# If two events separated by a spacetime interval of zero, can they both be said to be happening "now"?

Can the concept of c be validly expressed as "the rate at which an event propagates through space"?

There was a television program last year featuring Prof Brian Cox. The presenter asked him "Is it true that when we look at this particular star, we are seeing it as it was 18 million years ago?" And Professor Cox replied "Or 'now'." This was a turning point in my comprehension of relativity and I want to make sure I understood it correctly. Was his point that with simultaneity and time not being absolute, the idea of "now" is in effect meaningless?

This method of expression makes sense to me in that it explains length contraction and thought experiments such as the barn / ladder paradox: from the POV of the front of the barn, the event of the rear of the ladder crossing the threshold of the rear barn door hasn't reached it yet, while from the rear of the barn the same is true for the front of the ladder.

Is this a valid analogy, or is it over-simplistic and will lead to failure of comprehension elsewhere?

• events don't propogate because events are points in space-time. And in special relativity, in order for two events to be simultaneous in some frame, they have to be space-like separated. Thus if the spacetime interval separating two points is zero, they are lightlike separated and so there is no frame where they are simultaneous. Jul 22 '14 at 22:17
• It is the ability to know about events that propagates at no more than c. Jul 22 '14 at 23:01
• It seems to me that Brian Cox goes out of his way to mystify rather than clarify physics. I switch channels (or turn the TV off) when I see that he's the narrator of some program I thought I had wanted to watch. Jul 23 '14 at 10:59
• I've obviously misinterpreted a part of what he said (all 2 words of it). Can you think what he might have meant by the comment? Jul 23 '14 at 13:13

If two events both have a spacetime interval of zero, can they both be said to be happening “now”?

There is an interval associated with any two events but there is not an interval associated with an event.

From the Wikipedia article "Spacetime":

In spacetime, the separation between two events is measured by the invariant interval between the two events, which takes into account not only the spatial separation between the events, but also their temporal separation. The interval, $s^2$ between two events is defined as

$$s^2 = \Delta r^2 - c^2\Delta t^2$$

(for the remainder, we assume flat spacetime...)

Now, if two events have spacetime interval of zero then, in any inertial frame of reference, their separation in time is the same as their separation as space:

$$c^2\Delta t^2 = \Delta r^2$$

If two events are simultaneous in a reference frame, $\Delta t = 0$ in that reference frame.

So, if $\Delta t \ne 0$ in a reference frame, the two events are not simultaneous in that reference frame.

Thus, if two events with spacetime interval of zero are simultaneous in a reference frame, they must be co-located in that reference frame, i.e., $\Delta r = 0$.

To summarize, if two events have spacetime interval of zero and, in your reference frame, the events are spatially separated, if one event is "now", the other is not "now".

• "there is not an interval associated with an event". If there is no interval, there is no event. Nothing can happen in zero time. Jul 23 '14 at 12:51
• You are correct, the question was poorly worded and I've amended accordingly. Jul 24 '14 at 12:21

I'm not sure this is a complete answer to your question, but thinking about special relativity that way will get you into trouble.

Essentially, that way of interpreting special relativity attributes all of its weirdness to signal delay. Here's how I think you're interpreting the barn door experiment: The ladder is put stationary in the barn and is found to be the same length. The same ladder is then run through the barn at relativistic speed. Now, it takes a little extra time for the image of the back of the ladder crossing the back door to reach a person standing at the front door, and so the person at the front only sees the ladder crossing the back door when the front of the ladder is poking a little beyond the front door. Conclusion: the ladder is longer than the barn, even though the person moving with the ladder thinks it's exactly the length of the barn - the moving person's meter stick must have gotten longer.

The trouble is, that's backwards. Meter sticks get shorter when they move along the direction they measure. We got to this point assuming that the signal delay time of light was causing the problem, that we are forced to define "now" by what we see now. But that hardly seems satisfactory, since we can, in principle, factor in the finite light transmission speed in our calculations. What's interesting about special relativity is that all of the measurement discrepancies of special relativity persist, even if we know exactly how to factor in the time light took to get to us.

I won't try to explain length contraction, time dilation, etc. - I'd do a really bad job when compared to the many fantastic treatments of the subject out there. To be sure, though, it's much weirder and more interesting than signal delay.