Exploring the Geometry of
Spacetime via Geodesic Curves
Term paper
MTH201 Course
2020
Kaustav Sen, Arastu Pandey,
Ira Sharma, Prajakta Mane
Euclidean and Non-Euclidean Geometry:
Euclidean Geometry: Returning to the initial point of the journey would take 4 right-angle turns. Triangles look like triangles, and squares, like squares.
Non-Euclidean Geometry: Returning to the initial point of the journey would take 3 right-angle turns. Sum of all angles of a triangle is not 180o.
Bonus: The sides of the triangle are not even ‘straight’ (Or are they?).
The Geodesic:
The Geodesics are the ‘straight lines’ on the curved surface.
It is the shortest curve joining any two points on the surface.
They are the curves for which the tangent vectors stay
tangent when parallel transported along them.
Eg: A circle is a geodesic on the great sphere.
According to the above definition, for the ant living on the sphere, the curve 1 is a geodesic and the curve 2 isn’t. (For a third person (or a third ant), who can perceive the third dimension as well, neither of them are geodesics)
Meet our friend, the bug!
A simulation showing a bug travelling in a straight line according to his perspective, tracing a geodesic in 3D. (Also observe how many right angles it takes for the bug to reach the point of origin.)
Now let us move from 2D bug to the 3D animals, particularly, us humans. The way the straight traced by the ant in 2D is actually a geodesic in 3D, the same way can we think of the straight lines we humans travel in 3D to be geodesics in 4D?
Euler Lagrange Equation
Thank God for symmetry!!!
In the case Q=0 where P and R are explicit functions of v only, then :
For a surface of revolution having parametrization �x(u, v) = (f(v) cos u, f(v) sin u, g(v)), any meridian is a geodesic and a parallel is a geodesic if and only if f’(vO) = 0.
Hyperboloid
Meridian
Parallel
Other curves
c
Paraboloid
Meridian
Other curves
c
Funnel
Meridian
Other curves
c
The Space Time Curvature:
Spacetime is a four-dimensional continuum combining the familiar three dimensions of space with the dimension of time.
Massive objects warp the spacetime around them.
In the Einstein’s theory, the gravity is not a force but the effect the warping of spacetime around the massive bodies has on other objects.
The rubber sheet analogy:
Consider space-time as a rubber sheet that can be deformed. In any region distant from massive cosmic objects, space-time is uncurved—that is, the rubber sheet is absolutely flat. If one were to send out a ray of light or a test body, both the ray and the body would travel in perfectly straight lines, like a child’s marble rolling across the rubber sheet. However, the presence of a massive body curves space-time, as if a bowling ball were placed on the rubber sheet to create a cuplike depression. In the analogy, a marble placed near the depression rolls down the slope toward the bowling ball as if pulled by a force.
Home Project exploring the Rubber sheet analogy:
Bedsheet representing the uncurved
or flat spacetime without any mass.
Bedsheet with a mortar placed at the center showing how the massive bodies warp spacetime.
The path of the ball around the mortar crudely representing the geodesic.
Effects of Space-time Curvature:
The path of a geodesic is highly curved near a very massive body, resulting in decreasing its speed.
In space, light is bent in gravitational fields and travels along geodesics. Due to this, time passes slower in regions of high gravitational potential than it does in regions of low gravitational potential. This effect is called as Time Dilation.
Bending of waves in a gravitational field. Due to gravity, time passes more slowly at the bottom than at the top, causing the wave-fronts (shown in black) to gradually bend downward in order to keep the speed constant. The green arrow shows the direction of the apparent "gravitational attraction".
2. Black holes:
They’re the massive bodies that warp the spacetime such that even light will travel on the highly curved paths. The curvature becomes so high near them that the light bends into the path that all terminates at the blackhole’s singularity. In the rubber sheet analogy, it is a depression created by a tiny massive object so steep that nothing can escape it.
3. Wormholes:
A wormhole is considered to be a tunnel through which two distant regions of spacetime can be connected. Einstein proposed the existence of "bridges" through space-time which connect two different points in space-time, theoretically creating a shortcut that could reduce travel time and distance. The shortcuts are called Einstein-Rosen bridges, or wormholes.