Teleparallel Gravity and Its Modifications

Introduction to Teleparallel equivalent of general relativity Modiﬁed teleparallel gravity theories Conclusions

Teleparallel gravity and its modiﬁcations

Sebastian´ Bahamonde

[email protected] Department of Mathematics, University College London.

Particles and Fields Seminar - Ben Gurion University 04th December 2017

1 / 49 Introduction to Teleparallel equivalent of general relativity Modiﬁed teleparallel gravity theories Conclusions Outline

1 Introduction to Teleparallel equivalent of general relativity Basic concepts in teleparallel gravity Gauge structure: translation group Torsion, connection and action Teleparallel gravity vs General Relativity

2 Modified teleparallel gravity theories General properties Modified teleparallel theories of gravity: f(T,B) gravity New classes of modified theories of gravity f(Ti,B) Modified teleparallel theories of gravity: f(T,B,TG,BG)

3 Conclusions

2 / 49 Basic concepts in teleparallel gravity Introduction to Teleparallel equivalent of general relativity Gauge structure: translation group Modiﬁed teleparallel gravity theories Torsion, connection and action Conclusions Teleparallel gravity vs General Relativity Outline

3 Conclusions

3 / 49 Basic concepts in teleparallel gravity Introduction to Teleparallel equivalent of general relativity Gauge structure: translation group Modiﬁed teleparallel gravity theories Torsion, connection and action Conclusions Teleparallel gravity vs General Relativity Tetrad ﬁelds

Assuming that the manifold is differentiable: Deﬁne tetrads a (or vierbein) ea (or e ) which are the linear basis on the spacetime{ manifold.} { } At each point of the spacetime, tetrads gives us basis for vectors on the tangent space. Notation: Greek letters space-time indices; → µ Latin letters tangent space indices; Ea is the inverse of the tetrad. → µ a a µ Then for example ea = Ea dxµ and e = e µdx . µ n n Tetrads satisfy the orthogonality condition: Em e µ = δm ν m ν and Em e µ = δµ

4 / 49 Basic concepts in teleparallel gravity Introduction to Teleparallel equivalent of general relativity Gauge structure: translation group Modiﬁed teleparallel gravity theories Torsion, connection and action Conclusions Teleparallel gravity vs General Relativity Tetrad ﬁelds

c General basis ea satisﬁes [ea, eb] = f abec , where c µ ν { }c c f ab = Ea Eb (∂νe µ ∂νe ν) is the coefﬁcient of anholonomy (curl of the− basis). a Inertial frame (or holonomic frame) ea0 : f 0 cd = 0 (global property) Special Relativity in Minkowski: Linear frames (trivial frames) gives us a relation between ηab (tangent) and ηµν µ ν a b (spacetime): ηab = ηµνEa Eb and ηµν = ηabe µe ν. In trivial frames: Curvature and torsion are zero.

5 / 49 Basic concepts in teleparallel gravity Introduction to Teleparallel equivalent of general relativity Gauge structure: translation group Modiﬁed teleparallel gravity theories Torsion, connection and action Conclusions Teleparallel gravity vs General Relativity Tetrads and Special Relativity

a a a a b SR: e0µ = ∂µx0 and then x Λ bx . The new frame will be → Lorentz-rotated frame in SR

a a b a a b a e = Λ e0 = ∂ x + w x D x , (1) µ b µ µ bµ ≡ µ a a e where w bµ = Λ e∂µΛb is known as the spin connection which represents inertial effects. du0a Example free particles: inertial frame: dλ = 0. This Eq. is not local Lorentz invariant However, in the anholonomic frame, changing a a b du0a a b µ e µ = Λ be0µ, one gets dλ + w bµu u = 0 which is Lorentz covariant! Spin-connection in SR represent inertial effects and they are very important to the local Lorentz invariance. 6 / 49 Basic concepts in teleparallel gravity Introduction to Teleparallel equivalent of general relativity Gauge structure: translation group Modiﬁed teleparallel gravity theories Torsion, connection and action Conclusions Teleparallel gravity vs General Relativity Tetrads and Special Relativity

a a a a b SR: e0µ = ∂µx0 and then x Λ bx . The new frame will be → Lorentz-rotated frame in SR

