An introduction to the Penrose inequality I

Levi Lopes de Lima

Departament of MathematicsFederal University of Ceará

São Paulo - July, 2013

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 1 / 22

A tentative plan for the talk

1 A glimpse at SR: the positivity of energy and the causal structure of space-time.2 The conceptual framework of GR.3 The ADM formalism.4 The positive mass inequality.5 The Penrose inequality.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 2 / 22

A tentative plan for the talk

1 A glimpse at SR: the positivity of energy and the causal structure of space-time.2 The conceptual framework of GR.3 The ADM formalism.4 The positive mass inequality.5 The Penrose inequality.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 2 / 22

A tentative plan for the talk

1 A glimpse at SR: the positivity of energy and the causal structure of space-time.2 The conceptual framework of GR.3 The ADM formalism.4 The positive mass inequality.5 The Penrose inequality.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 2 / 22

A glimpse at Special Relativity

Special Relativity grew out of logical discrepancies between Newtonian Mechanics andElectromagnetism.

It relies on the following axioms:1 Equivalence of all inertial frames for the description of physical law.2 The speed of light (in vacuum) is the same in all inertial frames.

This leads to some amazing consequences:1 Space and time perceptions depend on the state of motion of the observer.2 No particle (massive or not!) can move faster than light.

The moral is that the speed of light is a fundamental physical parameter determining the wayspace and time are mixed together in the notion of space-time.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 3 / 22

A glimpse at Special Relativity

Special Relativity grew out of logical discrepancies between Newtonian Mechanics andElectromagnetism.

It relies on the following axioms:1 Equivalence of all inertial frames for the description of physical law.2 The speed of light (in vacuum) is the same in all inertial frames.

This leads to some amazing consequences:1 Space and time perceptions depend on the state of motion of the observer.2 No particle (massive or not!) can move faster than light.

The moral is that the speed of light is a fundamental physical parameter determining the wayspace and time are mixed together in the notion of space-time.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 3 / 22

A glimpse at Special Relativity

Special Relativity grew out of logical discrepancies between Newtonian Mechanics andElectromagnetism.

It relies on the following axioms:1 Equivalence of all inertial frames for the description of physical law.2 The speed of light (in vacuum) is the same in all inertial frames.

This leads to some amazing consequences:1 Space and time perceptions depend on the state of motion of the observer.2 No particle (massive or not!) can move faster than light.

The moral is that the speed of light is a fundamental physical parameter determining the wayspace and time are mixed together in the notion of space-time.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 3 / 22

A glimpse at Special Relativity

Special Relativity grew out of logical discrepancies between Newtonian Mechanics andElectromagnetism.

It relies on the following axioms:1 Equivalence of all inertial frames for the description of physical law.2 The speed of light (in vacuum) is the same in all inertial frames.

This leads to some amazing consequences:1 Space and time perceptions depend on the state of motion of the observer.2 No particle (massive or not!) can move faster than light.

The moral is that the speed of light is a fundamental physical parameter determining the wayspace and time are mixed together in the notion of space-time.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 3 / 22

A glimpse at Special Relativity

Special Relativity grew out of logical discrepancies between Newtonian Mechanics andElectromagnetism.

It relies on the following axioms:1 Equivalence of all inertial frames for the description of physical law.2 The speed of light (in vacuum) is the same in all inertial frames.

This leads to some amazing consequences:1 Space and time perceptions depend on the state of motion of the observer.2 No particle (massive or not!) can move faster than light.

The moral is that the speed of light is a fundamental physical parameter determining the wayspace and time are mixed together in the notion of space-time.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 3 / 22

A glimpse at Special Relativity

Special Relativity grew out of logical discrepancies between Newtonian Mechanics andElectromagnetism.

It relies on the following axioms:1 Equivalence of all inertial frames for the description of physical law.2 The speed of light (in vacuum) is the same in all inertial frames.

This leads to some amazing consequences:1 Space and time perceptions depend on the state of motion of the observer.2 No particle (massive or not!) can move faster than light.

