User:Rschwieb/Physics notes
The purpose of this page is to evolve how to think about various physical and mathematical physics concepts. Please keep comments in the comments section, as I would like to keep the main notes in my own words.
Notes
Mathematics
- Use of manifolds
Manifolds are "warped up Euclidean space". Since the old Euclidean global coordinates no longer work, the manifold is covered by an atlas whose coordinates can be smoothly translated. In this way, the manifold is covered by "global coordinates". Notions of "straight lines" and "parallel" are modified for the warped space.
- Tangent space
Because of the warped nature of manifolds, it becomes necessary to "flatten" the surface at points (the tangent space at the point), and do regular calculus in the small on these flat pieces. The infinitesimal contributions are then gathered together to talk about lengths of curves on the surface.
- Tensor/tensor field
An individual abstract tensor can be thought of as a generalized vector. It is an object independent of any basis. A tensor field can be thought of as a generalized vector field. The field itself is an association of tensors to the points of a manifold. Using vectors, for example, we can think of forces at each point as "vectors". When the quantity described is more complicated, as in the case of stress, then it may take more than a simple vector to describe the quantity. The metric tensor is a little different, as it is attaching a protractor to each point on the manifold, in order to measure angles.
- Covariance/contravariance
I'm not entirely sure yet if these are just artifacts of the way we do algebra, or if they are physically relevant.
- Metric tensor
As described before, a "protractor" at every point in the manifold, allowing for the measurement of tiny angles and lengths. This is the correct connection between local frames and their straight rulers and geodesics in warped space. We work in the small and pretend everything is flat, but then we have to take into account that the large is bent. Macdonald describes that tiny measured distances dictate the metric at a point, which dictate the Christoffel symbols in the large, which dictate the geodesics in the large warped space.
- Curvature
There are a few types of curvature which quantify how much a surface is curved at a point. Some are intrinsic to the surface (measurable by surface dwellers) and some are extrinsic, depending on a larger space. Containing the surface.
Physics
- Inertial frame
Hard to explain. Roughly, we would like to say "a reference frame in which an accellerometer reads zero". This is compatible with being in free space with no gravity. It is also true for a free-falling reference frame. In both cases, inertial objects move in straight lines. However, this would not include standing on the surface of the earth, as the accellerometer registers a sag. Maybe "a reference frame in which an accellerometer is motionless"?
- Postulates of general relativity
Roughly: inertial objects in inertial frames move with constant velocity in ("straight lines?"), and the speed of light is a constant observed in all inertial frames. The second postulate impels us to modify the Pythagorean theorem for spacetime, giving us the 3+1 metric signature.
- Einstein's equations
Einstein determined that the presense of energy/matter at a location has a direct connection with the metric at the location. The equations, derived from the postulates, will tell you the metric field for the region, if you can tell it how the mass/energy is distributed. (Conversely dictating a metric tensor should determine the mass/energy distribution?) Determination of the metric determines geodesics and Christoffel symbols. The equations are differential equations, so that a solution consists of a collection of functions.