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3. Ellis-Bronnikov drainhole

Ellis-Bronnikov drainhole

The Ellis drainhole is the oldest example of a traversable wormhole. It's a class of spherically symmetric static spacetimes parametrized by $a$, the radius of the throat.

1. History

2. Topology

The Ellis-Bronnikov drainhole has the topology $\mathbb{R}^2 \times S^2$.

3. Metrics and coordinates

Schwarzschild coordinates

Proper time coordinates

$$ds^2 = -dt^2 + dl^2 + (l^2 + a^2) (d\theta^2 + \sin^2 \theta d\varphi^2)$$

4. Tensor quantities

In Schwarzschild coordinates

In proper time coordinates

5. Symmetries

6. Stress-energy tensor

7. Curves

In Schwarzschild coordinates

The geodesic equation

In proper time coordinates

The geodesic equation

8. Equations

In Schwarzschild coordinates

In proper time coordinates

9. Causal structure

10. Asymptotic structure

11. Energy conditions

12. Limits and related spacetimes

13. Misc.

Bibliography