The moral is that the speed of light is a fundamental physical parameter determining the wayspace and time are mixed together in the notion of space-time.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 3 / 22

Lorentz-Minkowski’s space-time I

Let us start with the affine 4-dimensional space A4.

A point of A4 is called an event.

A rule that to each event associates its coordinates, say, s = (t , x , y , z), is an example of aninertial frame.

To each such frame we can associate the Lorentzian quadratic form

Q = −dt2 + dx2 + dy2 + dz2.

Here, we take c = 1.

In particular, given the frame, it makes sense to evaluate Q on a space-time segmentdetermined by two events.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 4 / 22

Lorentz-Minkowski’s space-time I

Let us start with the affine 4-dimensional space A4.

A point of A4 is called an event.

A rule that to each event associates its coordinates, say, s = (t , x , y , z), is an example of aninertial frame.

To each such frame we can associate the Lorentzian quadratic form

Q = −dt2 + dx2 + dy2 + dz2.

Here, we take c = 1.

In particular, given the frame, it makes sense to evaluate Q on a space-time segmentdetermined by two events.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 4 / 22

Lorentz-Minkowski’s space-time I

Let us start with the affine 4-dimensional space A4.

A point of A4 is called an event.

A rule that to each event associates its coordinates, say, s = (t , x , y , z), is an example of aninertial frame.

To each such frame we can associate the Lorentzian quadratic form

Q = −dt2 + dx2 + dy2 + dz2.

Here, we take c = 1.

In particular, given the frame, it makes sense to evaluate Q on a space-time segmentdetermined by two events.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 4 / 22

Lorentz-Minkowski’s space-time I

Let us start with the affine 4-dimensional space A4.

A point of A4 is called an event.

A rule that to each event associates its coordinates, say, s = (t , x , y , z), is an example of aninertial frame.

To each such frame we can associate the Lorentzian quadratic form

Q = −dt2 + dx2 + dy2 + dz2.

Here, we take c = 1.

In particular, given the frame, it makes sense to evaluate Q on a space-time segmentdetermined by two events.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 4 / 22

Lorentz-Minkowski’s space-time I

Let us start with the affine 4-dimensional space A4.

A point of A4 is called an event.

A rule that to each event associates its coordinates, say, s = (t , x , y , z), is an example of aninertial frame.

To each such frame we can associate the Lorentzian quadratic form

Q = −dt2 + dx2 + dy2 + dz2.

Here, we take c = 1.

In particular, given the frame, it makes sense to evaluate Q on a space-time segmentdetermined by two events.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 4 / 22

Lorentz-Minkowski’s space-time I

Let us start with the affine 4-dimensional space A4.

A point of A4 is called an event.

A rule that to each event associates its coordinates, say, s = (t , x , y , z), is an example of aninertial frame.

To each such frame we can associate the Lorentzian quadratic form

Q = −dt2 + dx2 + dy2 + dz2.

Here, we take c = 1.

In particular, given the frame, it makes sense to evaluate Q on a space-time segmentdetermined by two events.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 4 / 22

Lorentz-Minkowski’s space-time I

Let us start with the affine 4-dimensional space A4.

A point of A4 is called an event.

A rule that to each event associates its coordinates, say, s = (t , x , y , z), is an example of aninertial frame.

To each such frame we can associate the Lorentzian quadratic form

Q = −dt2 + dx2 + dy2 + dz2.

Here, we take c = 1.

In particular, given the frame, it makes sense to evaluate Q on a space-time segmentdetermined by two events.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 4 / 22

Lorentz-Minkowski’s space-time II

In general, different frames experience different space and time measurements for a givenspace-time segment.

However, Axiom 2 implies that being a null segment (Q = 0 ) does not depend on theparticular frame.

From this it is possible to show that the evaluation of Q on any space-time segment also doesnot depend on the frame used to measure it.

Thus, by Axiom 1, SR is the study of physical properties which remain invariant under lineartransformations preserving the Lorentzian form (the Lorentzian group).

In other words, SR encompasses the study of the geometry of Minkowski spaceL4 = (A4,Q).

We will see that free particles trace time-like geodesics in this geometry.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 5 / 22

Lorentz-Minkowski’s space-time II

In general, different frames experience different space and time measurements for a givenspace-time segment.

