# Timelike events

I've been thinking about the geometry of spacetime itself, and it has really been mind-blowing in some way. If space and time are so alike that we can treat them in a way that objects have a temporal extension as well as a spatial one, my question will be about this fact.

So let's suppose that we have a worldtube of an object that is extendended in bot space and time.
If we imagine the time component to be a line composed of timelike events that follow each other like on some sort of a historical timeline, does that mean that the endpoint of the cause (the previous event) is the beginning of the effect (the event that follows)?

For instance : If we imagine some event on the worldtube that lasts 4 second, and we consider it to be a length in time, which connects 2 points, will the event that follows, no matter how long it lasts, start from the endpoint of the event that I mentioned before. So is it valid to conclude that when saying 'cause comes before the effect' we mean that the effect starts in the same point where the cause ended. This question may be kind of metaphysical, but I'm also interested how would the geometry of space-time itself include the concept of change? Is the change, no matter what kind of physical phenomenon is in question, a point in time where the state of different properties swithces, or is the case there is more than one point in question, that two instants are sufficient for the change to occur? Of course, I'm seeking an answer from a geometrical perspective because I think everthing else is pretty much irrelevant when talking about spacetime, and I saw that many other forum members agree on this.

Regards, analyst

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WannabeNewton
Events do not have temporal duration. They indicate points in time not intervals of time. Causality in space-time is with regards to time-like intervals-there is no such thing as a "time-like event". If two events are time-like separated then they lie in the same (local) light cone and all (local) observers will agree on the relative temporal ordering of the two events. The forward progression of time is usually codified by the causal notion of being future-directed as applied to a smooth time-like vector field defined on the space-time; the vector field essentially acts as the forward "flow of time" (for space-times which do not possess such a vector field there is no well-defined i.e. no smoothly varying notion of the "flow of time"). These are the ways in which the metric, at the most basic level, encodes causality.

Events do not have temporal duration. They indicate points in time not intervals of time. Causality in space-time is with regards to time-like intervals-there is no such thing as a "time-like event". If two events are time-like separated then they lie in the same (local) light cone and all (local) observers will agree on the relative temporal ordering of the two events. The forward progression of time is usually codified by the causal notion of being future-directed as applied to a smooth time-like vector field defined on the space-time; the vector field essentially acts as the forward "flow of time" (for space-times which do not possess such a vector field there is no well-defined i.e. no smoothly varying notion of the "flow of time"). These are the ways in which the metric, at the most basic level, encodes causality.

I understand that, the basic point of my post was the notion of the time-like interval. So can it be said that when one timelike interval ends, that another one starts? I each interval is represented by a length than the following interval starts in the point where the first one ended and so on?

WannabeNewton
Sure.

Sure.
Ok, thanks. Glad to solve that out. So how does change occur, in a physical sense, regarding space-time? Is there a time point (or an event) which connects two time like intervals, for which can be said that it's a point of change in some properties of the body, or change includes different time points on the same timeline? I know it's not a strict physics question but I'm sure physics has an appropriate answer about this.

PeterDonis
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So let's suppose that we have a worldtube of an object that is extendended in bot space and time.
If we imagine the time component to be a line
If the world-tube is extended in space and time, then the "time component" can't be just a line. In fact, as you're using "time component" here, the "time component" is just the world tube itself. A single timelike line within the tube will be the worldline of one particular point within the object.

If we imagine some event on the worldtube that lasts 4 second
As WannabeNewton has pointed out, an event is a single point in spacetime; it doesn't have a duration in time or an extension in space. What you are imagining here is a segment of a timelike worldline that is 4 seconds long. The endpoints of the segment are events separated in time (along that segment) by 4 seconds.

will the event that follows, no matter how long it lasts, start from the endpoint of the event that I mentioned before.
Segments of worldlines can be contiguous, as you describe here, or they can be separate, or they can even overlap.

For example, consider the worldline of your wristwatch; it is a timelike line (if we consider your wristwatch to not have significant extension in space). Consider four events on this timelike line:

We can then label segments of your wristwatch's worldline by the events at their endpoints, e.g., segment AB, segment AC, segment BC, segment BD, etc. Then we can say, for example, that segments AB and BC are contiguous (the second starts where the first ends); segments AB and CD are separated (there is a finite gap between where the first ends and where the second starts); and segments AC and BD overlap (the second starts before the first ends). In other words, geometry along a timelike line works just like geometry along an ordinary line in Euclidean space. (Of course, things get more complicated when you consider spacetime geometry including both timelike and spacelike dimensions, because of the minus sign in the metric.)

So is it valid to conclude that when saying 'cause comes before the effect' we mean that the effect starts in the same point where the cause ended.
Not necessarily. Consider the example above: if some event in segment AB causes some event in segment CD, then cause and effect are not contiguous in time. However, the cause still comes before the effect; the relation of "before" and "after" along the timelike line is well-defined and doesn't depend on whether events or segments are contiguous.

