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# EVENTS and SPACETIME

...to suppress the pictures is to suppress a powerful source of suggestion. ... Pictorial representation is essential for discovery and rapid understanding...
J.L. Synge in The Hypercircle in Mathematical Physics (1957), p. 12

## Events and Worldlines

An EVENT is "an occurrence that happens in a small space and lasts a short time".
For example: the snapping of one's fingers, or the explosion of a firecracker, or the collision of two particles. The emphasis is on a small space and a short time.

It will be helpful to have a diagram. A diagram helps to summarize relationships between the events.

Consider the motion of a baseball thrown straight up into the air.

A Baseball Thrown Straight Up (in space)
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Occasionally, one might be interested in a complete record of the motion. Usually, one would plot the history of this baseball on a position vs. time graph. Each marked point on this graph represents a succession of events experienced by this baseball:

• the event "the baseball is at position x1 at time t1"
• the event "the baseball is at position x2 at time t2"
• etc.

Position vs. Time graph of the History of this Baseball
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Note: The ball is still being thrown straight up (in space). It's similar to the output you get, for example, from a paper-tape record of someone's heartbeat.

## The Spacetime Diagram

In relativity, it is customary to plot the history of this baseball on a Spacetime Diagram, where time runs upwards on this diagram.
Note: "Upwards (in space)" is now "to the right" on this diagram.

Spacetime Diagram of the History of this Baseball
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The curve that the baseball traced out on this diagram is called its worldline.

A WORLDLINE is "a curve representing the set of events experienced by an object", in other words, its "history".

## Spacetime

A SPACETIME is "the set of all (possible) events in the universe". It is mathematically modeled by an abstract 4-dimensional space. Each point in the spacetime represents an event. Spacetime has no general shape. Different models for the whole universe will have different shapes. It is believed that Spacetime has no edges. And thus, it is either infinite (like an ordinary 2-dimensional plane), or else finite (like a sphere, the 2-dimensional surface of a ball).

To the left, we have drawn a generic spacetime (say, all of the events that have happened, are happening, or will happen in our entire universe). We have drawn in one worldline (say, mine) and marked one event on that worldline (say, where I was standing at midnight on my 21st birthday).

### Why four dimensions?

We will find it convenient to label or locate events. To label an event, imagine laying down a four-dimensional grid. On this grid, we need to provide: four numbers: for instance, (t,x,y,z):
one (say, t) to identify the "time of occurrence" (that is, when it happened),

and

three (say, x, y, and z) to identify the "position of occurrence" (that is, where it happened).

The following analogy may be useful: Think of spacetime as you would a blackboard. Events in spacetime are like points on the blackboard. To label a point on the blackboard, you have to imagine laying down a grid. Realize that you have to give me two numbers (x,y).

x tells me how far along the horizontal.
y tells me how far up the vertical.
Thus, the blackboard is 2-dimensional.

ASIDE On Art: Since it is difficult (though not impossible) for us to visualize and to draw figures in four-dimensional space, one usually suppresses some dimensions and draws 2- or 3-dimensional diagrams. Often, a 2-dimensional diagram is sufficient.

ASIDE On Speech: I use the unqualified words "plane" and "surface" to mean "a figure whose dimension is one-less-than the dimension of the entire spacetime". That is, when I say "plane", I mean a 3-dimensional figure: a "3-plane". If I want to talk about an ordinary two-dimensional plane, I would say "2-plane".

Don't let the four-dimensionality distract you.

### The Role of Spacetime

Spacetime will turn out to be an essential ingredient of the Theory of Relativity. It is not essential for pre-Relativistic theories. But here, we have gone ahead and formulated these pre-Relativistic theories in terms of a spacetime model. In this way, as we work our way to the correct model, we can more-transparently see what revisions we are making in our understanding of space and time.

ASIDE On Reality: I'm not trying to claim that "Spacetime is real, and we really live in a four-dimensional world, and that there are these cones everywhere, etc..." and that you have to believe this. Spacetime is merely a convenient description of a particular aspect of the real world. It is a description of the "space and time" aspect of the world. I'm trying to describe "what it really is" by describing "what it is like". Thus,

We find it convenient to understand the "space and time" aspect of the world by describing it as if it were a certain abstract 4-dimensional space we call Spacetime.
We need to do this because relativity is notoriously counterintuitive, as history has demonstrated. In my experience, thinking about relativity in terms of geometric pictures is a good crutch we can use.

## examples

To get a feeling for these diagrams, consider the following examples:

An object at Rest
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An extended object (a meterstick) at Rest
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An object in Uniform Straight-Line Motion
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A faster object in Uniform Straight-Line Motion
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An extended object (a meterstick) in Uniform Straight-Line Motion
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A race two runners and a time-keeper
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A race: There and Back two runners and a time-keeper, a meterstick
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Speedy gets pulled over for speeding
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A simple harmonic oscillator (a mass on a spring)
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A damped harmonic oscillator (a mass on a spring with friction)
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Two people with masses on springs
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Two more people carrying masses on springs
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An object in uniform circular motion (The Earth orbiting the Sun)
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A flash of light(IMPORTANT EXAMPLE)
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The purpose of the animations was to give you some feeling for how these diagrams would represent the histories of objects in the real world. However, though our examples describe the time-evolution of three-dimensional objects in space, we should regard a spacetime diagram as we do a "map". We are concerned only with the finished picture... not with how it was drawn. It is to be regarded as an unchanging complete record of histories.
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