However, Axiom 2 implies that being a null segment (Q = 0 ) does not depend on theparticular frame.

From this it is possible to show that the evaluation of Q on any space-time segment also doesnot depend on the frame used to measure it.

Thus, by Axiom 1, SR is the study of physical properties which remain invariant under lineartransformations preserving the Lorentzian form (the Lorentzian group).

In other words, SR encompasses the study of the geometry of Minkowski spaceL4 = (A4,Q).

We will see that free particles trace time-like geodesics in this geometry.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 5 / 22

Lorentz-Minkowski’s space-time II

In general, different frames experience different space and time measurements for a givenspace-time segment.

However, Axiom 2 implies that being a null segment (Q = 0 ) does not depend on theparticular frame.

From this it is possible to show that the evaluation of Q on any space-time segment also doesnot depend on the frame used to measure it.

Thus, by Axiom 1, SR is the study of physical properties which remain invariant under lineartransformations preserving the Lorentzian form (the Lorentzian group).

In other words, SR encompasses the study of the geometry of Minkowski spaceL4 = (A4,Q).

We will see that free particles trace time-like geodesics in this geometry.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 5 / 22

Lorentz-Minkowski’s space-time II

In general, different frames experience different space and time measurements for a givenspace-time segment.

However, Axiom 2 implies that being a null segment (Q = 0 ) does not depend on theparticular frame.

From this it is possible to show that the evaluation of Q on any space-time segment also doesnot depend on the frame used to measure it.

Thus, by Axiom 1, SR is the study of physical properties which remain invariant under lineartransformations preserving the Lorentzian form (the Lorentzian group).

In other words, SR encompasses the study of the geometry of Minkowski spaceL4 = (A4,Q).

We will see that free particles trace time-like geodesics in this geometry.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 5 / 22

Lorentz-Minkowski’s space-time II

In general, different frames experience different space and time measurements for a givenspace-time segment.

However, Axiom 2 implies that being a null segment (Q = 0 ) does not depend on theparticular frame.

From this it is possible to show that the evaluation of Q on any space-time segment also doesnot depend on the frame used to measure it.

Thus, by Axiom 1, SR is the study of physical properties which remain invariant under lineartransformations preserving the Lorentzian form (the Lorentzian group).

In other words, SR encompasses the study of the geometry of Minkowski spaceL4 = (A4,Q).

We will see that free particles trace time-like geodesics in this geometry.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 5 / 22

Lorentz-Minkowski’s space-time II

In general, different frames experience different space and time measurements for a givenspace-time segment.

However, Axiom 2 implies that being a null segment (Q = 0 ) does not depend on theparticular frame.

From this it is possible to show that the evaluation of Q on any space-time segment also doesnot depend on the frame used to measure it.

Thus, by Axiom 1, SR is the study of physical properties which remain invariant under lineartransformations preserving the Lorentzian form (the Lorentzian group).

In other words, SR encompasses the study of the geometry of Minkowski spaceL4 = (A4,Q).

We will see that free particles trace time-like geodesics in this geometry.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 5 / 22

Lorentz-Minkowski’s space-time II

In general, different frames experience different space and time measurements for a givenspace-time segment.

However, Axiom 2 implies that being a null segment (Q = 0 ) does not depend on theparticular frame.

From this it is possible to show that the evaluation of Q on any space-time segment also doesnot depend on the frame used to measure it.

Thus, by Axiom 1, SR is the study of physical properties which remain invariant under lineartransformations preserving the Lorentzian form (the Lorentzian group).

In other words, SR encompasses the study of the geometry of Minkowski spaceL4 = (A4,Q).

We will see that free particles trace time-like geodesics in this geometry.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 5 / 22

Lorentz-Minkowski’s space-time II

In general, different frames experience different space and time measurements for a givenspace-time segment.

However, Axiom 2 implies that being a null segment (Q = 0 ) does not depend on theparticular frame.

From this it is possible to show that the evaluation of Q on any space-time segment also doesnot depend on the frame used to measure it.

