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.
Minkowski spacetime M is the model of spacetime geometry used in the theory of special relativity. Unlike most geometries which consider only spatial coordinates, Minkowski geometry includes time, and thus is concerned with processes which take place over time. Among other things, Minkowski spacetime is used to illustrate the effects of length contraction, time dilation, and relativity of simultaneity. M is a four dimensional coordinate system consisting of the usual three spatial dimensions as well as a special time dimension. We have chosen to model a rotating cube as a periodic time process to illustrate the behaviour of Minkowski spacetime geometry.
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:
View of the cube from the perspective of the moving observer:
In my model, the cube itself is also rotating at relativistic speeds. Due to the high rotational speeds of the cube and amount of time it takes light to travel from the rotating cube to the observers, the cube will also appear to warp. Closer points on the cube will be perceived at more recent times than farther points. Since the points on the cube are rotating around the cube's axis, the time delay affects the location and time at which points are viewed:
In this model, we have chosen to ignore the effects of acceleration due to rotation of the cube. We assume the equation used to rotate the cube gives the coordinates in the stationary observer's frame of reference. In assuming this, we are also ignoring any internal forces holding the rigid rotating object together.
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 = 3.0x108 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:
where β is the dimensionless ratio v/c, and travel is in the x direction. Distances perpendicular to the direction of relative travel are unaffected, so yA = yB and zA = zB. These transformation equations are valid for all speeds up to, but not including, the speed of light. Notice that the coordinates which are perpendicular to the direction of travel are unaffected. The value:
occurs frequently, and is called the Lorentz factor [Hal].
"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 = (x1, x2, x3, x4). 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:
This is found to be invariant for all observers, independent of their relative motion. This motivates the definition of the inner product or dot product of events v and w to be:
and interval between v and w is:
Minkowski spacetime M is defined to be a four dimensional linear space with a nondegenerate, symmetric, bilinear form g:
M is the mathematical model of the spacetime of relativity. By nondegenerate, we mean that if for all w, g(v,w) = 0, then v = 0. The form g is symmetric because g(v,w) = g(w,v) for all v and w, and indefinite because there exists vectors v not equal to 0 for which g(v,v) = 0. The quadratic form Q(v) is found by taking a vector v dot product with itself:
and is often denoted v2. Vectors v for which v·v = 0 are called null or lightlike vectors. A vector v is considered timelike if v·v < 0, or spacelike if v·v > 0. Lightlike vectors form a light cone or null cone, CN(x0), centered around an origin x0, which separates the spacelike and timelike vectors. Light cone viewed in three dimensions:
The null cone CN(x0), consists of all events x in M that are connectable to x0 by a light ray:
If the vector (x-x0) is spacelike, not even a photon can experience both events, because it would have to travel faster than the speed of light.
Lorentz transformations are used to relate the spacetime coordinates for an event observed by two observers in their respective frames of reference. If the reference frame S' is moving along the x-axis with speed v relative to the reference frame S, then an observer in S' reports the spacetime coordinates as x' = (x', y', z', t'), and an observer in S reports the spacetime coordinates as x = (x, y, z, t). The familiar equations which relate (x', y', z', t') to (x, y, z, t) are:
Lorentz transformations can also be used in matrix form. This allows for a more general definition of Lorentz transformations, which includes transformations when the direction of travel is not along the x-axis. Using the matrix form of Lorentz transformations allows the coordinate transformations to be calculated using matrix multiplication. If L is a Lorentz transformation matrix, then the transformation from x = (x, y, z, t) to x' = (x', y', z', t') is computed as x'T = LxT.
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 LThL = h, where h is the matrix:
and L is invertible:
Note that this definition of h allows the inner product of two vectors v and w to be calculated as:
The set LGH of general homogeneous Lorentz transformations includes all 4x4 matrices that satisfy LThL = h. The set LGH forms a group which is closed under the formation of products and inverses. Thus, if L1, L2 are in LGH, then L3 = L1L2 and L1-1 are also elements of LGH. If we enlarge the group of Lorentz transformations to include spacetime translations, we obtain the inhomogeneous Lorentz group or Poincaré group. By allowing spacetime translations, we allow two observers to use different spacetime origins.
We restrict our attention to a subset of LGH: the set of proper, orthochronous Lorentz transformations, L. A Lorentz transformation L in LGH is proper if det(L) = 1, and improper if det(L) = -1. By orthochronous we mean that L preserves the time orientation of timelike and lightlike vectors. By restricting ourselves to proper, orthochronous Lorentz transformations, we avoid reflections and time reversing transformations.
