Spacetime

In physics, spacetime is any mathematical model that combines space and time into a single continuum. It is a mathematical concept used to refer to all points of space and time and their relation to each other. Historically, space and time were thought of as separate entities. Time was thought to pass at the same rate for all observers, regardless of where they were or how fast they were moving. Similarly, measurements of distance were thought to be the same for everyone. When Albert Einstein developed the idea of spacetime, he showed that measurements of time and distance between the same events could differ for different observers. But at the same time, he showed that these measurements could be combined in a systematic way.

This combination simplified previously existing theories such as electromagnetism, and was central to the development of almost all fundamental physics since. Quantum field theory, which describes the fundamental particles that make up our universe, was invented by considering how quantum mechanics would work in a unified spacetime. Furthermore, considering how spacetime could be warped or bent led to the development of general relativity, which describes how gravity works.

The unification of space and time into spacetime is based on the assumption that the speed of light, $c$, is the same everywhere. No matter where an observer is or what velocity they are moving at, they will always measure the speed of light to be $c = 2.99 \times 10^{8} \text{m/s}$. Note that since $c$ has units of length/time and is constant, it can be thought of as a conversion factor between units of length and units of time. Considering how different observers measure events involving light, the relationships between space and time can be derived.

Tom and Georgia decide they want to calibrate their clocks. Since the speed of light is constant, they reason, it makes sense to calibrate their clocks by measuring how much time it takes light to move between two places.

Tom stands in a train, while Georgia stands on the ground. Inside the train there is a beam of light that bounces up and down forever between two mirrors. The height of the train (the distance between the mirrors) is given by $\Delta z$, and the train moves at velocity $v$ relative to the ground. Inside the train, Tom measures the time $\Delta t$ the light takes to move from the top of the train to the bottom of the train. Because the light moves at velocity $c$, it must be true that $c \Delta t = \Delta z$, the total distance traveled.

Georgia, standing on the ground, tries to measure the same thing. She measures the time it takes for the light to go from the top of the train to the bottom of the train, but in her perspective, the bottom of the train has moved a distance $v \Delta t$ by the time the light hits it. Again, because the light moves at velocity $c$, it must be true that $c \Delta t = \sqrt{\Delta z^{2} + v \Delta t^{2}}$. But unless $v$ or $\Delta t$ is zero, Georgia's equation is incompatible with Tom's equation.

This problem can be resolved if the amount of time that Georgia measures on the ground ($\Delta t_{G}$) is different from the amount of time that Tom measures on the train ($\Delta t_{T}$). Then the equations become $$c \Delta t_{T} = \Delta z$$ $$c \Delta t_{G} = \sqrt{\Delta z^{2} + v \Delta t_{G}^{2}}$$

Since there are two variables and two equations, the relationship between $\Delta t_{T}$ and $\Delta t_{G}$ can be derived. Squaring both equations and expressing in terms of $\Delta z^{2}$ gives

$$\Delta z^{2} = c^{2} \Delta t_{T}^{2}$$ $$\Delta z^{2} = c^{2} \Delta t_{G}^{2} - v \Delta t_{G}^{2}$$

$$\Delta t_{T}^{2} = \Delta t_{G}^{2} (1 - \frac{v^2}{c^2})$$

If $c$ and $\Delta z$ are the same for both observers, and $v \neq 0$, then it must be true that $\Delta t_{G} > \Delta t_{T}$

This example demonstrates that space and time individually vary dependent on who is measuring them, but the idea of spacetime runs deeper, capturing the systematic ways in which they vary.

From the example, we can see one reason why objects can't move at the speed of light. If we set $v = c$, the factor $1 - \frac{v^2}{c^2}$ would be zero. That would make converting from $\Delta t_{G}$ to $\Delta t_{T}$ impossible, since it would require dividing by zero.

To understand what spacetime is, it helps to think about what space is. The concept of space centers on being able agree on the distances between objects.