Thus, by Axiom 1, SR is the study of physical properties which remain invariant under lineartransformations preserving the Lorentzian form (the Lorentzian group).

In other words, SR encompasses the study of the geometry of Minkowski spaceL4 = (A4,Q).

We will see that free particles trace time-like geodesics in this geometry.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 5 / 22

Dynamics in SR I

Dynamics is implemented by using the Lagrangian formalism.

Notice that any time-like particle s ∈ I 7→ ψ(s) ∈ L4 can be reparameterized by its propertime τ , so that

Q(

dψdτ

,dψdτ

)≡ −1.

We then define the action of a free particle by

A(ψ) = α∫

dτ = α∫

Ldt , L = α√

1− v2, v =

√(dxdt

)2+

(dydt

)2+

(dzdt

)2.

Se v � 1 then

L ≈ α−αv2

2,

so we take α = −m, where m > 0 is the rest mass of the particle. Thus, in a given frame, theLagrangian becomes

L = −m√

1− v2

Clearly, this action is minimized on time-like segments.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 6 / 22

Dynamics in SR I

Dynamics is implemented by using the Lagrangian formalism.

Notice that any time-like particle s ∈ I 7→ ψ(s) ∈ L4 can be reparameterized by its propertime τ , so that

Q(

dψdτ

,dψdτ

)≡ −1.

We then define the action of a free particle by

A(ψ) = α∫

dτ = α∫

Ldt , L = α√

1− v2, v =

√(dxdt

)2+

(dydt

)2+

(dzdt

)2.

Se v � 1 then

L ≈ α−αv2

2,

so we take α = −m, where m > 0 is the rest mass of the particle. Thus, in a given frame, theLagrangian becomes

L = −m√

1− v2

Clearly, this action is minimized on time-like segments.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 6 / 22

Dynamics in SR I

Dynamics is implemented by using the Lagrangian formalism.

Notice that any time-like particle s ∈ I 7→ ψ(s) ∈ L4 can be reparameterized by its propertime τ , so that

Q(

dψdτ

,dψdτ

)≡ −1.

We then define the action of a free particle by

A(ψ) = α∫

dτ = α∫

Ldt , L = α√

1− v2, v =

√(dxdt

)2+

(dydt

)2+

(dzdt

)2.

Se v � 1 then

L ≈ α−αv2

2,

so we take α = −m, where m > 0 is the rest mass of the particle. Thus, in a given frame, theLagrangian becomes

L = −m√

1− v2

Clearly, this action is minimized on time-like segments.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 6 / 22

Dynamics in SR I

Dynamics is implemented by using the Lagrangian formalism.

Notice that any time-like particle s ∈ I 7→ ψ(s) ∈ L4 can be reparameterized by its propertime τ , so that

Q(

dψdτ

,dψdτ

)≡ −1.

We then define the action of a free particle by

A(ψ) = α∫

dτ = α∫

Ldt , L = α√

1− v2, v =

√(dxdt

)2+

(dydt

)2+

(dzdt

)2.

Se v � 1 then

L ≈ α−αv2

2,

so we take α = −m, where m > 0 is the rest mass of the particle. Thus, in a given frame, theLagrangian becomes

L = −m√

1− v2

Clearly, this action is minimized on time-like segments.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 6 / 22

Dynamics in SR I

Dynamics is implemented by using the Lagrangian formalism.

Notice that any time-like particle s ∈ I 7→ ψ(s) ∈ L4 can be reparameterized by its propertime τ , so that

Q(

dψdτ

,dψdτ

)≡ −1.

We then define the action of a free particle by

A(ψ) = α∫

dτ = α∫

Ldt , L = α√

1− v2, v =

√(dxdt

)2+

(dydt

)2+

(dzdt

)2.

Se v � 1 then

L ≈ α−αv2

2,

so we take α = −m, where m > 0 is the rest mass of the particle. Thus, in a given frame, theLagrangian becomes

L = −m√

1− v2

Clearly, this action is minimized on time-like segments.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 6 / 22

Dynamics in SR I

Dynamics is implemented by using the Lagrangian formalism.

