regions: one region of space gets realized, and then another, and then another.

For example: consider a region of space in which Harold is crowned king. As events transpire, a number of further regions of space get realized. Finally, a region of space arrives in which Harold is slain by William. The unfolding of time here is described by a sequence spatial regions leading from Harold’s crowning to Harold’s death.

Although there does not seem to be any one-true-description of unfolding, it happens that intertial observers do agree about the order in which (time-like separated) events unfold.

In particular, in the weak-gravitational regime of Harold and William, spacetime is approximately Minkowski. So, William will identify one possible sequence of spatial regions leading up to the death of Harold. An astronaut moving away from William with velocity v = c/2 will identify a different sequence of spatial regions. However, William and the astronaut will agree about the order in which time-like separated events occur: first Harold is crowned, and then Harold is slain.

Figure 1: An observer on Earth and an astronaut traveling away with velocity c/2 will describe two different foliations of spacetime into space-like hypersurfaces of simultaneity.

On the other hand, an accelerating observer will not generally agree about the order in which these events unfold.

Imagine that at the moment of Harold’s death, there is a second astronaut at rest relative to William, who begins to accelerate away at a constant rate. As the astronaut accelerates ever closer to the speed of light, her simultaneity hypersurfaces (the spatial regions that she uses to foliate spacetime) will tilt ever closer to 45 degrees, as dictated by the geometry of Minkowski spacetime. All these surfaces will intersect on some 2D surface I. Moreover, on the other side of the surface of I, the astronaut will judge the order of events to be the reverse of what William judges: first Harold is slain by William, and then Harold is crowned king. The astronaut will describe some events as unfolding in the wrong direction.

Figure 2: The simultaneity hypersurfaces of an accelerating astronaut. These surfaces tilt as the astronaut accelerates, so that events to the left of I are judged to unfold in the wrong direction.

So, the order in which time unfolds is not a fact that all observers agree about.

According to one common definition of objectivity, a claim is objectively true or false if and only if all observers agree about that claim. But in our example, the claim, ‘time unfolds from Harold’s crowning towards Harold’s death’ can only be valuated according to the subjective judgement of one observer. Other observers, such as the accelerating astronaut, are equally correct to valuate the claim differently.

Thus, in this sense, there is no objective fact of the matter as to the order in which time unfolds.

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## 11 thoughts on “Can Time Unfold in the Wrong Direction?”

1. Bryan

Well, I’m arguing there is no objective right or wrong direction for unfolding. So how about: Yes, time can unfold in the ‘wrong’ direction as much as it can unfold in the ‘right’ direction.

2. Justin

Interesting. I would have thought that you wanted to answer: No, time can no more unfold in the ‘wrong’ direction than it can unfold in the ‘right’ direction.

3. Justin

Of the many things I love about you, Jonathan, that you almost always spike the ball when set up is perhaps at the forefront!

4. wolfgang

Bryan,

notice that the accelerated observer sees the events in the same sequence as any other observer.
In other words, your definition (and it is a definition) of simultaneous is just not very helpful for the accelerated observer.

5. wolfgang

One more thing, your accelerated observer uses a coordinate system with a coordinate singularity (the same event has two different coordinates)…

6. Bryan

Thanks for the comments Wolfgang. Your observations are interesting and correct. First: If a signal is sent to the astronaut from the event of Harold’s crowning and the event of his death, then a) the astronaut won’t ever receive the signals if he keeps accelerating, because they’ll stay behind his event horizon; and b) if the astronaut stops accelerating, then he will receive the signals in the correct order.

So the Universe does ‘censor’ the bad judgments my astronaut will make using the standard simultaneity relation in a screwy reference frame.

Second: The accelerated observer’s description of the order events does match the Rindler coordinate system, and so is singular at the origin. But should we really *demand* a non-singular coordinate description? After all, many metrics (e.g. Schwarzschild) provide singular descriptions of spacetime, but remain perfectly reasonable in regions *outside* the singularity.

7. wolfgang

>> should we really *demand* a non-singular coordinate description?

Not necessarily. It just does not make always sense to use them beyond the singularity (like in your example).

8. Anonymous

Sorry, but this example makes no sense to me. What you are doing is to create a coordinate system that you then use from two separate perspectives, finding them to clash, when those happenings can’t meet for any of the observers.

If there was a real possibility for you to see ‘times arrow’ count backwards, as you seem to assume here? Then ‘time reversibility’ should be possible in all accelerations. You can use gravity to show that this isn’t the case.

Although all mass, according to the equivalence principle is ‘gravitationally constantly uniformly accelerating’ there is no such thing in SpaceTime as you observing any other ‘clock’, as defined by gravity, to ‘tick’ backwards relative your clock. So either you, or the equivalence principle, should be wrong here.

Yoron.