Spherical Coordinates and Projections,Time
Spherical coordinates can be a little challenging to understand at first. Spherical coordinates determine the position of a point in three-dimensional space based on the distance ρ from the origin and two angles θ and ϕ. If one is familiar with polar coordinates, then the angle θ isn’t too difficult to understand as it is essentially the same as the angle θ from polar coordinates. But some people have trouble grasping what the angle ϕ is all about.
Relationship between spherical and Cartesian coordinates
Spherical coordinates are defined as indicated in the following figure, which illustrates the spherical coordinates of the point P.
The coordinate ρ is the distance from P to the origin. If the point Q is the projection of P to the xy-plane, then θ is the angle between the positive x-axis and the line segment from the origin to Q. Lastly, ϕ is the angle between the positive z-axis and the line segment from the origin to P.
We can calculate the relationship between the Cartesian coordinates (x,y,z) of the point P and its spherical coordinates (ρ,θ,ϕ) using trigonometry. The pink triangle above is the right triangle whose vertices are the origin, the point P, and its projection onto the z-axis. As the length of the hypotenuse is ρ and ϕ is the angle the hypotenuse makes with the z-axis leg of the right triangle, the z-coordinate of P (i.e., the height of the triangle) is z=ρcosϕ. The length of the other leg of the right triangle is the distance from P to the z-axis, which is r=ρsinϕ. The distance of the point Q from the origin is the same quantity.
The cyan triangle, shown in both the original 3D coordinate system on the left and in the xy-plane on the right, is the right triangle whose vertices are the origin, the point Q, and its projection onto the x-axis. In the right plot, the distance from Q to the origin, which is the length of hypotenuse of the right triangle, is labeled just as r. As θ is the angle this hypotenuse makes with the x-axis, the x- and y-components of the point Q (which are the same as the x- and y-components of the point P) are given by x=rcosθ and y=rsinθ. Since r=ρsinϕ, these components can be rewritten as
x=ρsinϕcosθ and y=ρsinϕsinθ.
In summary, the formulas for Cartesian coordinates in terms of spherical coordinates are:
x=ρsinϕcosθ
y=ρsinϕsinθ
z=ρcosϕ.
Time
Time is the progression of events from the past to the present into the future. Basically, if a system is unchanging, it is timeless. Time can be considered to be the fourth dimension of reality, used to describe events in three-dimensional space. It is not something we can see, touch, or taste, but we can measure its passage.
Synchronized clocks remain in agreement. Yet we know from Einstein’s special and general relativity that time is relative. It depends on the frame of reference of an observer. This can result in time dilation, where the time between events becomes longer (dilated) the closer one travels to the speed of light. Moving clocks run more slowly than stationary clocks, with the effect becoming more pronounced as the moving clock approaches light speed. Clocks in jets or in orbit record time more slowly than those on Earth, muon particles decay more slowly when falling, and the Michelson-Morley experiment confirmed length contraction and time dilation.
Thinking of past and future brings us to another problem that has foxed scientists and philosophers: why time should have a direction at all. In every day life it’s pretty apparent that it does. If you look at a movie that’s being played backwards, you know it immediately because most things have a distinct time direction attached to them: an arrow of time. For example, eggs can easily turn into omlettes but not the other way around, and milk and coffee mix in your cup but never separate out again.
The most dramatic example is the history of the entire Universe, which, as scientists believe, started with the Big Bang around thirteen billion years ago and has been continually expanding ever since. When we look at that history, which includes our own, it’s pretty clear which way the arrow of time is pointing.
One thing we have neglected to say so far is that Einstein’s theory, which describes the macroscopic world so admirably well, doesn’t work for the microscopic world. To describe the world at atomic and subatomic scales, we need to turn to quantum mechanics, a theory that’s fundamentally different from Einstein’s. Reconciling the two, creating a theory of quantum gravity, is the holy grail of modern physics. When Schrödinger and Heisenberg formulated quantum mechanics in the 1920s, they ignored Einstein’s work and treated time in Newton’s spirit, as an absolute that is ticking away in the background. This already gives us a clue as to why the two theories might be so hard to reconcile. The status of time in quantum mechanics has also created profound problems within the theory itself and has lead to “decades of muddle and subtlety,” as Davies puts it.