Beyond the Black Hole Horizon - md.spacegrant.org · •We focus on the curvature of...
Transcript of Beyond the Black Hole Horizon - md.spacegrant.org · •We focus on the curvature of...
![Page 1: Beyond the Black Hole Horizon - md.spacegrant.org · •We focus on the curvature of spacetimeinside the black hole horizon •Mathematicians have proved that this curvature is characterized](https://reader031.fdocuments.us/reader031/viewer/2022042005/5e6f741358ba2449ff72a89d/html5/thumbnails/1.jpg)
Beyond the Black Hole Horizon
Max Coplan, Dr. James Overduin
Towson University
![Page 2: Beyond the Black Hole Horizon - md.spacegrant.org · •We focus on the curvature of spacetimeinside the black hole horizon •Mathematicians have proved that this curvature is characterized](https://reader031.fdocuments.us/reader031/viewer/2022042005/5e6f741358ba2449ff72a89d/html5/thumbnails/2.jpg)
Black Holes
•Are real!
![Page 3: Beyond the Black Hole Horizon - md.spacegrant.org · •We focus on the curvature of spacetimeinside the black hole horizon •Mathematicians have proved that this curvature is characterized](https://reader031.fdocuments.us/reader031/viewer/2022042005/5e6f741358ba2449ff72a89d/html5/thumbnails/3.jpg)
Nobel Prize
![Page 4: Beyond the Black Hole Horizon - md.spacegrant.org · •We focus on the curvature of spacetimeinside the black hole horizon •Mathematicians have proved that this curvature is characterized](https://reader031.fdocuments.us/reader031/viewer/2022042005/5e6f741358ba2449ff72a89d/html5/thumbnails/4.jpg)
Inside the Horizon
• Black-hole interiors are the only place in nature we can never observe — by definition!
• The only way to “see” beyond the event horizon is through mathematics
• Existing visualizations in the realistic case (mass and spin but no electric charge) use special coordinates known as Boyer-Lindquist coordinates
• This brings out the singularity (ring) and horizon structure (spherical inner horizon, ellipsoidal ergosphere):
![Page 5: Beyond the Black Hole Horizon - md.spacegrant.org · •We focus on the curvature of spacetimeinside the black hole horizon •Mathematicians have proved that this curvature is characterized](https://reader031.fdocuments.us/reader031/viewer/2022042005/5e6f741358ba2449ff72a89d/html5/thumbnails/5.jpg)
Invariants
• Simplicity of the preceding picture is due to the choice in coordinates
• But coordinate-dependent quantities can be misleading! Consider Greenland vs. U.S.A. on different projections
• To be sure of drawing true conclusions, we need to express results in terms of invariants: quantities whose value is the same regardless of coordinates
![Page 6: Beyond the Black Hole Horizon - md.spacegrant.org · •We focus on the curvature of spacetimeinside the black hole horizon •Mathematicians have proved that this curvature is characterized](https://reader031.fdocuments.us/reader031/viewer/2022042005/5e6f741358ba2449ff72a89d/html5/thumbnails/6.jpg)
Objective
• We focus on the curvature of spacetime inside the black hole horizon
• Mathematicians have proved that this curvature is characterized by at most seventeen invariants for the most general possible black holes (those with mass, spin, and charge)
• We use a powerful symbolic computational tool to calculate and plot all seventeen of these curvature invariants for the first time
• The results differ dramatically from the simple picture above!
