A 12-lecture mini-course providing an overview of advanced topics relevant to current research in General Relativity. The course will focus exclusively on classical aspects of gravity and, towards the end, on the quantum behaviour of particles/fields in gravitational backgrounds. The course will run from 20 March to 28 April 2023. Video recordings are available from the ICTS website for the series, https://www.icts.res.in/lectures/AGR2023
Tuesday 18 April 2023
Notes updated to 17.04.2023
The lecture notes have been updated to 17.04.2023 and are available in the shared folder.
The notes are now updated to modify a figure and add a paragraph. The figure I drew in class for focusing of null geodesics in the future/past was a little misleading because my geodesics started from a common point on S. However these are supposed to be geodesics from a set S to a point Q (as I defined earlier) so they can in general start from different points of S. The figure has been modified accordingly. The added para algebraically derives the fact that for a sphere, one set of null geodesics focuses in the future and the other in the past. It's explicitly done for a circle in the x-y plane in 2+1 d spacetime, but the generalisation to a (D-2)-sphere in (D-1)-dim spacetime is obvious.
On page 44, it is shown that a non-prompt null geodesic can be connected by a timelike path. So a particle can reach the same destination as a photon that started from the same destination as that photon in the same amount of time even if it is slower than the photon?
Yes of course that can happen if the photon takes a longer route! In the example you are referring to, we are comparing a null path that "winds" partially around one side of the cylinder (this path can correspond to a photon's trajectory) with a timelike path that takes a shorter route. So there is no surprise. Now one can find an even shorter (in terms of proper time) null route if we allow zigzag paths. This is the famous twin paradox: consider the special case when the initial and final spatial points are the same. Then the proper time for a route that is everywhere null but changes direction half-way and returns to the original spatial point, is zero -- which means it is much less than the proper time for a time-like observer who simply stays at rest at the original point. However if you carefully read the definition of promptness, even such a null route is not prompt. There is simply no prompt path from a point P to a point in J^+(P) but not in \del J^(P).
The term "twin paradox" was coined to give a publicly accessible example of how time progresses differently in different reference frames. As students of Physics, we don't need to use publicly accessible examples for our own understanding. We have facts that can be expressed in figures and equations. The relevant fact here is that a timelike geodesic maximises the proper time between events, while a zig-zag null geodesic (always at 45 degrees, but changing spatial direction so it can reach the same final space-time point) minimises proper time, in fact sets it to zero. Because of the zig-zag, this corresponds to accelerated motion at the points where the direction changes. But the total accumulated proper time is zero. In words this is an extreme twin paradox: one of the twins sits literally in one place, while the other travels literally at the speed of light for some time and then returns at the speed of light. When they meet again, the twin who travelled has aged by exactly zero while the one who stayed home has aged by whatever is the maximum proper time between the events (that depends on how far the moving twin went). This ideal case is obviously extreme but we can make it realistic by asking the moving twin to travel as fast as possible away from earth for a reasonable distance (these numbers are determined by practicality) and then return. Comparing her clock with that of her twin who stayed on earth, a calculable time difference will be seen -- which is the difference between the elapsed time on earth and the proper time taken by the moving twin, which obviously is not zero in the practical case. The only thing needed is to know how to compute proper times of timelike geodesics which - in Minkowski spacetime - is simply done by integrating the geodesic equation along the trajectory.
The notes are now updated to modify a figure and add a paragraph. The figure I drew in class for focusing of null geodesics in the future/past was a little misleading because my geodesics started from a common point on S. However these are supposed to be geodesics from a set S to a point Q (as I defined earlier) so they can in general start from different points of S. The figure has been modified accordingly.
ReplyDeleteThe added para algebraically derives the fact that for a sphere, one set of null geodesics focuses in the future and the other in the past. It's explicitly done for a circle in the x-y plane in 2+1 d spacetime, but the generalisation to a (D-2)-sphere in (D-1)-dim spacetime is obvious.
Sir,
ReplyDeleteWhy Compactness of manifold requires in order to define a notion of Causality?
I did not require compactness in order to define causality. Please check the notes.
DeleteOn page 44, it is shown that a non-prompt null geodesic can be connected by a timelike path. So a particle can reach the same destination as a photon that started from the same destination as that photon in the same amount of time even if it is slower than the photon?
ReplyDeleteYes of course that can happen if the photon takes a longer route! In the example you are referring to, we are comparing a null path that "winds" partially around one side of the cylinder (this path can correspond to a photon's trajectory) with a timelike path that takes a shorter route. So there is no surprise. Now one can find an even shorter (in terms of proper time) null route if we allow zigzag paths. This is the famous twin paradox: consider the special case when the initial and final spatial points are the same. Then the proper time for a route that is everywhere null but changes direction half-way and returns to the original spatial point, is zero -- which means it is much less than the proper time for a time-like observer who simply stays at rest at the original point. However if you carefully read the definition of promptness, even such a null route is not prompt. There is simply no prompt path from a point P to a point in J^+(P) but not in \del J^(P).
DeleteSir, Can you provide literature about how we study the twin paradox in Spacetime(General Relativity)?
ReplyDeleteThe term "twin paradox" was coined to give a publicly accessible example of how time progresses differently in different reference frames. As students of Physics, we don't need to use publicly accessible examples for our own understanding. We have facts that can be expressed in figures and equations. The relevant fact here is that a timelike geodesic maximises the proper time between events, while a zig-zag null geodesic (always at 45 degrees, but changing spatial direction so it can reach the same final space-time point) minimises proper time, in fact sets it to zero. Because of the zig-zag, this corresponds to accelerated motion at the points where the direction changes. But the total accumulated proper time is zero. In words this is an extreme twin paradox: one of the twins sits literally in one place, while the other travels literally at the speed of light for some time and then returns at the speed of light. When they meet again, the twin who travelled has aged by exactly zero while the one who stayed home has aged by whatever is the maximum proper time between the events (that depends on how far the moving twin went). This ideal case is obviously extreme but we can make it realistic by asking the moving twin to travel as fast as possible away from earth for a reasonable distance (these numbers are determined by practicality) and then return. Comparing her clock with that of her twin who stayed on earth, a calculable time difference will be seen -- which is the difference between the elapsed time on earth and the proper time taken by the moving twin, which obviously is not zero in the practical case. The only thing needed is to know how to compute proper times of timelike geodesics which - in Minkowski spacetime - is simply done by integrating the geodesic equation along the trajectory.
ReplyDelete