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3-20-2015

The , its invariants, and their use in the classification of

Jesse Hicks Utah State University

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Recommended Citation Hicks, Jesse, "The , its invariants, and their use in the classification of spacetimes" (2015). Presentations and Publications. Paper 8. https://digitalcommons.usu.edu/dg_pres/8

This Presentation is brought to you for free and open access by DigitalCommons@USU. It has been accepted for inclusion in Presentations and Publications by an authorized administrator of DigitalCommons@USU. For more information, please [email protected]. Introduction A Little Background The Idea The Example Conclusion

The Riemann curvature tensor, its invariants, and their use in the classification of spacetimes

Jesse Hicks

Utah State University

March 20, 2015

Jesse Hicks Utah State University The Riemann curvature tensor, its invariants, and their use in the classification of spacetimes Introduction A Little Background The Idea The Example Conclusion

“ tells how to move; matter tells spacetime how to ” - John Wheeler The Einstein field equations: 1 R − g R + Λg = κT αβ 2 αβ αβ αβ

Solutions are components gαβ of tensor g

Equivalence Problem: g1 ≡ g2?

Jesse Hicks Utah State University The Riemann curvature tensor, its invariants, and their use in the classification of spacetimes Introduction A Little Background The Idea The Example Conclusion

Let M be an n-dimensional differentiable .

Definition

A field g on M is a mapping p 7→ gp, where p ∈ M and gp : TpM × TpM → R is a symmetric, non-degenerate, on the to p on M.

Jesse Hicks Utah State University The Riemann curvature tensor, its invariants, and their use in the classification of spacetimes Introduction A Little Background The Idea The Example Conclusion

Given a {Xα (p)} of TpM, g(p) has components

g(p)αβ = gp(Xα (p), Xβ (p))

In a neighborhood U of a point,

gαβ

are C ∞ functions of coordinates

Definition A spacetime is a 4-dimensional manifold with metric tensor having signature (3, 1), i.e. a Lorentzian signature.

Jesse Hicks Utah State University The Riemann curvature tensor, its invariants, and their use in the classification of spacetimes Introduction A Little Background The Idea The Example Conclusion

We refer to gαβ as the metric.

The metric defines a unique

g λµ Γ λ = (∂ g + ∂ g − ∂ g ) α β 2 α βµ β αµ µ αβ

λ = Γβ α

λ Γα β allows us to define a , called the

...which returns a new tensor when applied to a given tensor.

Jesse Hicks Utah State University The Riemann curvature tensor, its invariants, and their use in the classification of spacetimes Introduction A Little Background The Idea The Example Conclusion

On a 4-dimensional manifold M

X : C ∞(M, R) → C ∞(M, R)

1 2 3 4 X = X (x)∂x1 + X (x)∂x2 + X (x)∂x3 + X (x)∂x4 The components of the covariant derivative of X are

ν ν ν β X ;α = ∂αX + Γα βX

When performing repeated covariant differentiation and taking the difference, we get

ν ν ν ν ν λ ν λ µ X ;α;β − X ;β;α = (∂βΓµ α − ∂αΓµ β + Γλ βΓµ α − Γλ αΓµ β)X

Jesse Hicks Utah State University The Riemann curvature tensor, its invariants, and their use in the classification of spacetimes Introduction A Little Background The Idea The Example Conclusion

Define

ν ν ν ν λ ν λ Rµ αβ = ∂βΓµ α − ∂αΓµ β + Γλ βΓµ α − Γλ αΓµ β

ν Rµ αβ are the components of the Riemann curvature tensor!

ν Rµ αβ 6= 0 indicates curvature.

ν If all Rµ αβ = 0, the spacetime is flat.

Jesse Hicks Utah State University The Riemann curvature tensor, its invariants, and their use in the classification of spacetimes Introduction A Little Background The Idea The Example Conclusion

ν The Rµ αβ change when changing coordinates,

BUT certain contractions and products and combinations of its contractions do NOT!

Called Riemann invariants Example

ν µα From Rµα := Rµ αν is created the Ricci R := g Rµα

Jesse Hicks Utah State University The Riemann curvature tensor, its invariants, and their use in the classification of spacetimes Introduction A Little Background The Idea The Example Conclusion

From J. Carminati and R. Mclenaghan’s paper “Algebraic invariants of the Riemann tensor in a four-dimensional Lorentzian space”

Jesse Hicks Utah State University The Riemann curvature tensor, its invariants, and their use in the classification of spacetimes Introduction A Little Background The Idea The Example Conclusion

continued...