3 Conclusions

7 / 49 Basic concepts in teleparallel gravity Introduction to Teleparallel equivalent of general relativity Gauge structure: translation group Modiﬁed teleparallel gravity theories Torsion, connection and action Conclusions Teleparallel gravity vs General Relativity Gauge theory of translations - Teleparallel gravity

TG is a gauge theory of translations: a ﬁeld ψ transforms covariantly under a translation gauge xa x a + ξa(xµ) if a → 0 gauge potential Ba = ∂ ξa is introduced. µ − µ The translational gauge covariant derivative can be written a as eµψ = e µ∂aψ, and then

a a a e µ = ∂µx + B µ , (2)

a where e µ cannot be a trivial frame. This mean that one needs to replace a Minkowski metric to a Riemannian metric giving us

a b µν ab µ ν gµν = ηabe µe ν , g = η E aE b . (3)

8 / 49 Basic concepts in teleparallel gravity Introduction to Teleparallel equivalent of general relativity Gauge structure: translation group Modiﬁed teleparallel gravity theories Torsion, connection and action Conclusions Teleparallel gravity vs General Relativity Gauge theory of translations - Teleparallel gravity

a a a e µ = ∂µx + B µ , (2)

a where e µ cannot be a trivial frame. This mean that one needs to replace a Minkowski metric to a Riemannian metric giving us

a b µν ab µ ν gµν = ηabe µe ν , g = η E aE b . (3)

a a a e µ = ∂µx + B µ , (2)

a where e µ cannot be a trivial frame. This mean that one needs to replace a Minkowski metric to a Riemannian metric giving us

a b µν ab µ ν gµν = ηabe µe ν , g = η E aE b . (3)

Let us now consider inertial effects in TG. a a b By performing a Lorentz transformation x Λ bx one ﬁnds that the tetrad becomes →

a a a b a a a e µ = ∂µx + w bµx + B µ = Dµx + B µ , (4)

a a c where w bµ = Λ c∂µΛb is the spin connection (pure gauge-like) which describes inertial effects in the frame described. Here, Dµ is the Lorentz covariant derivative. a Proper frames: w bµ = 0.

9 / 49 Basic concepts in teleparallel gravity Introduction to Teleparallel equivalent of general relativity Gauge structure: translation group Modiﬁed teleparallel gravity theories Torsion, connection and action Conclusions Teleparallel gravity vs General Relativity Gauge theory of translations - Inertial effects

Let us now consider inertial effects in TG. a a b By performing a Lorentz transformation x Λ bx one ﬁnds that the tetrad becomes →

a a a b a a a e µ = ∂µx + w bµx + B µ = Dµx + B µ , (4)

Let us now consider inertial effects in TG. a a b By performing a Lorentz transformation x Λ bx one ﬁnds that the tetrad becomes →

a a a b a a a e µ = ∂µx + w bµx + B µ = Dµx + B µ , (4)

Like any gauge theory, the ﬁeld strength is the covariant rotational of the gauge potential

T a = D Ba D Ba . (5) µν µ ν − ν µ a a a Since e µ = Dµx + B µ, one can rewrite the ﬁeld strength as T a = ∂ ea ∂ ea + wa eb wa eb . (6) µν µ ν − ν µ bµ ν − bν µ This expression coincides with the Torsion tensor!. It can be proved that the torsion tensor transforms covariantly under both diffeomorphisms and local Lorentz transformations and only represent gravitational effects.

