1. Relativity and Wormholes
By Joaquin P Noyola
PHYS 4117 – Learning by Seminar; Spring 2006

2. Outline Brief Introduction to Relativity What are wormholes?
Several wormhole systems
Space Travel, Really?
Current Reasearch

3. Inertial Frames
…are coordinate systems which obey Newton’s laws and are not accelerating.
…with respect to an observer…and from now on, when I say a frame I’ll be referring to an inertial frame.
static (x, y, z) coordinate system
moving (x’, y’, z’) coordinate system

4. Special Relativity
Time is the fourth dimension and is measure in meters of light travel, not seconds.
Distance is now measured by the interval
Now distance incorporates not only the change in spatial position (x, y, z), but also the change in time from one place to another within a given frame. From one frame to another, spatial coordinates of an event change as well as the temporal coordinate (i.e. the time), but when we compute the interval, its value remains the same.
which is invariant from one inertial frame to another.

5. General Relativity
The interval is no longer valid due to acceleration, and Δs2  ds2
We call the resulting equation the metric.
Now a change in spacetime position, the interval, becomes vanishingly small. Where f, g, h, and j are function of the gravitational field Φ, explain f, g, h, and j w/ space curvature
Comparing it to the interval in special relativity

6. Even light follows this curved trajectory
Embedded diagram of a potential well
Gravitational fields cause spacetime to bend, creating a potential well
Path of lowest energy
Because of this bending, the object’s “straight-line” trajectory becomes curved to account for the curvature of space.
Even light follows this curved trajectory

7. Tidal Forces
Forces caused by the difference in a gravitational field from one point to another.
Fhead = 10g
Tidal force = 90g
Since this force appears to pull in both directions (like tension on a rope), it will tear you apart if it is big enough. Tidal forces might also squeeze you into a funnel-like shape because field lines are not parallel but rather converge towards their source.
Ffeet = 100g

8. Wormholes

9. What are Wormholes? They are tunnels through spacetime

10. What are Wormholes?
Wormholes are spheres, although represented as 3D tunnels in embedded diagrams.
They are caused by gravitational forces described by a metric.
Great variety! static throat, dynamic throat, spinning, not spinning, with little or great tidal forces, etc.
,more about that in a minute

11. Wormholes are not Black holes!
Wormholes are two-mouth tunnels, but black holes are a one way trip.
As seen on the figure, the wormhole connects two parts of the universe through its throat, while the black hole just pulls everything towards its singularity

12. What do they look like?
How they might look from far away. Indistinguishable from black holes. The black spot you see in the middle is just that, a black spot not a black hole and I’ll explain it in the next slide. The picture clearly illustrates what is called gravitational lensing, which is just the bending of light by a gravitational field as explained earlier. We can also appreciate the boundary between the light coming from the background (behind the wormhole) and the light coming out of the wormhole from the other side.
We know that in an embedded diagram a wormhole looks like any other 3D tunnel, and so it is easy to visualize how the black spot forms. But recall that a wormhole is not a 3D tunnel, the mouth is a sphere and the throat is in hyperspace (i.e. as 4 spatial dimensions). Therefore, the wormhole looks just same from any side you look at it; no matter where you look at from, you will still see the same black spot!

13. What causes the black spot?
It is caused by the bending of light towards the wall of the throat as it passes through the wormhole.
Remember that a wormhole is also a bending of space so for light, the path of lowest energy is towards the walls of the throat, the walls act as a potential well. The first figure illustrates how incoming light rays bend toward the walls of the throat, causing the central axis to lack any light, forming the black spot. Figure two shows the black spot (as red circles) at different locations in the throat. As we can see, the black spot becomes smaller as we approach the other side; making it seem as if it was getting further away from us as we approach it!

14. What types of wormholes are there?
The Schwartzchild wormhole
The Morris-Thorne wormhole
The Visser wormhole

15. The Schwartzchild Wormhole
It is actually a Black Hole-Wormhole-White Hole system.
ds2 = - ( 1 - rs / r ) dt2 + ( 1 - rs / r )-1 dr2 + r2 dΩ2
Where rs = 2 G M / c2
It has tremendous tidal forces in its throat and outside its mouth due to the black hole. The throat is dynamic.

17. The Morris-Thorne Wormhole
Huge tidal forces at the mouth, but no horizons.
Less mass required, but still planet-size chunks.
Throat is fairly stable compared to the Schwartzschild wormhole
ds2 = - e2Φ(r)dt2 + dr2 + R(r)dΩ2

18. The Visser Wormhole
Require cutting two similar holes in space-time and sewing the edges together.
It uses a lot of negative mass, therefore it has smaller tidal forces.
Than the Morris-Thorne

19. Space Travel, Really?

20. Basic Idea
We want to create a traversable and stable wormhole for use in space travel.
We could create new wormholes or just enlarge existing wormholes

21. Problems - Engineering
All wormholes require exotic (negative) matter to keep their throat open.
It takes to much energy to just open them, leave alone maintain them.
Fine tuning problems; too much precision needed.
At the moment it is beyond our reach, but there is no telling about whether future generations will do it.
1 in 10^30 for proton, 1 in 10^60 for humans

22. Current Research
Mostly theoretical, no significant experimental approach has been taken directly on the subject.
Exotic matter experiments have shown that this material is obtainable.

23. Conclusion
We have not seen wormholes directly, but our known laws of physics do not ban their existence. It is not outrageous to believe they exist.
Regarding space travel, maybe only future generations will be capable of telling whether it is possible or not.

24. And that is all!