Notice that any time-like particle s ∈ I 7→ ψ(s) ∈ L4 can be reparameterized by its propertime τ , so that

Q(

dψdτ

,dψdτ

)≡ −1.

We then define the action of a free particle by

A(ψ) = α∫

dτ = α∫

Ldt , L = α√

1− v2, v =

√(dxdt

)2+

(dydt

)2+

(dzdt

)2.

Se v � 1 then

L ≈ α−αv2

2,

so we take α = −m, where m > 0 is the rest mass of the particle. Thus, in a given frame, theLagrangian becomes

L = −m√

1− v2

Clearly, this action is minimized on time-like segments.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 6 / 22

Dynamics in SR I

Dynamics is implemented by using the Lagrangian formalism.

Notice that any time-like particle s ∈ I 7→ ψ(s) ∈ L4 can be reparameterized by its propertime τ , so that

Q(

dψdτ

,dψdτ

)≡ −1.

We then define the action of a free particle by

A(ψ) = α∫

dτ = α∫

Ldt , L = α√

1− v2, v =

√(dxdt

)2+

(dydt

)2+

(dzdt

)2.

Se v � 1 then

L ≈ α−αv2

2,

so we take α = −m, where m > 0 is the rest mass of the particle. Thus, in a given frame, theLagrangian becomes

L = −m√

1− v2

Clearly, this action is minimized on time-like segments.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 6 / 22

Dynamics in SR II

The 4-velocity of ψ is

V =dψdτ

= �(1, v),

where

v =(

dxdt,

dydt,

dzdt

)is the 3-velocity and

� =1√

1− v2.

The 3-momentum is

p =∂L∂v

= �mv .

The 4-momentum isP = (E ,p),

whereE = p · v− L = �m

is the energy. This is Einstein’s famous equation!

Notice that we can apply Noether’s theorem to conclude that E and p are conserved in eachframe. This is consistent with Axiom 1.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 7 / 22

Dynamics in SR II

The 4-velocity of ψ is

V =dψdτ

= �(1, v),

where

v =(

dxdt,

dydt,

dzdt

)is the 3-velocity and

� =1√

1− v2.

The 3-momentum is

p =∂L∂v

= �mv .

The 4-momentum isP = (E ,p),

whereE = p · v− L = �m

is the energy. This is Einstein’s famous equation!

Notice that we can apply Noether’s theorem to conclude that E and p are conserved in eachframe. This is consistent with Axiom 1.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 7 / 22

Dynamics in SR II

The 4-velocity of ψ is

V =dψdτ

= �(1, v),

where

v =(

dxdt,

dydt,

dzdt

)is the 3-velocity and

� =1√

1− v2.

The 3-momentum is

p =∂L∂v

= �mv .

The 4-momentum isP = (E ,p),

whereE = p · v− L = �m

is the energy. This is Einstein’s famous equation!

Notice that we can apply Noether’s theorem to conclude that E and p are conserved in eachframe. This is consistent with Axiom 1.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 7 / 22

Dynamics in SR II

The 4-velocity of ψ is

V =dψdτ

= �(1, v),

where

v =(

dxdt,

dydt,

dzdt

)is the 3-velocity and

� =1√

1− v2.

The 3-momentum is

p =∂L∂v

= �mv .

The 4-momentum isP = (E ,p),

whereE = p · v− L = �m

is the energy. This is Einstein’s famous equation!

Notice that we can apply Noether’s theorem to conclude that E and p are conserved in eachframe. This is consistent with Axiom 1.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 7 / 22

Dynamics in SR II

The 4-velocity of ψ is

V =dψdτ

= �(1, v),

where

v =(

dxdt,

dydt,

dzdt

)is the 3-velocity and

� =1√

1− v2.

The 3-momentum is

p =∂L∂v

= �mv .

The 4-momentum isP = (E ,p),

whereE = p · v− L = �m

is the energy. This is Einstein’s famous equation!

Notice that we can apply Noether’s theorem to conclude that E and p are conserved in eachframe. This is consistent with Axiom 1.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 7 / 22

Dynamics in SR II

The 4-velocity of ψ is

V =dψdτ

= �(1, v),

where

v =(

dxdt,

dydt,

dzdt

)is the 3-velocity and

� =1√

1− v2.