Consider two people looking at a stick on a line.

The stick begins at $x_{b}$ and ends at $x_{e}$. Each person measures position along the $x$ axis by counting the number of tick marks from them. Person A says that the beginning of the stick is at $x_{b} = 2$ and the end of the stick is at $x_{e} = 3$, while person B says that the beginning of the stick is at $x_{b} = 4$ and the end of the stick is at $x_{e} = 5$. Despite the fact that they are looking at the same stick, they disagree on where the beginning and the end of the stick are, because they have different reference frames. They view the same object from different perspectives. However, there is a measure that they can agree on. The distance between the beginning and the end of the stick $\Delta x = x_{e} - x_{b} = 1$ is the same for both of them, and indeed would be the same for an observer at any point along the line.

Quantities on which every observer can agree are called invariant. In one dimension, $\Delta x$ is an invariant.

But this does not hold in two dimensions. Consider the case depicted below, with two people looking at a rod.

Each person considers the $(x,y$) coordinates of the beginning and end of the rod using their own coordinate system. Unlike the case in one dimension, $\Delta x$ and $\Delta y$ are not invariant, because each observer can have perspectives that are rotated relative to each other- they are only defined relative to the particular frame of reference of an observer. If you ask each of them how long ($\Delta x$) and how wide ($\Delta y$) the rod is, they will disagree from each of their perspectives.

But if you ask them what the Euclidean distance is between the two ends of the rod, they will agree. Specifically, they will always calculate $d^{2} = \Delta x^{2} + \Delta y^{2}$ the same.

Similarly, in three dimensions $d^{2} = \Delta x^{2} + \Delta y^{2} + \Delta z^{2}$ is the same for all observers. This is the invariant that defines 3-d space.

Where space encodes the invariant distances between objects, spacetime describes the invariant intervals between events. An event is anything that happens at a particular point of space and instant of time. Instead of describing points using coordinates $(x, y, z)$, events are described using coordinates $(x, y, z, c t)$, where $c$ is the speed of light. The reason for using $c$ specifically will be clear later, but note that $c$ has units of length/time and $t$ has units of time, so $c t$ has units of length, just like $x$, $y$, and $z$.

Newton thought that $\Delta t$ itself was an invariant, even though $\Delta x$, $\Delta y$, and $\Delta z$ were not. He thought that no matter where one was or how fast one was moving, the time between two events would always be measured exactly the same. This meant that there were two invariants, $\Delta t$ and $d^{2} = \Delta x^{2} + \Delta y^{2} + \Delta z^{2}$.

Einstein overthrew this idea by realizing that both $\Delta t$ and $d^{2}$ are quantities relative to particular observers, and that the relevant invariant quantity is a combination of the two. Special relativity holds that $\Delta s^{2} = \Delta x^{2} + \Delta y^{2} + \Delta z^{2} - \Delta (ct)^{2}$ is invariant. Notice that time has a minus sign, while all the spatial terms are positive.

Time is often called "the fourth dimension." This is true, but can be misleading. If time were the 4th dimension in exactly the same way that $x$, $y$ and $z$ were the first three dimensions, then the invariant quantity describing this alternate spacetime would be $\Delta s_{alt}^{2} = \Delta x^{2} + \Delta y^{2} + \Delta z^{2} + \Delta (ct)^{2}$. For this reason, physicists refer to spacetime as 3+1-dimensional, where 3 represents the number of space dimensions and 1 represents the number of time dimensions. Writing it this way emphasizes that time is different from space due to the minus sign in the equation for $\Delta s^{2}$. The minus sign is very important.

Einstein said that he regretted calling his principle the "principle of relativity," since "principle of invariance" better captures the importance of the idea. The important part is not that time and space individually are relative, but that the way in which they differ for different observers always leaves $\Delta s^{2}$ the same.

