Mathematical Relativity and the Nature of the Universe Priscila Reyes Advisor: Dr. Vehbi Emrah Paksoy Department of Mathematics Halmos College of Natural Science and Oceanography NSU Mathematics Colloquium Series November 28, 2016
Mathematical Relativity and the Nature of the Universe
Priscila Reyes
Advisor: Dr. Vehbi Emrah Paksoy
Department of Mathematics
Halmos College of Natural Science and Oceanography
NSU Mathematics Colloquium Series
November 28, 2016
Brief Introduction to Mathematical Background
Tensor Algebra
โข Let ๐ = < ๐1, โฆ , ๐๐ > be a vector space with dual ๐ โ=< ๐1, โฆ , ๐๐ >โข Where ๐ โ ๐ =< ๐๐ โ๐๐ | 1 โค ๐ โค ๐ โค ๐ >
โข Then ๐๐ ๐ ๐ is all possible tensors by
๐๐ ๐ ๐ = ๐ โ ๐โโฏ๐โ๐
โโ๐
โโโฏโ๐
โ
r times s times
โข ๐๐ ๐ ๐ is also known as an ๐, ๐ โ tensor
Lorentz Spaces
โข Let (๐, ๐) be an inner product space. If I=1, then (๐, ๐) is a Lorentz space
โข Ex. ๐น4, ๐ =
1 0 0 00 1 0 00 0 1 00 0 0 โ1
Define
โข V - a vector space
โข g - an inner product on V
โข I - the number of negative eigenvalues of g
Causal Character
Space-like, Light-like, and Time-like
โข Let (๐, ๐) be a Lorentzian manifold of dimension n
โข A vector v in ๐ is calledโข Space-like if ๐(๐, ๐) > 0โข Time-like if ๐ ๐, ๐ < 0โข Light-like if ๐ ๐, ๐ = 0
โข A vector is called causal if they are time-like or light-like
โข A subspace ๐ of ๐ is calledโข Space-like if all vectors in ๐ are space-like
โข (If it is positive definite)
โข Time-like if it contains a time-like vector
โข Light-like if it has a light-like vector but no time-like vectorsโข (If it is positive semi-definite)
The Light Cone
โข In the light cone (shown on the right), vectors areโข Space-like if they lie outside of the cones
โข Time-like if they lie inside of the cones
โข Light-like if they lie at the boundaries of the cone
โข Vectors pointing towards the positive z-axis are said to be future-pointing
โข Vectors pointing towards the negative z-axis are said to be past-pointing
โข The point of convergence of the two cones is called the present
โข Events outside of the cones cannot be experienced by an observer
Reverse Cauchy-Schwarz Inequality
โข Let ๐ be a Lorentz space and u, v be causal vectors. Then,
|g ๐ฎ, ๐ฏ | โฅ | ๐ | ร | ๐ |
โข This is due to the properties of the orthogonal complement of a subspace and the properties of causal vectors
โข The triangle inequality also has an equivalent in Lorentz spaces called the wrong-way triangle inequality
โข The relational operator is reversed
Note
โข The Cauchy-Schwarz inequality is๐ โ ๐ โค | ๐ | ร | ๐ |
โข The triangle inequality is
| ๐ + ๐| โค ๐ + | ๐ |
The Twin Paradox in Lorentz Spaces
โข Assume that you have two twins, A and B (both aged 21), in a Lorentz space
โข Twin A stays on earth
โข Twin B goes on a rocket at 24
25the speed of light for 7 years and
then returns to earth
โข What should the age of the twins be at twin Bโs return?
The Twin Paradox in Lorentz Spaces (cont.)
