Planes of Simultaneity

As discussed on the previous page, events that are outside each other’s light cones—i.e., events that have spacelike separation—cannot affect each other. That’s probably a good thing, because there is no absolute fact of the matter which of the two events occurred first! Whether the one event occurs before the other, or vice versa, depends on the reference frame. Similarly, there is no fact of the matter whether two spacelike-separated events occur at the same time. Whether the events are regarded as simultaneous depends on the observer’s frame of reference.

To visualize how this works, let’s return again to the example of the alien cow abduction, and draw a spacetime diagram from the farmer’s perspective. Since the farmer considers himself to be at rest, his own worldline is vertical. If he also regards his own location as the center of the horizontal space axis, his worldline will be exactly the same vertical line as the time axis in his diagram. Horizontal lines (or flat planes, if we are making a 3D spacetime diagram) represent sets of events the farmer regards as simultaneous. These are called planes of simultaneity. In real-life spacetime, planes of simultaneity are actually hyperplanes: three-dimensional “slices” of four-dimensional spacetime. In a three-dimensional spacetime diagram (like the one shown to the right), planes of simultaneity are represented as two-dimensional planes; and in a two-dimensional diagram like the ones shown below, they are represented simply as one-dimensional lines.

Now, let’s consider how to represent the alien’s time axis and planes of simultaneity within the farmer’s spacetime diagram. If the alien considers herself to be at rest in the center of her own space axis, then her time axis will be the same as her worldline, which is represented in the farmer’s spacetime diagram as a slanted line (since the alien spaceship is moving at high speed in his frame of reference).

What about her planes of simultaneity? In order to figure out which sets of events the alien considers to be simultaneous, we can use the Lorentz transformations—the equations Hendrik Lorentz formulated to describe length contraction and time dilation, which Einstein later incorporated into his own special theory of relativity. These equations imply that the alien’s planes of simultaneity should intersect her time axis at exactly twice the angle between her time axis and the light cone at the point of intersection. (See appendix E [link coming soon] for an explanation of the equations and how they imply this.) This is only true in spacetime diagrams where the speed of light corresponds to a 45-degree slope (i.e., a slope of 1). This simple way of visualizing planes of simultaneity is another benefit of adopting that slope convention. In other words, her planes of simultaneity are geometric reflections of her time axis over the light cone. For example, if the alien’s time axis (her worldline) is slanted at an angle of 55 degrees (10 degrees above the 45-degree slope of the light cone), then her planes of simultaneity will be slanted at 35 degrees (10 degrees below the light cone).

Now, consider two events A and B that are simultaneous from the farmer’s perspective, as shown in the figure below. Since the alien’s planes of simultaneity are slanted, she doesn’t regard A and B as simultaneous events. From her perspective, B occurs before A, because her planes of simultaneity slope upward to the right. Conversely, the alien regards A and C as happening simultaneously; but in the farmer’s reference frame A happens before C.

On the other hand, imagine what would happen if the alien were moving the opposite direction. In that case, her planes of simultaneity would slope the other way (upward to the left), and she would regard A as earlier than B. So in some reference frames, event A happens before B; in other frames, B happens before A; and in yet another frame (the farmer’s), A and B are simultaneous. According to special relativity, there is no absolute fact of the matter whether events A and B are simultaneous or whether one occurred before the other. Since A and B have spacelike separation (i.e., their spacetime interval is positive), the order in which they occur is just a matter of perspective.

In contrast, the temporal order of events with timelike or lightlike separation is absolute—it does not depend on the observer’s reference frame. In other words, when two events are within each other’s light cones, there is an objective fact of the matter which event occurs first. To see why, consider two events D and E that have timelike separation: E occurs within the future light cone of D, and D is inside the past light cone of E. (See figure below.) The farmer and the alien both agree that D happened before E. No matter how fast the alien goes—even if her spaceship is going nearly the speed of light—her planes of simultaneity won’t be slanted enough to make E occur before D. Since her spaceship can’t go faster than the speed of light, the slope of her worldline (in the farmer’s spacetime diagram) will always be at least slightly above 45 degrees, and therefore her planes of simultaneity must be at least slightly below 45 degrees. That’s why all observers agree about the order of events with timelike or lightlike separation.

But suppose, for the sake of argument, that the alien spaceship could travel faster than light. While going faster than light, the alien’s time axis (her worldline) would be angled less than 45 degrees from horizontal, and her planes of simultaneity would be tilted higher than 45 degrees. In that case, she might regard E as happening before D. But E is in the future light cone of D, which means that causal processes can go from D to E: what happens at D can affect what happens at E. Thus, if the alien were moving faster than light, she would see effects happen before their causes! Time would run backwards, from her perspective.

Is it really possible to go faster than light, and thus travel back in time? Probably not, for the reasons mentioned on the previous page: no physical object can travel faster than light, so far as we know. There are some possible exceptions in general relativity and quantum mechanics, as we’ll see later. But there is a way in which it would be possible to time-travel into the future, as we’ll see on the next page.