Consider a beam of light as it leaves a flashlight, travels in the $x$ direction, and hits a wall. The light travels a distance $\Delta x$ at speed $c$. What is the interval $\Delta s^{2}$ between the event of the beam leaving the flashlight and the event of the beam hitting the wall?

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If the light is moving at speed $c$, the time $\Delta t$ it takes to get across the room is $\Delta t = \frac{\Delta x}{c}$. If we assume that $\Delta y = 0$ and $\Delta z = 0$, the interval is $$\Delta s^{2} = \Delta x^{2} - (c \frac{\Delta x}{c})^{2}$$ $$\Delta s^{2} = \Delta x^{2} (1 - \frac{c^{2}}{c^{2}})$$ $$\Delta s^{2} = 0$$

With a bit of extra algebra, one can show that the interval between any two events that a beam of light crosses through have an interval of 0 between them.

^[1] From these problems we see that $\Delta s^{2}$ can be positive, negative, or 0 for distinct events, unlike $d^{2}$ which is always positive for distinct objects.

Events with $\Delta s^{2} = 0$ are called lightlike-separated; events with $\Delta s^{2} < 0$ are called timelike-separated. These events are causally connected. They define the light-cone, depicted here. Here, time is represented as the vertical direction and two spatial dimensions are represented horizontally (since representing all 3+1 dimensions would be difficult). The surface of the cone is all the points with $\Delta s^{2} = 0$ relative to the point at the center of the cone. The interior of the cone covers all the timelike-separated events. The events outside the cone, with $\Delta s^{2} > 0$, are called spacelike-separated. These events are causally unrelated. Nothing outside the light cone can affect the point at the center of the cone, or vice versa

Space can be defined by the transformations that leave distance invariant. I.E. there are ways to change the coordinate system such that $d^{2}$ stays the same. These transformations correspond to translating (that is, moving from one point to another), rotating, and reflecting. In the 1D spatial example above, the only difference between the perspectives of A and B is a simple translation. B is two steps to the left from A. In the 2D example, moving from A's perspective to B's perspective takes both moving from point A to point B and rotating.

The set of transformations that leave the spacetime interval unchanged are known as the Lorentz transformations. The Lorentz transformations include translations, reflections, and rotations (since those keep the spatial part of the interval invariant and don't affect the time part). They also include boosts, or transformations between the perspectives of observers moving at different velocities. For instance, if a dispatcher on Earth wanted to communicate with a spaceship moving rapidly away from Earth to let it know what was in front of it, the dispatcher would need to account for the fact that the spaceship is moving. The spaceship experiences time more slowly than Earth. The example from the first section is one example of a boost; general boosts can involve relative velocities in any direction.

^[2]

Spacetime can be bent in the same way that space can be bent. Mathematically, bent space is represented using invariant distances other than the standard one discussed above. In addition to space bending, time also "bends" such that time intervals can vary for different observers even when they aren't moving relative to each other. For instance, time flows at different rates for different observers depending on how far away they are from the surface of the Earth. Accounting for this difference is critical to the function of GPS satellites.

The equations that describe specifically how spacetime bends in the presence of mass are known as the Einstein field equations, and are the mathematical basis of Einstein's theory of general relativity. The curvature of space has varied and interesting consequences: for instance, light will curve around massive objects such as the sun. Detecting this behavior was one of the first experimental confirmations of general relativity. Black holes are an example of extremely curved spacetime, where spacetime is curved to the point that light cannot escape. It is even possible that our entire universe is slightly curved at the largest length scales, though cosmologists (physicists who study the structure and history of the whole universe) agree that is is at least very close to flat.

1. Aainsqatsi, K. Light cone in 2D space plus a time dimension. Retrieved May 4, 2016, from https://en.wikipedia.org/wiki/Light_cone#/media/File:World_line.svg
2. Johnstone, . Illustration of spacetime curvature. Retrieved May 4, 2016, from https://commons.wikimedia.org/wiki/File:Spacetime_curvature.png