โข The age of twin B in his own proper time is 21+7+7=35 years
โข However, because of Lorentz contraction (which results in time dilation) twin Aโs age has to be calculated as follows:
14
1 โ๐ฃ2
๐2
=14
1 โ
242
252๐2
๐2
=14
252 โ 242
252
=14ร25
(25โ24)(25+24)=
14โ25
7= 50
Thus, the age of twin A is 21+50=71
7 years
7 years
Trajectory in
time of twin ATrajectory in
time of twin B
Observers
Definition of an Observer
โข An observer is a future-pointing time-like curve ๐พ(๐) such that ๐ ๐พโ, ๐พโ = โ1
โข ๐พ: ๐ผโM
โข The image of ๐พ is world line
โข The world line is essentially the history of an observer
โข ๐ is proper time
โข An observer is said to be free-falling if itโs a geodesic
โข A geodesic is the equivalent of a straight line in a manifold
Spacetimes
Background for the Orientability of a Spacetime:Differential Forms
โข Let M be a manifold of dimension n and let ๐ โ ๐โข Note that ๐๐
โ=< ๐๐ฅ1 , โฆ , ๐๐ฅ๐ >, the tangent space of the dual of M at point
p
โข ๐๐ฅ๐ are 1-forms
โข ๐๐ฅ๐ โง ๐๐ฅ๐ = 1
2(๐๐ฅ๐ โ๐๐ฅ๐ โ ๐๐ฅ๐ โ๐๐ฅ๐) is a 2-form
โข Then, in general, ๐ค = ๐ค๐1, โฆ ๐๐๐๐ฅ๐1 โง โฏโง ๐๐ฅ๐๐ is a k-form, where
1 โค ๐, ๐2 < โฏ < ๐๐ โค ๐
Requirements for the Orientability and Time-Orientability of a Spacetime
โข A manifold M is orientable if it has a nowhere vanishing n-form
โข A Lorentzian manifold M is time-oriented if there exists a causal vector field X
โ X is time-like or light-like
โข X is time-like if X(p) is time-like
โข X is light-like if X(p) is light-like
Definition of a Spacetime
โข A spacetime is a 4-dimensional Lorentzian manifold (๐, ๐, ๐ป) which is connected, oriented, and time oriented together with a Levi-Civita connectionโข The covariant derivative ๐ป is an affine
connection and it determines the curvature of the spacetime as well as its geodesics
โข In particular, Christoffel symbols, denoted ฮ๐๐๐
are included in the Levi-Civita connection and physically represent how much an object deviates from being flat
โข Interesting spacetimes are solutions to Einsteinโs Field Equations (EFE)
The Shape and Composition of a Spacetime
โข The Riemann curvature R is a particular (0,4)-tensor
โข If R=0, the spacetime is said to be flat
โข The Ricci tensor Ric is a particular (2,0) symmetric tensor
โข If Ric = 0, the spacetime is said to be Ricci-flat
โข A Ricci-flat spacetime represents a vacuum (there is no matter present)
โข The scalar curvature of a spacetime S is the trace of the Ricci tensor
โข The Einstein tensor ๐บ = ๐ ๐๐ โ1
2๐๐ is a (2,0)-tensor
โข The Einstein tensor expresses the curvature of the spacetime
Minkowski Spacetime
โข ๐ = โ4
