Spacetime surgery - why are there unglueable points?
Answers
Answered by
0
In The time travel paradox by S. Krasnikov (2002), Deutsch-Politzer spacetime is constructed by making two cuts and rejoining the manifold by gluing opposite "banks" of the cuts... omitting the "corner" points.
See figure below - cut along dashed lines; corner points are the circles at the ends of the dashed lines; the identification is above upper line to below lower line.
Krasnikov then says "The corner points cannot cannot be glued back into the spacetime..."
See figure below - cut along dashed lines; corner points are the circles at the ends of the dashed lines; the identification is above upper line to below lower line.
Krasnikov then says "The corner points cannot cannot be glued back into the spacetime..."
Answered by
4
HEY MATE ⭐⭐⭐
HERE'S THE ANSWER ✌
_______________
⬇⬇⬇⬇⬇
Physically, the question of what happens precisely at the corner points isn't especially meaningful. Instead, you could think about phenomena in a neighborhood of the points (e.g. flux through a circle or tube enclosing the singularity), and use that to characterize the point itself.
By definition, a manifold should be locally Euclidean. Any open neighborhood (with the metric topology) containing those endpoints will not look Euclidean, because of 'creases' that develop around them (i.e. curvature singularities).
These singularities are apparent from the fact that a small closed circular loop around a corner point has twice the normal circumference. Alternatively, using the Lorentzian metric structure (or causal structure), the singularity can be deduced from observing that a closed loop around a corner point must have at least two components during which time is increasing, instead of one.
To determine whether the singular corner points can be glued back into the spacetime, one can ask whether the sharply concentrated intrinsic curvature at the end-points can be redistributed so that the manifold is well defined everywhere (in particular, if we sent a massless point-like particle directly at one of the corner points, we should have a mathematically consistent, physically motivated rule to determine what the particle does after it reaches the corner point). If we require spacetime to be asymptotically flat, i.e. Minkowskian far from the altered region, then on physical grounds it's not hard to see that the corner singularities cannot be smoothed out without also removing the handle entirely.
The reason for this is essentially the discontinuity in behavior of null geodesics near the corners: whether the trajectory winds nn times around the handle before it continues to future null infinity, or n+1n+1 times.
Removing this discontinuity would require either closing off the ends of the handle, or shrinking all of spacetime to a torus so that all trajectories are periodic in time (up to a boundary), and there is no sharp transition between trajectories with different winding numbers around the handle.
✔✔✔✔✔✔✔
__________________
HOPE IT HELPS ☺☺☺☺☺☺
HERE'S THE ANSWER ✌
_______________
⬇⬇⬇⬇⬇
Physically, the question of what happens precisely at the corner points isn't especially meaningful. Instead, you could think about phenomena in a neighborhood of the points (e.g. flux through a circle or tube enclosing the singularity), and use that to characterize the point itself.
By definition, a manifold should be locally Euclidean. Any open neighborhood (with the metric topology) containing those endpoints will not look Euclidean, because of 'creases' that develop around them (i.e. curvature singularities).
These singularities are apparent from the fact that a small closed circular loop around a corner point has twice the normal circumference. Alternatively, using the Lorentzian metric structure (or causal structure), the singularity can be deduced from observing that a closed loop around a corner point must have at least two components during which time is increasing, instead of one.
To determine whether the singular corner points can be glued back into the spacetime, one can ask whether the sharply concentrated intrinsic curvature at the end-points can be redistributed so that the manifold is well defined everywhere (in particular, if we sent a massless point-like particle directly at one of the corner points, we should have a mathematically consistent, physically motivated rule to determine what the particle does after it reaches the corner point). If we require spacetime to be asymptotically flat, i.e. Minkowskian far from the altered region, then on physical grounds it's not hard to see that the corner singularities cannot be smoothed out without also removing the handle entirely.
The reason for this is essentially the discontinuity in behavior of null geodesics near the corners: whether the trajectory winds nn times around the handle before it continues to future null infinity, or n+1n+1 times.
Removing this discontinuity would require either closing off the ends of the handle, or shrinking all of spacetime to a torus so that all trajectories are periodic in time (up to a boundary), and there is no sharp transition between trajectories with different winding numbers around the handle.
✔✔✔✔✔✔✔
__________________
HOPE IT HELPS ☺☺☺☺☺☺
Similar questions