In the case of full 4-D spacetime, "before" and "after" are defined by light cones: in other words, if we have some event C which is a cause, and some event E which is its effect, it must be possible for either a light signal or some timelike object to travel from C to E. We express this by saying that E is within C's future light cone, or C is within E's past light cone.

how would the geometry of space-time itself include the concept of change?
It doesn't. Spacetime as a geometric object just is; it doesn't change. We can model what we call "change" within spacetime by looking at timelike (or lightlike) lines and establishing "before" and "after" relationships along them, but as far as spacetime itself is concerned, all those relationships just exist; they don't change.

WannabeNewton
Glad to solve that out. So how does change occur, in a physical sense, regarding space-time? Is there a time point (or an event) which connects two time like intervals, for which can be said that it's a point of change in some properties of the body, or change includes different time points on the same timeline? I know it's not a strict physics question but I'm sure physics has an appropriate answer about this.
Sorry I honestly have no idea what you're asking.

PeterDonis
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So how does change occur, in a physical sense, regarding space-time?
It doesn't "occur", because nothing "occurs" in the spacetime viewpoint; spacetime just is.

Consider the example of your wristwatch again. We picked out four events on your wristwatch's worldline, A, B, C, and D. Your wristwatch had a different reading at each of those events. That fact alone means your wristwatch "changed" from event to event. In other words, in the spacetime viewpoint, "change" just means physical quantities, like your wristwatch's reading, are different at different events along a timelike (or lightlike) line. That's all it is. Nothing has to "occur".

It doesn't "occur", because nothing "occurs" in the spacetime viewpoint; spacetime just is.

Consider the example of your wristwatch again. We picked out four events on your wristwatch's worldline, A, B, C, and D. Your wristwatch had a different reading at each of those events. That fact alone means your wristwatch "changed" from event to event. In other words, in the spacetime viewpoint, "change" just means physical quantities, like your wristwatch's reading, are different at different events along a timelike (or lightlike) line. That's all it is. Nothing has to "occur".
Ok, I got this, no completely but I'll ask your for further detail. Can a change occur in a single time event (point along the timelike line), or does it connect two different time points along the timelike line? And I know that in spacetime changes do not occur, they 'are', but still if by change we mean 'being substantially different in another instant', than this question makes sense.

Let's take the previously mentioned 4 seconds of some body's worldtube. If we split those 4 seconds with a point on half, we get two intervals of 2 seconds. It can be concluded that in the middle point the first 2 seconds end and the second 2 seconds begin. So that point serves as a instant of change. Is this valid reasoning, and can an act of being different in spacetime occur in the same point of time?

PeterDonis
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Can a change occur in a single time event (point along the timelike line), or does it connect two different time points along the timelike line?
Since a change just *is* the fact of the values of some physical parameters being different at two different points along a timelike (or lightlike) line, obviously it requires comparing those values at two different points. You can't have different values of physical parameters at a single point.

Let's take the previously mentioned 4 seconds of some body's worldtube. If we split those 4 seconds with a point on half, we get two intervals of 2 seconds. It can be concluded that in the middle point the first 2 seconds end and the second 2 seconds begin.
Yes, but that's not a change in physical parameters. See below.

So that point serves as a instant of change.
No, because, as above, physical parameters can't take on two different values at a single point. The "change" from the first interval to the second interval at the midpoint is not a change in physical parameters; it's just a change in arbitrary labeling.

Nugatory
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Let's take the previously mentioned 4 seconds of some body's worldtube. If we split those 4 seconds with a point on half, we get two intervals of 2 seconds. It can be concluded that in the middle point the first 2 seconds end and the second 2 seconds begin.
If the body in question is a clock that we zeroed at the beginning of the interval, than that midpoint is the single event "the clock reads 00:00:02". Yes, it's the end of the two-second interval between "the clock reads 00:00:00" and "the clock reads 00:00:02" and the beginning of the two-second interval between "the clock reads 00:00:02" and "the clock reads 00:00:04" but it's the same event either way.

Since a change just *is* the fact of the values of some physical parameters being different at two different points along a timelike (or lightlike) line, obviously it requires comparing those values at two different points. You can't have different values of physical parameters at a single point.

No, because, as above, physical parameters can't take on two different values at a single point. The "change" from the first interval to the second interval at the midpoint is not a change in physical parameters; it's just a change in arbitrary labeling.
Ok, so if change is the state of fact between two (or more) time points, there exists no such thing as an instant of change. So the neccessity for the 'change' is having two points. But since no two points on the timelike line can be adjacent (right next to each other), how does that reflect on what you said? Let's imagine that some body changes one of its intrinsic properties, so that it has the property A at one time point, and the property B on some point after the point A. What happens in between, does there exist an interval AB with some intermidiate or boundary properties? I always conceptualized change as having state A in one time point, and having state B in another point which is adjacent to the first time point. But if there are infinite points in between, that makes it tricky.