โข g = ๐๐ฅ โ ๐๐ฅ + ๐๐ฆ โ ๐๐ฆ + ๐๐ง โdz โ dt โ ๐๐ก
=
1 0 0 00 1 0 00 0 1 00 0 0 โ1
โข ๐ปโ๐
โ๐= 0
โ Minkowski spaces model flat spacetime
โ Represent no gravitation
โข Oriented by ฮฉ = ๐๐ฅ โง ๐๐ฆ โง ๐๐ง โง ๐๐ก
โข Time-oriented by โt =โโt
โข Note that g(โt, โt) = -1
โข Both flat and Ricci-flat
Einstein-deSitter Spacetime
โข ๐ = โ3 ร (0,โ)
โข g = ๐ก4
3(๐๐ฅ โ ๐๐ฅ + ๐๐ฆ โ ๐๐ฆ +๐๐ง โ dz) โ dt โ ๐๐ก
=
๐ก4
3 0 0 0
0 ๐ก4
3 0 0
0 0 ๐ก4
3 00 0 0 โ1
โข Oriented by ฮฉ = ๐๐ฅ โง ๐๐ฆ โง ๐๐ง โง ๐๐ก
โข Time-oriented by โt
โข Not flat, but Ricci-flat
Schwarzchild Spacetime
โข ๐ ๐ = 1 โ๐๐
๐, ๐๐ > 0
(0, ๐๐ ) (๐๐ , โ)
โข ๐ = (๐๐ , โ) ร ๐2 ร (0,โ)
r (๐, ๐) t
0 โค ๐ < ๐0 โค ๐ < 2๐
โข Oriented by ฮฉ = ๐๐ โง ๐๐ โง ๐๐ โง ๐๐ก
โข Time-oriented by โtโข Represents an uncharged, static
black hole
โข Not flat, but Ricci-flat
The Schwarzchild Metric
โข g = 1 โ๐๐
๐
โ1๐๐ โ ๐๐ + +๐2๐๐โจ๐๐ + ๐2๐ ๐๐2๐๐๐ โ ๐๐ โ
1 โ๐๐
๐๐๐ก โ ๐๐ก
=
1 โ๐๐
๐
โ10 0 0
0 ๐2 0 00 0 ๐2๐ ๐๐2๐ 0
0 0 0 โ 1 โ๐๐
๐
The Schwarzchild Radius
โข Letโs look at an observer in proper time in this spacetime:
๐พ t = r ๐ , ฮธ ๐ , ๐ ๐ , ๐ก ๐ = (๐พ1, ๐พ2, ๐พ3, ๐พ4)๐2๐พ๐
๐๐2= โฮ๐๐
๐ ๐๐พ๐
๐๐
๐๐พ๐
๐๐
โข Then, the radial acceleration ๐2๐
๐๐2= โฮ๐ก๐ก
๐ (๐๐ก
๐๐)2
โ๐2๐
๐๐2= โ
1
2(1 โ
๐๐
๐)(
๐๐
๐2)(๐๐ก
๐๐)2
The Schwarzchild Radius (cont.)
โข At rest, we have ๐๐
๐๐=
๐๐
๐๐=
๐๐
๐๐= 0
โ ๐พโ = 0,0,0,๐๐ก
๐๐= 0 โ๐ + 0 โ๐ + 0 โ๐ +
๐๐ก
๐๐โ๐ก
โข g(๐พโ, ๐พโ) = โ1
โ โ1 =๐๐ก
๐๐
2(โ(1 โ
๐๐
๐))
โข๐2๐
๐๐2=
โ๐๐
2๐2
The Schwarzchild Radius (cont.)
โข๐2๐
๐๐2=
โ๐บ๐
๐2=
โ๐๐
2๐2
โข โ๐๐ = 2๐บ๐ = The Schwarzchild radius
Note:
โข ๐ ๐ =โ๐บ๐
๐2๐
โข F=mg(r)
Physical Description of the Schwarzchild Radius
โข The Schwarzchild radius is the event horizon of a black hole
โข GM(sun) = 1.5km
โข GM(earth) = 4.5mm
โข If the radius of either the sun or the earth were to shrink to these sizes while keeping their original masses, they would become black holes!
Kerr Spacetime
โข g =
๐ด
๐ต0 0 0
0 ๐ด 0 0
0 0 ๐ฟ๐ ๐๐2๐โ2๐บ๐๐๐๐ ๐๐2๐
๐ด
0 0โ2๐บ๐๐๐๐ ๐๐2๐
๐ดโ(1 โ
2๐บ๐๐
๐ด)
โข ๐ด = ๐2 + ๐2๐๐๐ 2๐
โข ๐ต = ๐2 โ 2๐บ๐๐ + ๐2
โข ๐ฟ = ๐2 + ๐2 +2๐บ๐๐๐๐ ๐๐2๐
๐ด
โข This is the metric given in Boyer-Lindquist coordinatesโข Many others exist
โข ๐ด > 2๐บ๐๐
โข ๐ต > 0
โข Conditions above imposed to have well-defined time direction
โข ๐, ๐, ๐ are spherical coordinates and t is time
โข Represents the history outside of an uncharged, rotating black hole with mass m and angular momentum per unit mass a
โข Note that when a equals 0 , this actually collapses to Schwarzchild spacetime
โข Kerr spacetime is not flat, but it is Ricci-flat
Kerr Spacetime (cont.)
โข What if a > Gm?โข This is physically impossible. The black hole will spit out matter.