Jesse Hicks Utah State University The Riemann curvature tensor, its invariants, and their use in the classification of spacetimes Introduction A Little Background The Idea The Example Conclusion

g o ? / g˜

  ν ˜ ν Rµ αβ Rµ αβ

  invariants o ? / invariants

THE KEY: Obvious when two metrics are NOT equivalent

Jesse Hicks Utah State University The Riemann curvature tensor, its invariants, and their use in the classification of spacetimes Introduction A Little Background The Idea The Example Conclusion

In Einstein , A. Z. Petrov gave a “complete” classification of spacetimes with according to .

In Exact Solutions of Einstein’s Equations, Stephani et al compiled many famous spacetimes.

The G¨odelmetric SHOULD be in Petrov’s classification, as it’s a with 5.

Jesse Hicks Utah State University The Riemann curvature tensor, its invariants, and their use in the classification of spacetimes Introduction A Little Background The Idea The Example Conclusion

The G¨odelMetric g = −a2 dt dt − a2ex dt dz + a2 dx dx + a2 dy dy − a2ex dz dt − 1/2 a2e2 x dz dz

g = 1/4 a2 dr dr + 1/4 a2 dr ds + 3/4 a2 dr du + 1/4 a2(2 es/2+u/2+v/2+r/2 + 3) dr dv + 1/4 a2 ds dr + 1/4 a2 ds ds − 1/4 a2 ds du − 1/4 a2(1 + 2 es/2+u/2+v/2+r/2) ds dv + 3/4 a2 du dr − 1/4 a2 du ds + 1/4 a2 du du − 1/4 a2(2 es/2+u/2+v/2+r/2 − 1) du dv + 1/4 a2(2 es/2+u/2+v/2+r/2 + 3) dv dr − 1/4 a2(1 + 2 es/2+u/2+v/2+r/2) dv ds − 1/4 a2(2 es/2+u/2+v/2+r/2 − 1) dv du − 1/4 a2(4 es/2+u/2+v/2+r/2 − 1 + 2 es+u+v+r ) dv dv

−r+s+u+v r+s+u+v r−s+u+v Using t = 2 , x = 2 , y = 2 , z = v

Jesse Hicks Utah State University The Riemann curvature tensor, its invariants, and their use in the classification of spacetimes Introduction A Little Background The Idea The Example Conclusion

Here G¨odelis again:

g = −a2 dt dt − a2ex dt dz + a2 dx dx + a2 dy dy − a2ex dz dt − 1/2 a2e2 x dz dz

It’s been shown that (33.17) with  = −1 is the ONLY metric in Petrov with equivalent Killing vectors:

1 4 −2 x3 2 2 −x3 2 4 3 3 g˜ = 2 dx dx + k22 e dx dx − e dx dx + k22 dx dx + 2 dx4 dx1 − e−x3 dx4 dx2

Jesse Hicks Utah State University The Riemann curvature tensor, its invariants, and their use in the classification of spacetimes Introduction A Little Background The Idea The Example Conclusion

Compute their respective Riemann invariants:

G¨odel,we have R = −1 and

3 r = m1 = 0 1 16a4 1 3 m = r = 2 8 2 64a6 96a 1 21 m = r = 3 8 3 1024a8 96a 1 1 m = − w = 4 10 1 6a6 768a 1 1 m = w = 5 10 2 36a6 576a

Jesse Hicks Utah State University The Riemann curvature tensor, its invariants, and their use in the classification of spacetimes Introduction A Little Background The Idea The Example Conclusion

Petrov’s (33.17),  = −1, we have R = −2 and k22

1 1 r1 = 2 m1 = − 3 4k22 12k22 r2 = 0 1 m2 = 4 1 36k22 r = 3 4 1 64k22 m3 = 4 1 36k22 w1 = 2 6k22 m4 = 0 1 1 w2 = 3 m5 = − 5 36k22 108k22

These are different metrics. Petrov incorrectly normalized.

Jesse Hicks Utah State University The Riemann curvature tensor, its invariants, and their use in the classification of spacetimes Introduction A Little Background The Idea The Example Conclusion

A corrected and complete classification is in a database.

A classifier has been coded in Maple that makes comparisons against database.

Many metrics from Stephani’s compilation have been identified in Petrov.

Software has been written to help find explicit equivalences.

Using that software, all homogeneous spaces of dimension 3-5 have explicit equivalences to metrics in Petrov.

Jesse Hicks Utah State University The Riemann curvature tensor, its invariants, and their use in the classification of spacetimes Introduction A Little Background The Idea The Example Conclusion

Many thanks to my advisor Dr. Ian Anderson and the Differential Geometry Group at USU!

Thank You!

Jesse Hicks Utah State University The Riemann curvature tensor, its invariants, and their use in the classification of spacetimes