10 / 49 Basic concepts in teleparallel gravity Introduction to Teleparallel equivalent of general relativity Gauge structure: translation group Modiﬁed teleparallel gravity theories Torsion, connection and action Conclusions Teleparallel gravity vs General Relativity Translation Field Strength

Like any gauge theory, the ﬁeld strength is the covariant rotational of the gauge potential

3 Conclusions

11 / 49 Basic concepts in teleparallel gravity Introduction to Teleparallel equivalent of general relativity Gauge structure: translation group Modiﬁed teleparallel gravity theories Torsion, connection and action Conclusions Teleparallel gravity vs General Relativity Connection in Teleparallel gravity

Let us now introduce the so-called “Weitzenbock¨ connection”: Weitzenbock¨ connection

ρ ρ a ρ a a b Γ˜ µν = Ea Dµe ν = Ea (∂µe ν + w bµe ν) . (7)

By using this connection, one can express the torsion tensor as follows Torsion tensor

T ρ = Γ˜ρ Γ˜ρ . (8) µν νµ − µν

12 / 49 Basic concepts in teleparallel gravity Introduction to Teleparallel equivalent of general relativity Gauge structure: translation group Modiﬁed teleparallel gravity theories Torsion, connection and action Conclusions Teleparallel gravity vs General Relativity Connection in Teleparallel gravity

Let us now introduce the so-called “Weitzenbock¨ connection”: Weitzenbock¨ connection

ρ ρ a ρ a a b Γ˜ µν = Ea Dµe ν = Ea (∂µe ν + w bµe ν) . (7)

By using this connection, one can express the torsion tensor as follows Torsion tensor

T ρ = Γ˜ρ Γ˜ρ . (8) µν νµ − µν

Let us now introduce the so-called “Weitzenbock¨ connection”: Weitzenbock¨ connection

ρ ρ a ρ a a b Γ˜ µν = Ea Dµe ν = Ea (∂µe ν + w bµe ν) . (7)

By using this connection, one can express the torsion tensor as follows Torsion tensor

T ρ = Γ˜ρ Γ˜ρ . (8) µν νµ − µν

Let us now introduce the so-called “Weitzenbock¨ connection”: Weitzenbock¨ connection

ρ ρ a ρ a a b Γ˜ µν = Ea Dµe ν = Ea (∂µe ν + w bµe ν) . (7)

By using this connection, one can express the torsion tensor as follows Torsion tensor

T ρ = Γ˜ρ Γ˜ρ . (8) µν νµ − µν

ρ The Weitzenbock¨ connection Γ˜ νµ is related to the ρ Levi-Civita connection Γ νµ via Relationship between connections

ρ ρ ρ Γ˜ νµ = Γ νµ + K µν , (9)

ρ 1 ρ ρ ρ where K µν = 2 (Tµ ν + Tν µ T µν) is the contorsion tensor. − In this connection, it is easy to verify that the spacetime is globally ﬂat: Curvature in Teleparallel gravity

Ra (ωa ) = ∂ ωa ∂ ωa +ωa ωc ωa ωc 0 . (10) bµν bµ µ bν − ν bµ cµ bν − cν bµ ≡ 13 / 49 Basic concepts in teleparallel gravity Introduction to Teleparallel equivalent of general relativity Gauge structure: translation group Modiﬁed teleparallel gravity theories Torsion, connection and action Conclusions Teleparallel gravity vs General Relativity Connection in TG

ρ The Weitzenbock¨ connection Γ˜ νµ is related to the ρ Levi-Civita connection Γ νµ via Relationship between connections

ρ ρ ρ Γ˜ νµ = Γ νµ + K µν , (9)

ρ 1 ρ ρ ρ where K µν = 2 (Tµ ν + Tν µ T µν) is the contorsion tensor. − In this connection, it is easy to verify that the spacetime is globally ﬂat: Curvature in Teleparallel gravity

ρ The Weitzenbock¨ connection Γ˜ νµ is related to the ρ Levi-Civita connection Γ νµ via Relationship between connections

ρ ρ ρ Γ˜ νµ = Γ νµ + K µν , (9)

