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# The Light Cone: Introduction

These documents comprise an elementary introduction to the Theory of Relativity. The intended audience is a class of nonscience undergraduates in (PHY 105) Science for the 21st Century a course offered by the Department of Physics at Syracuse University. (Eventually, these documents will be expanded to include more technical explanations for more advanced audiences.)
Admittedly, this is an ambitious plan to explain the Theory of Relativity in three one-hour lectures. Comments and criticisms are welcome. Rob Salgado (salgado@physics.syr.edu)

## Spacetime

The Theory of Relativity is really a physical theory about time and space. In fact, it actually says that there is no such thing as time and there is no such thing as space. Instead, it is a theory of "a new thing called Spacetime". Spacetime is the history of an entire universe, that is, the set of all of the events that happened in that universe.

Our plan is to follow a pseudo-historical development of how we came to understand "Spacetime", the spacetime of General Relativity. We do so by considering a sequence of models for spacetime. Each model represents "how we understood space and time" at various stages in our human history. Specifically, we will consider:

• Aristotle's view
• Galileo's view
• Einstein's view
Each model recovers the notions of "space" and "time" in slightly (but quite significantly) different ways. At issue:

# How is the nature of space and of time to be viewed?

### "Absolutely" ("Universally", "Publicly") or "Relatively" ("Personally", "Privately")

It is not philosophy, but physical experiment, that decides which one models the real world best. Indeed, it has been our gradual understanding of the Laws of Physics that has forced us to replace one model with another.

Throughout, I will be exercising the use of my poetic license.

## Causal Structure and the Light Cone

In general, a causal structure provides each event in spacetime its own notion of ordering events into "those events to its past", "those events in its present", and "those events to its future". Each model for spacetime has a different causal structure.

At the root of Einstein's Spacetime is a particular causal structure. Unfortunately, the nature of the causal structure of Spacetime and its (experimentally verified) consequences are often beyond our everyday experience. Hence, it may appear quite counterintuitive. This is the source of much of the confusion in this subject.

The Light Cone is a mathematical model that very neatly encodes this causal structure of Einstein's Spacetime. Each event in Spacetime has a double-cone attached to it, where the vertex (i.e., the tip) corresponds to the event itself. By convention, "time" runs vertically in all of our diagrams. The upward-directed cone opens to enclose the directions pointing towards events to the future of this event. The downward-directed cone directions pointing towards events to the past of this event.

In my opinion, understanding the Light Cone is an important first step towards understanding the Theory of Relativity. (Some mathematics is a necessary crutch to clarify and correct our physical intuition.)

## Flat and Curved Spacetimes

An important second step will be understanding how the Light Cones are arranged throughout Spacetime.

If the Light Cones are arranged in a neat grid in a uniform manner, we say that "Spacetime is flat". The spacetime of Special Relativity is a flat spacetime.

If the Light Cones are arranged in a non-uniform manner, we say that "Spacetime is curved". The spacetime of General Relativity is generally a curved spacetime.

Einstein's Theory of Gravity relates the curvature of Spacetime to the presence of matter. As an interesting example, we will consider the Spacetime around a Black Hole.

## Credit where credit is due

### The causal past:

This presentation of the spacetime-models is heavily based on Geroch (General Relativity from A to B, which was based on a course for nonscience undergraduates). The presentation of the Schwarzschild black hole is based on Geroch and on Ellis/Williams (Flat and Curved Space-Times).

### The causal future:

The next planned revision of these web documents will include "Track-2" material. In particular, I will try to further develop "Galilean geometry" and "Minkowski geometry" based on Yaglom (A Simple Non-Euclidean Geometry and Its Physical Basis.) and introduce "4-vectors and -tensors", with much inspiration from Burke (Spacetime, Geometry, Cosmology) and (Applied Differential Geometry). For now, follow the "Technical Comments" links for the Track 2 material.

Of course, I am solely responsible for any errors in the contents of these documents.

Here are my references for the Light Cone

## Now, let's begin.

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