Wormholes and its classification.

1. Classification
of
Wormholes

2.  At the beginning of the 20th century, Albert Einstein
revolutionized the idea that space and time do not exist as
separate , rather it is a combination of space and time to a
single abstract universe i.e. space, consisting of three
dimensions and time is said to have only one dimension.
Einstein said we can think of these space-time as fabric.

3. In physics, a wormhole is a hypothetical topological
feature of space-time that is essentially a "shortcut"
through space and time. A wormhole has at least two
mouths which are connected to a single throat. If the
wormhole is traversable, matter can 'travel' from one
mouth to the other by passing through the throat.
Traveling through a wormhole takes less time than
traveling between the same starting and ending
regions in normal space .

4. The name "wormhole" comes
from the following analogy used
to explain the phenomenon:
imagine that the universe is the
skin of an apple, and a worm is
travelling over its surface, then
distance from one side of the
apple to the other is equal to half of
the apple's circumference. The worm has to travel is
considerably less, if it burrows a wormhole directly
through the apple, instead of staying on the apple's
surface.

6. History of
Wormhole
 Einstein first proposed
wormholes in 1935. He wrote
a paper with Nathan Rosen in
which they showed that
general relativity allowed for
what they called “bridges.”
They theorized that there
could be places where
space/time is folded that
allowed transfer of matter
from one point to another in
the universe, specifically in
black holes.

7. Weight added to a piece of
fabric will cause sag
towards the center.
Super massive black holes
cause a tear in space-time
and this tear is the
wormhole.
These two regions would
meet and form a
wormhole connection.

8. Types of Wormholes
Lorentzan wormholes (general relativity)
Metric with signature (-1, +1, +1, +1)
Mainly studied by experts in Einstein gravity
Euclidean wormholes (particle physics)
Metric with signature (+1, +1, +1, +1)
Mainly studied in quantum gravity theory

9. However, in 1962 John A. Wheeler and Robert W.
Fuller showed that Einstein-Rosen bridge space-time
structure was highly unstable in field-free space, and
that it will pinch off too quickly before a single photon
could be transmitted through it. This work lead to find
different kinds of wormholes. The possibility of
traversable wormholes in general relativity gathered
pace by the publication of a paper in 1987,
“Wormholes in space-time and their use for
Interstellar travel: A tool for teaching general
relativity”, by Michael Morris and Kip Thorne which is
known to us a Morris-Thorne wormhole.

10. “Morris-Thorne
Wormholes”
Simplified their analysis by first assuming the
existence of a suitably well-behaved geometry.
 The solution must every where obey the Einstein
field equations. It is assumed that GR is correct.
 The matter and fields that generate the wormhole’s
space-time curvature must have a physically
reasonable stress-energy tensor or not.

11. The metric for a static spherically symmetric space-time
is given by
b(r) called the “shape” function since it determines
the spatial shape of the wormhole.

Φ(r) called the “red shift” function since it
determines the gravitational red shift.

12.  As is the throat radius, so =
Traversability criteria: For traversable wormhole,
there should not be any horizons present i.e. must
be finite everywhere.

13. Einstein Field Equations
Einstein field equation, in an orthonormal
frame, (with c = G = 1) we obtain the following stress-energy
scenario which are

14. Where is the energy density, is the radial
pressure, and is the lateral pressure measured in
the orthogonal direction to the radial direction.

15. Stress-Energy at the Throat
at the throat, the tension exceeds the total
mass-energy density.
Materials with the property is called, “exotic”.
fundamental condition in wormhole physics is the violation
of the NEC. i.e. . Matter that violates the NEC
is denoted as exotic matter.

16. Thank
you