Distinguishing spacelike and timelike events

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In summary, the authors are saying that two events that are simultaneous in one frame of reference may not be simultaneous in another frame of reference.
  • #1
amiras
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I am having difficulties understanding some lines in the book.

"An event can be later than another spacelike separated event in one inertial frame and earlier in another."

So for example let's take the thought experiment where lighting strikes two sides of the train, and observers in different frames of reference do not agree which side of the train was stroked first.

So events of the lighting strikes are spacelike separated?

But it is also written: "Information can be received at P (point) only from events inside or on its past light cone, but not from events outside it."

But spacelike separated events do not lie in the light cone. I find somehow contradictory.
 
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  • #2
Hi amiras! Just because two events are simultaneous in one frame, doesn't mean they will be simultaneous in another as you noted. However this will not deter the potentially space-like nature of the separation of the events since the space-time interval is a Lorentz invariant.

For example let's say we have an observer ##O## standing equidistant from two ends of a train of length ##2L## and that this train is moving with some velocity ##v## in the ##+x## direction relative to an observer ##O'## standing on the sidetracks. Now imagine that at the moment ##O## passes ##O'## (which is the event that both observers label as the origin of their coordinate systems), ##O## sends out a light pulse in both directions. In ##O##'s frame, the pulses coincide with the ends of the train at events ##P_1 = (t,-L)## and ##p_2 = (t,L)## so they are simultaneous in ##O##'s frame. The space-time interval between these two events in ##O##'s frame is given by ##\Delta s^{2} = (2L)^{2} > 0## i.e. it is space-like.

Now let's boost to the frame of ##O'##. Now for the event corresponding to the light pulse hitting the back end of the train we have ##t'_1 = \gamma(t - \frac{vL}{c^{2}})## and for the front we have ##t'_2 = \gamma(t + \frac{vL}{c^{2}})##. In other words ##t'_2 = \gamma(t + \frac{vL}{c^{2}}), t'_1 = \gamma(t - \frac{vL}{c^{2}}), t'_1 - t'_2 = -\frac{2vL}{c^{2}}## meaning in the frame of ##O'## the light pulse hitting the back does so before the light pulse hitting the front. This is intuitively obvious because the constancy of the speed of light in all inertial frames will imply that ##O'## sees the back light pulse travel less to hit the end of the train since the end is "catching up" with this pulse whereas the front of the train is "moving away" from the forward light pulse. However as noted above, the space-time interval is a Lorentz invariant so boosting to another frame won't change the value of ##\Delta s^{2}## meaning we will still have ##\Delta s^{2} > 0## in the frame of ##O'## (you can verify this yourself if you wish). So events do not need to be simultaneous to be space-like separated is what they are saying.

All the lightcone is saying is that you cannot send a signal from the event ##p_1 = (t,-L)## that will be received at event ##p_2 = (t,L)## (this would require the signal to travel instantaneously). In other words the light cone of ##p_1## does not contain ##p_2##. Even though the events don't look simultaneous in the frame of the observer standing on the side-tracks, they are still space-like separated as mentioned above and as such the same geometric meaning of the light cone carries over.
 
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  • #3
Of course! Thank you for the answer, it is much clearer to me.

EDIT: Sorry, I somehow managed to add word "not" into a sentence. Every time I read what you wrote, it gets more and more clear to me (maybe because you edit and add something new every time :))

"light cone of p1 does not contain p2" - I somehow wrongly assumed that they are both contained, but that would not make sense. The way I understand it, is that future light cone contains the possible events including p1 and p2, but the timeline can only pass only through one of them (p1 or p2).
 
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  • #4
It is still not clearer? What else needs clarifying, please feel free to ask my friend :)!
 
  • #5


The concept of spacelike and timelike events is an important one in understanding the fundamental principles of spacetime in physics. To understand this concept, we must first understand the concept of spacetime itself. In the theory of relativity, spacetime is considered as a four-dimensional continuum, with three dimensions of space and one dimension of time. This means that events, or occurrences in the universe, are described by their position in space and time.

Now, a spacelike event is one that is separated from another event by a distance in space that is greater than the distance in time. In other words, the time interval between the two events is shorter than the spatial distance between them. On the other hand, a timelike event is one that is separated from another event by a distance in time that is greater than the distance in space. In this case, the spatial distance between the two events is shorter than the time interval between them.

In the example you provided, the lightning strikes on both sides of the train are spacelike separated events. This means that the time interval between the two strikes is shorter than the spatial distance between them. In one frame of reference, the observer on the train may see the strike on one side first, while in another frame of reference, the observer on the ground may see the strike on the other side first. This is because the two events are separated by a distance in space that is greater than the distance in time.

Now, in regards to the statement about information being received from events only inside or on the past light cone, this is referring to the concept of causality. The past light cone of an event is the region of spacetime from which light emitted from that event can reach a given point in spacetime. This means that any information or influence from an event can only reach a point if it is within or on the past light cone of that event.

In the example of the lightning strikes, the information about the strike on one side of the train can only reach a point if it is within or on the past light cone of that event. This is because the event is spacelike separated from the other strike, meaning the information cannot travel faster than the speed of light between the two events. This is not contradictory, as the concept of the light cone is specific to the speed of light and does not apply to spacelike separated events.

In summary, the concept of spacelike and tim
 

1. What is the difference between spacelike and timelike events?

Spacelike events refer to events that are separated by a distance in space, while timelike events are separated by a duration in time. This means that spacelike events occur in different locations at the same time, while timelike events occur at different times in the same location.

2. How can we distinguish between spacelike and timelike events?

One way to distinguish between spacelike and timelike events is by calculating the spacetime interval between them. If the interval is positive, the events are spacelike and if it is negative, they are timelike. Another way is by using the Lorentz transformation equations, which take into account the effects of time dilation and length contraction.

3. Can spacelike and timelike events occur simultaneously?

No, spacelike and timelike events cannot occur simultaneously. This is because in order for two events to occur at the same time, they must have a spacetime interval of zero. Since spacelike and timelike events have positive and negative intervals, respectively, they cannot occur at the same time.

4. How do spacelike and timelike events affect causality?

Spacelike events do not have a causal relationship, meaning that one event cannot influence the other. This is because there is no way for information to travel between the events faster than the speed of light. Timelike events, on the other hand, can have a causal relationship as information can travel between them at or below the speed of light.

5. Are there any real-life examples of spacelike and timelike events?

Yes, there are many examples of spacelike and timelike events in everyday life. An example of a spacelike event would be two people on opposite sides of the world who experience a sunrise at the same time. An example of a timelike event would be two people experiencing the same event, such as a concert, at different times due to time zone differences.

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