Nugatory
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I always conceptualized change as having state A in one time point, and having state B in another point which is adjacent to the first time point. But if there are infinite points in between, that makes it tricky.
There are infinite points on the number line between 1.0002 and 1.0003 but that doesn't stop us from saying that those two points are separated by a distance of .0001, nor stop us from knowing, for any point, exactly which ones are to its left and which are to its right. What doesn't work is the notion of "adjacent" or "two different points with zero distance between them" - you can have two points that are very near one another, but never adjacent.

If an object starts changing state at time A and finishes changing state at time B, that's two different events with some interval between them. Both the start event and the finish event are single points; draw a line through either of them and you'll have no trouble saying that any other point on that line is on one side or the other.

There are infinite points on the number line between 1.0002 and 1.0003 but that doesn't stop us from saying that those two points are separated by a distance of .0001, nor stop us from knowing, for any point, exactly which ones are to its left and which are to its right. What doesn't work is the notion of "adjacent" or "two different points with zero distance between them" - you can have two points that are very near one another, but never adjacent.

If an object starts changing state at time A and finishes changing state at time B, that's two different events with some interval between them. Both the start event and the finish event are single points; draw a line through either of them and you'll have no trouble saying that any other point on that line is on one side or the other.
But what is happening then in the interval AB, could you give me an example of some physical quality based on this scenarion, that its start event was the property A and its finish state was the property B and the change occurs in between. Will it have intermediary states between those two in each point?

Nugatory
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But what is happening then in the interval AB, could you give me an example of some physical quality based on this scenarion, that its start event was the property A and its finish state was the property B and the change occurs in between. Will it have intermediary states between those two in each point?
Yes. For example...

At noon I drop an egg on the floor and it breaks when it hits the floor. If I have a really accurate clock and a high speed camera, I can construct a detailed timeline: Event A is "the bottom surface of the egg hit the floor and the shell started cracking at 12:00:00.000", event B is "the shell is well and thoroughly smashed flat and the top surface of the egg hit the floor at 12:00:00.001". And if my equipment is good enough, I can identify events in between these two events, events at which the bottom half of the shell is smashed but the top half is still moving, smashing itself down into the floor.

However, for most practical purposes I can say that the egg broke at a single point in space-time, namely that particular spot on the floor at exactly noon. That doesn't mean that the two events A and B are "adjacent" or that they both happened at the same time, just that they're so close in time that I'm not going to care about whether some third event happened between A and B or after B, so I can say "after the egg broke" without any ambguity.

You might want to a side trip through "Zeno's paradox" (google will find plenty of hits) for an example of the problems that you will come across if you focus too much on the points between points... Just remember that an event is a single point, and if any process takes enough time for you to care about, then you must think about it as two events marking the start and end of that time.

Yes. For example...

At noon I drop an egg on the floor and it breaks when it hits the floor. If I have a really accurate clock and a high speed camera, I can construct a detailed timeline: Event A is "the bottom surface of the egg hit the floor and the shell started cracking at 12:00:00.000", event B is "the shell is well and thoroughly smashed flat and the top surface of the egg hit the floor at 12:00:00.001". And if my equipment is good enough, I can identify events in between these two events, events at which the bottom half of the shell is smashed but the top half is still moving, smashing itself down into the floor.

However, for most practical purposes I can say that the egg broke at a single point in space-time, namely that particular spot on the floor at exactly noon. That doesn't mean that the two events A and B are "adjacent" or that they both happened at the same time, just that they're so close in time that I'm not going to care about whether some third event happened between A and B or after B, so I can say "after the egg broke" without any ambguity.

You might want to a side trip through "Zeno's paradox" (google will find plenty of hits) for an example of the problems that you will come across if you focus too much on the points between points... Just remember that an event is a single point, and if any process takes enough time for you to care about, then you must think about it as two events marking the start and end of that time.
Thank you very much on a good example. Could this be generalized to all physical processes wrt to time, let's say the change in temperature, in every temporal point some body will have a different value of temperature, of course if it is undergoing thermal change. So we use the physical value A as the start of the interval of 'change' and the value B as its end and in between at every point there are intermediary states?

Nugatory
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Thank you very much on a good example. Could this be generalized to all physical processes wrt to time, let's say the change in temperature, in every temporal point some body will have a different value of temperature, of course if it is undergoing thermal change. So we use the physical value A as the start of the interval of 'change' and the value B as its end and in between at every point there are intermediary states?
In classical and relativistic physics and when you're dealing with real physical quantities that cannot change discontinuously, yes.