1. Ayan Banerjee
Jadavpur University

2. Space-time
 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. What is Wormhole ?
 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.
 Traversable wormholes would allow travel in both directions
from one part of the universe to another part of that same
universe (called intra-universe) very quickly or would allow travel
from one universe to another (called inter-universe) .
 Traveling through a wormhole takes less time than traveling
between the same distance 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.

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

6. History of the Wormhole
 Einstein introduced his general theory of relativity, in
1916, which still remains the standard model for
gravitation. In the year1935, Einstein and his collaborator
Nathan Rosen investigated the possibility of obtaining an
atomistic theory of matter and electricity. They wanted to
know how general relativity would treat individual
particles, and hoped somehow it can account for quantum
phenomena. This phenomenon known as an "Einstein-
Rosen bridge".

7.  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.

8. 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.

9.  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.

10.  As is the throat radius, so =
 Traversability criteria: For traversable wormhole, there
should not be any horizons present i.e. must be finite
everywhere.
Einstein Field Equations
Einstein field equation, in an orthonormal
frame, (with c = G = 1) we obtain the following
stress-energy scenario.

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

12. 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.

14. Research Zone
 Stabilizing a wormhole with exotic matter.
 Minimize Exotic matter.

15. Thank you