![Page 7: Beyond the Black Hole Horizon - md.spacegrant.org · •We focus on the curvature of spacetimeinside the black hole horizon •Mathematicians have proved that this curvature is characterized](https://reader031.fdocuments.us/reader031/viewer/2022042005/5e6f741358ba2449ff72a89d/html5/thumbnails/7.jpg)
General Relativity
• In GR (Einstein, 1916), gravity = shape of spacetime• Shape of spacetime is described mathematically by
the metric tensor, a generalization of Pythagoras' theorem
• All physical quantities are then constructed out of the metric and its first and second derivatives with respect to space and time:
![Page 8: Beyond the Black Hole Horizon - md.spacegrant.org · •We focus on the curvature of spacetimeinside the black hole horizon •Mathematicians have proved that this curvature is characterized](https://reader031.fdocuments.us/reader031/viewer/2022042005/5e6f741358ba2449ff72a89d/html5/thumbnails/8.jpg)
Implementation
• Two main challenges:• Translating tensor language of general relativity to computational syntax
(easy)
• Simplifying the resulting expressions (hard)
![Page 9: Beyond the Black Hole Horizon - md.spacegrant.org · •We focus on the curvature of spacetimeinside the black hole horizon •Mathematicians have proved that this curvature is characterized](https://reader031.fdocuments.us/reader031/viewer/2022042005/5e6f741358ba2449ff72a89d/html5/thumbnails/9.jpg)
Translation
• Our inspiration is a Mathematica code in James Hartle's Gravity (Addison-Wesley, 2003)
• Given a metric, our code computes the inverse metric, Christoffel symbols, Riemann and Ricci Tensors
![Page 10: Beyond the Black Hole Horizon - md.spacegrant.org · •We focus on the curvature of spacetimeinside the black hole horizon •Mathematicians have proved that this curvature is characterized](https://reader031.fdocuments.us/reader031/viewer/2022042005/5e6f741358ba2449ff72a89d/html5/thumbnails/10.jpg)
Issue: Pages and pages of results
![Page 11: Beyond the Black Hole Horizon - md.spacegrant.org · •We focus on the curvature of spacetimeinside the black hole horizon •Mathematicians have proved that this curvature is characterized](https://reader031.fdocuments.us/reader031/viewer/2022042005/5e6f741358ba2449ff72a89d/html5/thumbnails/11.jpg)
Simplification
• The more difficult challenge is to simplify the resulting expressions! For example, messy combinations of sin(%) and cos(2%) terms in *+,above --- we would like to express all results in terms of powers of a single trigonometric function
• To do this we constructed a "trig simplifier":
![Page 12: Beyond the Black Hole Horizon - md.spacegrant.org · •We focus on the curvature of spacetimeinside the black hole horizon •Mathematicians have proved that this curvature is characterized](https://reader031.fdocuments.us/reader031/viewer/2022042005/5e6f741358ba2449ff72a89d/html5/thumbnails/12.jpg)
Simplification resultsA known result !
![Page 13: Beyond the Black Hole Horizon - md.spacegrant.org · •We focus on the curvature of spacetimeinside the black hole horizon •Mathematicians have proved that this curvature is characterized](https://reader031.fdocuments.us/reader031/viewer/2022042005/5e6f741358ba2449ff72a89d/html5/thumbnails/13.jpg)
Example: Weyl Invariants
![Page 14: Beyond the Black Hole Horizon - md.spacegrant.org · •We focus on the curvature of spacetimeinside the black hole horizon •Mathematicians have proved that this curvature is characterized](https://reader031.fdocuments.us/reader031/viewer/2022042005/5e6f741358ba2449ff72a89d/html5/thumbnails/14.jpg)
![Page 15: Beyond the Black Hole Horizon - md.spacegrant.org · •We focus on the curvature of spacetimeinside the black hole horizon •Mathematicians have proved that this curvature is characterized](https://reader031.fdocuments.us/reader031/viewer/2022042005/5e6f741358ba2449ff72a89d/html5/thumbnails/15.jpg)
Physical Implications
![Page 16: Beyond the Black Hole Horizon - md.spacegrant.org · •We focus on the curvature of spacetimeinside the black hole horizon •Mathematicians have proved that this curvature is characterized](https://reader031.fdocuments.us/reader031/viewer/2022042005/5e6f741358ba2449ff72a89d/html5/thumbnails/16.jpg)
Conclusions
• Curvature inside real black holes is (1) both positive and negative, (2) definitely not constant, and (3) ridiculously complex!
• The interiors of black holes are much more complex and beautiful than we ever imagined
• The Weyl invariants are especially contorted. Interesting since these describe curvature associated with gravitational waves
• Spacetime is mostly warped ("gravito-electric" dominated) in regions of positive curvature, but mostly twisted/dragged ("gravito-magnetic" dominated) in regions of negative curvature
• This may explain certain puzzles in astrophysics
• More remains to be understood about the physical importance of these invariants
• But before that, it is necessary to find the invariants, and that is what we have done here!