**Minkowski Spacetime**

by Andrea Mantler.

Last update = July 26, 2002.

**Note:** This is an html version of Andrea's original report. Some of the mathematical symbols used in the original report are not available in html. The reader is asked to keep in mind that the Greek letters beta, lambda, sigma, and eta which appear in some of the gif formulas are written as b, L, S, and H in the text, as there is no html support for these symbols.

The java applet which illustrates these relativistic phenomena will soon be made available for download.

It has been observed that the speed of light is a constant value for all observers, independent of their relative motion. Consequences of this are that time appears to slow down and lengths contract for a moving observer. In order to demonstrate these effects, we have chosen to model the appearance of a rigid, rotating cube which is being viewed by two observers. One observer is moving at high speeds with respect to the rotating cube, and the other is stationary with respect to the rotating cube. Time dilation and length contraction cause the cube to be deformed along the direction in which the cube and the traveling observer are moving relative to one another. View of the cube from the perspective of the stationary observer:

The constancy of the speed of light requires a change to the usual linear transformation between moving and stationary observers. If observer B is moving relative to observer A with velocity v, and both A and B observe a photon, the photon will be moving at the speed of light in both A's and B's reference frame. The speed of light in a vacuum has the same value c = 3.0x10^{8} m/s in all directions and in all inertial reference frames [Hal]. An inertial frame of reference is one where the observer is moving at a constant velocity. A consequence of this principle is that distance and time coordinates of A and B are related by:

"Points" in Minkowski spacetime are called *events*. An event **x** is represented by three spatial coordinates, and a time coordinate: **x** = (x, y, z, t), or equivalently **x** = (x_{1}, x_{2}, x_{3}, x_{4}). A *worldline* is a continuous sequence of events that represents a physical particle as it moves through space and time. The *interval* [Feyn] between two events **x** = (x, y, z, t) and **x'** = (x', y', z', t') is defined to be:

A Lorentz transformation matrix **L** has several properties:

- The rows of
**L**are mutually orthogonal unit vectors. - The columns of
**L**are mutually orthogonal unit vectors. - The matrix
**L**satisfies**L**, where^{T}hL = h**h**is the matrix:

We restrict our attention to a subset of **L _{GH}**: the set of proper, orthochronous Lorentz transformations, L. A Lorentz transformation

Lorentz transformations are analogous to rotations in space and time [Feyn]. Like three dimensional rotations, the determinant of **L** in **L _{GH}** is equal to one, and the rows and columns are orthogonal vectors. In fact, an important subgroup

Returning to the equations transforming coordinate systems from one reference frame traveling with respect to another along the x-axis:

**L = R**,^{-1}LR

and an event **x** observed by the stationary observer is related to the event **x'** observed by the moving observer by:

**x' = L**._{x}= R^{-1}LR_{x}

The colour of an object depends on the wavelength of the light leaving the object. Lengths contract when transformed using a Lorentz transformation, so the colour the moving observer sees may not be the same as the colour the stationary observer sees. Note: the application I have written does not take possible colour shifts into consideration.

The same is true for our rotating cube. We see the corners of the cube as they were some time in the past. For example. it we are looking at the cube from above, we would see the top face at a more recent time than the bottom face, and thus at different places in the rotation period. This makes the cube look like it is being "wrung out", with faster rotations giving more twist:

When observing the rotating cube from the side, the vertical edges remain nearly straight but the faces become convex or concave, depending on whether they are approaching or receding:

Again, the greater the rotational velocity, the greater the effect. If we view the cube from an angle of 45 degrees up from the horizontal, we get a combination of the previous two warping effects:

At any instant in time, the observer sees all events with which it is "connectable by a light ray". When the observer is located at event **x _{o}**, he can see another event which occurred previously at

Note that when the observer is moving, we must first make sure that all of the events are represented in the coordinate system of one of the two inertial reference frames. It is not valid to calculate the interval between two events when they are in two separate reference frames. Intervals between two events remain the same when transformed using a Lorentz transformation, so it does not matter which coordinate system is used. Since there are many points on the cube which make up the composite image for only one observer, we have chosen to transform the spacetime location of the observer to the coordinate system of the rotating cube. This saves calculating the Lorentz transformations for each point in each iteration of Newton'w method. Once all of the events composing the image are found, one matrix multiplcation is done on each event, transforming the image into the observer's reference frame.

When experimenting with rotating the cube at various speeds, we noticed that the cube appears to "jitter" when rotated near and faster than the speed of light.(Note: the mathematical model we are using does not prevent the cube from rotating faster than the speed of light.) The approaching edge of the cube would jump back and forth, rather than rotate smoothly.

The reason for this apparently jerky motion is clear when we visualize the portion of the observer's light cone which is in the observer's past, and the worldline of one of the vertices, in three dimensions. At any instant in time, the obersver sees anything that intersects its past light cone. Narmally, the worldline of a vertex intersects with the past light cone of the observer in only one location. When the vertex is rotating faster than the speed of light, it is able to intersect with the past light cone in multiple locations. In the following diagrams, the blue surface represents a portion of the past light cone, and the red line is the worldline of a vertex being rotated at 1.5 times the speed of light. The three dimensions being depicted include the x, y, and t dimensions.

At this speed, we can clearly see the worldline of the photon pass through the past light cone of the observer in three distinct locations. Newton's method iterates until the first point of intersection is found. Jerky motion results when, on successive images, Newton's method chooses different points of intersection. If we were to draw all intersections, we would see several images superimposed on one another, rather than jittery motion.

At speeds closer to the speed of light, the worldline of the vertex may closely parallel the past light cone for short distances. Jerky motion occurs more on the approaching edge, because those segments of the vertex's worldline approximate the curve of the past light cone at speeds near the speed of light:

When the worldline of the vertex follows the shape of the past light cone, there are an infinite number of intersections for that segment of the worldline. If all the intersections were drawn, we would likely see a blurred approaching edge.

[Hal] **Halliday, David, et al.**, Fundamentals of Physics, Fourth Edition. John Wiley & Sons, Inc., New York, 1993.

[Hill] **Hill, F. S. Jr.**, Computer Graphics. Macmillan, New York, 1990.

[Nab] **Naber, Gregory L.**, The Geometry of Minkowski spacetime: an introduction to the mathematics of the special theory of relativity. Springer-Verlag, New York, 1992.

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