โข What if a < Gm?โข Let B=0
โข โ ๐2 โ 2๐บ๐๐ + ๐2 = 0
โข Use quadratic formula to find
๐ ยฑ=๐บ๐ ยฑ 4(๐บ2๐2 โ ๐2)
2= ๐บ๐ ยฑ ๐บ2๐2 โ ๐2
โข ๐+ is the event horizon
โข If ๐ < ๐+, there is no coming back
Kerr Spacetime (Singularity)
โข At A=0, we have
๐2 + ๐2๐๐๐ 2๐ = 0
r = 0 ๐ =๐
2
โข This represents the singularity of the black holeโข It cannot be seen by an observer
โข It has no future
โข This is known as cosmic censorship
With a choice of spherical
coordinates, this is a circle
๐ฅ2 + ๐ฆ2 = ๐2
at z = 0
r=0
๐=๐/2๐+
Kerr Limit of Stationarity
โข If ๐ด < 2๐บ๐๐, t stops representing the time dimension and r becomes the time dimension in its place
โข This is because in this case the sign of ๐๐ก โ ๐๐ก will stop being negative
โข This implies that we are not modeling a stationary object anymore, and it does change with time
Kerr Limit of Stationarity (cont.)
โข If ๐ด < 2๐บ๐๐, ๐ = ๐๐ ๐ก๐๐ก is the surface and ๐๐ ๐ก๐๐ก is the limit of stationarity
โข It Is the largest solution of A=2Gmr
โ ๐2 + ๐2๐๐๐ 2๐ = 2๐บ๐๐
โ ๐2 โ 2๐บ๐๐ + +๐2๐๐๐ 2๐ = 0 (Use quadratic formula)
โ ๐๐ ๐ก๐๐ก = ๐บ๐ + ๐บ2๐2 โ ๐2๐๐๐ 2๐ > ๐บ๐ + ๐บ2๐2 โ ๐2 = ๐+โข The surface between ๐+ and ๐๐ ๐ก๐๐ก is known as the ergosphere
Einstein Field Equation
Statement of Einstein Field Equation
โข The Einstein field equation is actually a set of equations that can be collapsed into one
โข Let ๐,โณ, ๐น be a relativistic model. Let T and E be stress-energy tensors of โณ and ๐น, respectively
โข โณ is a matter model
โข F is the electro-magnetic field
โข Then, ๐,โณ,๐น obeys the Einstein Field Equation if๐บ + ฮ๐ = ๐ + ๐ธ
Geometry (notice G - the Einstein
tensor)
Matter, energy, and
electromagnetism
Statement of Einstein Field Equation
โข ๐บ + ฮ๐ = ๐ + ๐ธโข Implies that the geometry of spacetime determines matter, energy, and
electromagnetism and vice-versa
โข ฮ is the cosmological constant
History of the Cosmological Constant
โข Originally, Einstein included ฮ because the universe was thought to be static
โข Between 1929 and 1989, the equation was simplified to be
๐บ = ๐ + ๐ธ
โข ฮ was not needed anymore due to the discovery of the expanding universe
โข Currently, we have added back ฮ to account for dark matter and dark energy
โข Energy density in a vacuum implies the existence of dark energy
โข However, in a paper published on November 7, 2016 by Erik P. Verlinde, the concept of dark matter is challenged
โข Verlinde renders it unnecessary as he explains that gravity may be an emergent phenomenon due to vacuum entanglements
โข Sachs, R. K., & Wu, H. (1977). General relativity for mathematicians. New York: Springer-Verlag.
โข Verlinde, E. P., (2016). Emergent gravity and the dark universe
โข Images:
โข https://commons.wikimedia.org/wiki/File:Eddington_A._Space_Time_and_Gravitation._Fig._4.jpg
โข https://commons.wikimedia.org/wiki/File:World_line.svg
โข https://commons.wikimedia.org/wiki/File:Light-clock.png
โข https://commons.wikimedia.org/wiki/File:Gravity_well_plot.svg
โข https://commons.wikimedia.org/wiki/File:Spacetime_lattice_analogy.svg
โข https://commons.wikimedia.org/wiki/File:Schematic_Kerr_Black_Hole.jpg
โข https://commons.wikimedia.org/wiki/File:Albert_Einstein_Head.jpg
Thank you for listening!