ρ 1 ρ ρ ρ where K µν = 2 (Tµ ν + Tν µ T µν) is the contorsion tensor. − In this connection, it is easy to verify that the spacetime is globally ﬂat: Curvature in Teleparallel gravity

ρ The Weitzenbock¨ connection Γ˜ νµ is related to the ρ Levi-Civita connection Γ νµ via Relationship between connections

ρ ρ ρ Γ˜ νµ = Γ νµ + K µν , (9)

ρ 1 ρ ρ ρ where K µν = 2 (Tµ ν + Tν µ T µν) is the contorsion tensor. − In this connection, it is easy to verify that the spacetime is globally ﬂat: Curvature in Teleparallel gravity

The teleparallel action is formulated based on a gravitational scalar called the torsion scalar T Z 2 4 STG = T + 2κ m e d x . (11) − L

2 µ where κ = 8πG, e = det(ea ) = √ g, m matter − L Lagrangian and T = 1 T ρ T µν + 1 T ρ T νµ T λ T νµ. 4 µν ρ 2 µν ρ − λµ ν T and the scalar curvature R differs by a boundary term B as R = T + B so: − Equivalence between field equations The teleparallel field equations are equivalent to the Einstein field equations.

14 / 49 Basic concepts in teleparallel gravity Introduction to Teleparallel equivalent of general relativity Gauge structure: translation group Modiﬁed teleparallel gravity theories Torsion, connection and action Conclusions Teleparallel gravity vs General Relativity Teleparallel action

The teleparallel action is formulated based on a gravitational scalar called the torsion scalar T Z 2 4 STG = T + 2κ m e d x . (11) − L

3 Conclusions

15 / 49 Basic concepts in teleparallel gravity Introduction to Teleparallel equivalent of general relativity Gauge structure: translation group Modiﬁed teleparallel gravity theories Torsion, connection and action Conclusions Teleparallel gravity vs General Relativity Teleparallel gravity vs General Relativity

Two completely equivalent ways of understanding gravity: Connections and strength ﬁelds G.R. = Levi-Civita connection = Curvature with vanishing torsion ⇒ ⇒ TG = Weitzenbock¨ connection = Torsion with vanishing curvature⇒ (ﬂat). ⇒

How gravity is explained in both theories? GR = Geometry (curvature of space-time) = geodesic equations⇒ ⇒ TG = Forces = Force equations as Maxwell eqs (no geodesic⇒ eq.). ⇒

16 / 49 Basic concepts in teleparallel gravity Introduction to Teleparallel equivalent of general relativity Gauge structure: translation group Modiﬁed teleparallel gravity theories Torsion, connection and action Conclusions Teleparallel gravity vs General Relativity Teleparallel gravity vs General Relativity

How gravity is explained in both theories? GR = Geometry (curvature of space-time) = geodesic equations⇒ ⇒ TG = Forces = Force equations as Maxwell eqs (no geodesic⇒ eq.). ⇒

Gauge structure GR = NO (only diffeomorphism) ⇒ TG = Gauge theory of the translations ⇒ Must have the equivalence principle? GR = YES ⇒ TG = Can survive with or without ⇒ Can we separate inertia with gravity? GR = NO (mixed) = No tensorial expression for the gravitational⇒ energy-momentum⇒ density TG = YES = gravitational energy-momentum density is a tensor.⇒ ⇒

17 / 49 Basic concepts in teleparallel gravity Introduction to Teleparallel equivalent of general relativity Gauge structure: translation group Modiﬁed teleparallel gravity theories Torsion, connection and action Conclusions Teleparallel gravity vs General Relativity Teleparallel gravity vs General Relativity

Geometrical differences Curvature = how the tangent spaces roll along the curve. Torsion =⇒ how tangent spaces twist about a curve when ⇒ they are parallel transported

Curvature Torsion

Figure: Transporting a vector in a closed trajectory creates a different vector Figure: Open parallelogram