Lorentz transformations are analogous to rotations in space and time [Feyn]. Like three dimensional rotations, the determinant of L in LGH is equal to one, and the rows and columns are orthogonal vectors. In fact, an important subgroup R of LGH consists of the group corresponding to rotations in three dimensions. A matrix R in LGH has the form:
where [Ri,j]i,j=1,2,3 is a unimodular orthogonal matrix, which is a rotation matrix in three dimensions. [Ri,j] satisfies det[Ri,j] = 1, [Ri,j]T = [Ri,j]-1, and [Ri,j][Ri,j]T = [Ri,j][Ri,j]-1 = I3. R corresponds to a physical rotation of the spatial coordinates within a given frame of reference, and thus is called a rotation matrix. R is the rotation subgroup of L [Nab].
Returning to the equations transforming coordinate systems from one reference frame traveling with respect to another along the x-axis:
we can write these equations in matrix form, suitable for use in Minkowski spacetime geometry. In this case, the Lorentz transformation matrix can be written as:
with the inverse:
This form of the matrix L is often called a boost in the x-direction [Nab]. Transforming x = (x, y, z, t) to x' = (x', y', z', t') gives:
Note that LLT = LL-1 = I4, the 4x4 identity matrix. Lorentz transformations with the direction of motion along the y- or z-axis are also straight forward. When the direction of travel is along the y-axis, L is:
and along the z-axis, L is:
When the relative direction of travel is not along one of the coordinate axes, the Lorentz transformation L is slightly more complicated to compute. We first use a rotation matrix R in R which rotates the coordinate system so that the direction of travel is aligned with one of the coordinate axes, say the x-axis. We then use Lx to transform the coordinates from the stationary to moving observer, and finally use R-1 to return the coordinate system to its original alignment. Thus, the matrix L is found using:
- L = R-1LR,
and an event x observed by the stationary observer is related to the event x' observed by the moving observer by:
- x' = Lx = R-1LRx.
To be able to see the details of what is happening to the rotating object, we have equipped our observers with both a telescope and a recording device. The observer is initially very far away from the rotating object. The telescope is used to enlarge the view of the rotating object, and is controlled so that the appearance of the rotating object remains a constant, visible size. The cube is rotating very fast. The recording device on the observer is used to record what the observer sees. The recording is played back at slower speeds so that it is possible for a human to perceive what apparent transformations of the object are taking place.
Mechanics of Perceived Light
The lighting model we are using is the standard graphical representation of diffuse radiation, and does not take specular reflection into account. Diffuse radiation, or scattering, occurs when the incident light slightly penetrates the surface and is reemitted uniformly in all directions [Hill]. This type of light depends on the intensity of the light source, the reflection coefficient of the surface, and the angle at which the light hits the object. The percentage of light reflected at the observer determines the brightness of the object. It is not affected by what speed the observer is traveling at the time when he intersects the light ray reflected by the object. Therefore the shade of each face is calculated using the direction of the normal vectors in the object's frame of reference, not the observer's.
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.
Visual Effects of Rotating an Object at High Speeds
Light takes a non-zero amount of time to travel from the object to the observer. For example, it takes approximately eight minutes for the light from the sun to reach earth, and much longer for light from stars in other solar systems. Thus, we observe a more recent image of the sun than other stars. When we look into the night sky, the image we perceive is composed of light rays that were emitted from different stars at vastly differing times. We see an image which is created by the convergance of all the light rays which happen to reach us at the same time. Since the stars are moving with repect to us, we see them where they used to be and not where they are now.
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 xo, he can see another event which occurred previously at xp only if there is a photon whose worldline passes through both xo and xp. Thus xo and xp must satisfy
Q(xo - xp) = 0. xo is on xp's future directed light cone, and xp is on xo's past directed light cone. To determine the composite image of the cube that the abserver sees, we must find all events xp on the worldlines of the points on the cube where Q(xo - xp) = 0 holds, and xp must occur before xo. In the software model, we have used Newton's iterative method to find the correct events corresponding to locations on the surface of the cube.
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.
[Feyn] Feynman, Richard P., Six not-so-easy pieces: Einstein's relativity, symmetry, and space-time. Addison-Wesley Pub., Reading, Mass., 1997.
[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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