Geometry, Vectors, and Relativity Link
Geometry and Vectors Link
Triangle Angles Link
- The sum of the interior angles of a triangle is 180 degrees.
Sine and Cosine Link
- The text introduces the relationship between the sine and cosine of an angle, the notation of points in terms of their coordinates, the concept of vectors, the Euclidean metric, and the formula for finding the x-coordinate of a point in a 2D plane.
Dot Product Link
- It also provides a formula for calculating a number from two vectors, which is applicable in both Euclidean and spacetime geometry. The text uses the dot product formula and Pythagoras's theorem to prove Eqn (4), which gives the cosine of the angle between two vectors. It also explains how to find the projection of a vector onto a line using the dot product formula.
Perpendicular Vectors Link
- The metric function for two perpendicular vectors is zero. The projection of a vector onto another vector is given by the formula OS = g[(x,y),(cos A, sin A)], where (x,y) are the coordinates of the vector, A is the angle between the vector and the line, and g is the dot product.
Angle Between Vectors Link
- The angle between two vectors is the angle between the lines that pass through the origin and the two vectors. We can use the formula for the projection of a vector onto another vector to find the angle between the two vectors. We also know that the sum of the interior angles of a triangle is 180 degrees. This means that if we have three vectors, we can use the formula to find the angle between the first and second vectors, the second and third vectors, and the third and first vectors. We can then use the fact that the sum of the interior angles of a triangle is 180 degrees to find the angle between the first and third vectors. Finally, we can use the fact that the metric function for two perpendicular vectors is zero to find the angle between two perpendicular vectors.
Spacetime Geometry Link
Coordinates and Light Cone Link
- The text discusses the change of coordinates for a rotation in space, the similarity between spacetime equations and the geometry of a two-dimensional slice through four-dimensional spacetime, and the concept of the light cone. The text introduces the Minkowskian metric, a new formula for g that accounts for the possibility of negative values for x2–t2 and divides vectors in spacetime into three classes: spacelike, timelike, and lightlike. To find the x-coordinate of the event P in Figure 7, we need to draw a line through P at right angles to the x-axis.
Projection of Vectors Link
- In order to find the x-coordinate of the event P in Figure 7, we need to draw a line through P at right angles to the x-axis. However, if we use an arbitrary spacelike vector OG instead of the x-axis, the same method will project OP onto OG, and the event S will be considered to happen at the same place as P. The spacetime metric does not give us the length of OS, as SP is not perpendicular to OG. The length of the projection OS can be found by using Eqn (14 a) to find OS2=OT2 – OU2. We then note that (OT/OS, OU/OS) is a unit vector and find the unit vector in the direction of OG by dividing ( u,w) by its length. Finally, we use the spacetime metric to find the length of OS.
Special Relativity Link
Lorentz Transformation Link
- The text discusses the Lorentz transformation equations for changing coordinates between the Earth's reference frame and the ship's reference frame in Special Relativity.
Doppler Effect and Uncertainty Principle Link
- It also covers the Doppler effect, the uncertainty principle, and the use of radar to measure time.
Time Dilation and Twin Paradox Link
- Time dilation occurs when an observer on Earth and an astronaut tethered to a spaceship measure different amounts of time passing due to their relative velocities. The twin paradox is resolved by the fact that a two-way voyage is not symmetrical; the space travelling twin takes a detour from the straight-line path of the twin who stays at home. This detour causes the space travelling twin to age less than the twin who stays at home.
High Speed Effects Link
- High speed travel affects the perception of space and time due to the Doppler shift of starlight, special relativity, and aberration. The Doppler shift causes a "starbow" effect with a circular rainbow-like appearance, while special relativity causes the forward hemisphere of space to appear squashed together and the backward hemisphere to expand. Aberration causes the familiar constellations to appear pushed forward in the sky, crowded around the direction in which the ship is travelling.