18 / 49 Basic concepts in teleparallel gravity Introduction to Teleparallel equivalent of general relativity Gauge structure: translation group Modiﬁed teleparallel gravity theories Torsion, connection and action Conclusions Teleparallel gravity vs General Relativity Teleparallel gravity vs General Relativity

Geometrical differences Curvature = how the tangent spaces roll along the curve. Torsion =⇒ how tangent spaces twist about a curve when ⇒ they are parallel transported

Curvature Torsion

Figure: Transporting a vector in a closed trajectory creates a different vector Figure: Open parallelogram

Geometrical differences Curvature = how the tangent spaces roll along the curve. Torsion =⇒ how tangent spaces twist about a curve when ⇒ they are parallel transported

Curvature Torsion

Figure: Transporting a vector in a closed trajectory creates a different vector Figure: Open parallelogram

Geometrical differences Curvature = how the tangent spaces roll along the curve. Torsion =⇒ how tangent spaces twist about a curve when ⇒ they are parallel transported

Curvature Torsion

Figure: Transporting a vector in a closed trajectory creates a different vector Figure: Open parallelogram

Geometrical differences Curvature = how the tangent spaces roll along the curve. Torsion =⇒ how tangent spaces twist about a curve when ⇒ they are parallel transported

Curvature Torsion

Figure: Transporting a vector in a closed trajectory creates a different vector Figure: Open parallelogram

Geometrical differences Curvature = how the tangent spaces roll along the curve. Torsion =⇒ how tangent spaces twist about a curve when ⇒ they are parallel transported

Curvature Torsion

Figure: Transporting a vector in a closed trajectory creates a different vector Figure: Open parallelogram

When it is assumed that for all frames the spin connection a is zero, w bc = 0, then one refers to pure tetrad teleparallel theories. In this case, the torsion tensor becomes

T a = ∂ ea ∂ ea = f a (12) µν µ ν − ν µ µν Therefore, the torsion tensor and the scalar torsion T are not invariant under Local Lorentz transformations. However, for standard teleparallel gravity, the theory turns out to be quasi-local Lorentz invariant (up to a boundary term)

19 / 49 Basic concepts in teleparallel gravity Introduction to Teleparallel equivalent of general relativity Gauge structure: translation group Modiﬁed teleparallel gravity theories Torsion, connection and action Conclusions Teleparallel gravity vs General Relativity Pure tetrad teleparallel

When it is assumed that for all frames the spin connection a is zero, w bc = 0, then one refers to pure tetrad teleparallel theories. In this case, the torsion tensor becomes

Since the spin connection is irrelevant for the solution, the ﬁeld equations of pure tetrad teleparallel gravity yields correct solutions for the tetrad and for the metric of spacetime. Einstein and Weitzbrock¨ presented teleparallel gravity a within this formalism, assuming w bc = 0. For this historical reason, mostly many authors use this formalism too.

20 / 49 Basic concepts in teleparallel gravity Introduction to Teleparallel equivalent of general relativity Gauge structure: translation group Modiﬁed teleparallel gravity theories Torsion, connection and action Conclusions Teleparallel gravity vs General Relativity Pure tetrad teleparallel

20 / 49 General properties Introduction to Teleparallel equivalent of general relativity Modified teleparallel theories of gravity: f(T,B) gravity Modified teleparallel gravity theories New classes of modified theories of gravity f(T ,B) Conclusions i Modified teleparallel theories of gravity: f(T,B,TG,BG) Outline

3 Conclusions

21 / 49 General properties Introduction to Teleparallel equivalent of general relativity Modified teleparallel theories of gravity: f(T,B) gravity Modified teleparallel gravity theories New classes of modified theories of gravity f(T ,B) Conclusions i Modified teleparallel theories of gravity: f(T,B,TG,BG) Modified teleparallel theories of gravity

As there are many modifications in GR, it is also possible to construct modifications to teleparallel gravity. If ones modifies pure tetrad teleparallel gravity, one finds theories which are not invariant under local LT. Two approaches for constructing modified teleparallel theories: How to construct modified teleparallel theories?

a 1 Do NOT assume w bc = 0, (like covariant f(T ) gravity) a 2 Use pure tetrad (w bc = 0) but choosing “good tetrads”. Both approaches give the same field equations but (1) is more theoretically correct. However, there is still a debate a about how to find w bc. Mostly all the literature use (2). 22 / 49 General properties Introduction to Teleparallel equivalent of general relativity Modified teleparallel theories of gravity: f(T,B) gravity Modified teleparallel gravity theories New classes of modified theories of gravity f(T ,B) Conclusions i Modified teleparallel theories of gravity: f(T,B,TG,BG) Modified teleparallel theories of gravity

3 Conclusions

23 / 49 General properties Introduction to Teleparallel equivalent of general relativity Modified teleparallel theories of gravity: f(T,B) gravity Modified teleparallel gravity theories New classes of modified theories of gravity f(T ,B) Conclusions i Modified teleparallel theories of gravity: f(T,B,TG,BG) Motivating from GR: f(R) gravity

A well studied modiﬁcation of GR is f(R) gravity, which has the following action (change R f(R)) → f(R) gravity action Z 2 4 S = (f(R) + 2κ m)√ g d x . f(R) L −

Here, f is an arbitrary (sufﬁciently smooth) function of the Ricci scalar. Ricci scalar depends on second derivatives of the metric tensor Fourth order theory. → Very famous and explore theory which mimics cosmological observations.

24 / 49 General properties Introduction to Teleparallel equivalent of general relativity Modified teleparallel theories of gravity: f(T,B) gravity Modified teleparallel gravity theories New classes of modified theories of gravity f(T ,B) Conclusions i Modified teleparallel theories of gravity: f(T,B,TG,BG) Motivating from GR: f(R) gravity

A well studied modiﬁcation of GR is f(R) gravity, which has the following action (change R f(R)) → f(R) gravity action Z 2 4 S = (f(R) + 2κ m)√ g d x . f(R) L −

In analogy with f(R) gravity, one can consider in the Teleparallel framework, the f(T ) gravity (change T f(T )) → f(T ) gravity action Z 2 4 S = (f(T ) + 2κ m)e d x . f(T ) L The torsion scalar T depends on the first derivatives of the tetrads Second order theory. → T is not invariant under local LT = f(T ) is also not invariant under local LT. ⇒ Not equivalency between f(R) and f(T ) Field equations of f(T ) = Field equations of f(R) 6 25 / 49 General properties Introduction to Teleparallel equivalent of general relativity Modified teleparallel theories of gravity: f(T,B) gravity Modified teleparallel gravity theories New classes of modified theories of gravity f(T ,B) Conclusions i Modified teleparallel theories of gravity: f(T,B,TG,BG) f(T ) gravity - pure tetrad

The relationship between R and T are Relationship between R and T 2 R = T + ∂ (eT µ) = T + B. − e µ − Inspired by the later discussion, we deﬁne the action (Bahamonde et. al Phys.Rev. D92,10, 104042 (2015)) f(T,B) gravity action Z 2 4 Sf(T,B) = f(T,B) + 2κ Lm e d x ,

where f is a function of both of its arguments and Lm is a matter Lagrangian.

26 / 49 General properties Introduction to Teleparallel equivalent of general relativity Modified teleparallel theories of gravity: f(T,B) gravity Modified teleparallel gravity theories New classes of modified theories of gravity f(T ,B) Conclusions i Modified teleparallel theories of gravity: f(T,B,TG,BG) f(T,B) gravity

where f is a function of both of its arguments and Lm is a matter Lagrangian.

It is possible to show that this theory has the correct limits: Limiting cases

1 When f(T,B) = f(T ), one obtains f(T ) gravity 2 When f(T,B) = f( T + B) = f(R), one obtains the teleparallel equivalent of f(R) gravity− 3 Other new theories such as f(T,B) = T + F (B).

Additionally, we showed that Lorentz invariance and conservation