The 3-momentum is

p =∂L∂v

= �mv .

The 4-momentum isP = (E ,p),

whereE = p · v− L = �m

is the energy. This is Einstein’s famous equation!

Notice that we can apply Noether’s theorem to conclude that E and p are conserved in eachframe. This is consistent with Axiom 1.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 7 / 22

Dynamics in SR III

We see thatE2 = |p|2 + m2,

so thatE =

√|p|2 + m2 ≥ |p|.

In words, P is future directed and time-like.

On the other hand,p = Ev,

so that |p| = Ev , where v = |v|.

Thus, v ≥ 1 leads to a contradiction unless v = 1 and m = 0.

This shows that no particle (massive or not!) is allowed to move faster than light.

Moreover, the positivity of energy (which always holds true for massive particles) is closelyrelated to the causal character of the 4-momentum vector P. In fact,

E = −Q(Vobs,P),

where Vobs is the 4-velocity of the (co-moving) observer.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 8 / 22

Dynamics in SR III

We see thatE2 = |p|2 + m2,

so thatE =

√|p|2 + m2 ≥ |p|.

In words, P is future directed and time-like.

On the other hand,p = Ev,

so that |p| = Ev , where v = |v|.

Thus, v ≥ 1 leads to a contradiction unless v = 1 and m = 0.

This shows that no particle (massive or not!) is allowed to move faster than light.

Moreover, the positivity of energy (which always holds true for massive particles) is closelyrelated to the causal character of the 4-momentum vector P. In fact,

E = −Q(Vobs,P),

where Vobs is the 4-velocity of the (co-moving) observer.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 8 / 22

Dynamics in SR III

We see thatE2 = |p|2 + m2,

so thatE =

√|p|2 + m2 ≥ |p|.

In words, P is future directed and time-like.

On the other hand,p = Ev,

so that |p| = Ev , where v = |v|.

Thus, v ≥ 1 leads to a contradiction unless v = 1 and m = 0.

This shows that no particle (massive or not!) is allowed to move faster than light.

Moreover, the positivity of energy (which always holds true for massive particles) is closelyrelated to the causal character of the 4-momentum vector P. In fact,

E = −Q(Vobs,P),

where Vobs is the 4-velocity of the (co-moving) observer.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 8 / 22

Dynamics in SR III

We see thatE2 = |p|2 + m2,

so thatE =

√|p|2 + m2 ≥ |p|.

In words, P is future directed and time-like.

On the other hand,p = Ev,

so that |p| = Ev , where v = |v|.

Thus, v ≥ 1 leads to a contradiction unless v = 1 and m = 0.

This shows that no particle (massive or not!) is allowed to move faster than light.

Moreover, the positivity of energy (which always holds true for massive particles) is closelyrelated to the causal character of the 4-momentum vector P. In fact,

E = −Q(Vobs,P),

where Vobs is the 4-velocity of the (co-moving) observer.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 8 / 22

Dynamics in SR III

We see thatE2 = |p|2 + m2,

so thatE =

√|p|2 + m2 ≥ |p|.

In words, P is future directed and time-like.

On the other hand,p = Ev,

so that |p| = Ev , where v = |v|.

Thus, v ≥ 1 leads to a contradiction unless v = 1 and m = 0.

This shows that no particle (massive or not!) is allowed to move faster than light.

Moreover, the positivity of energy (which always holds true for massive particles) is closelyrelated to the causal character of the 4-momentum vector P. In fact,

E = −Q(Vobs,P),

where Vobs is the 4-velocity of the (co-moving) observer.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 8 / 22

Dynamics in SR III

We see thatE2 = |p|2 + m2,

so thatE =

√|p|2 + m2 ≥ |p|.

In words, P is future directed and time-like.

On the other hand,p = Ev,

so that |p| = Ev , where v = |v|.

Thus, v ≥ 1 leads to a contradiction unless v = 1 and m = 0.

This shows that no particle (massive or not!) is allowed to move faster than light.

Moreover, the positivity of energy (which always holds true for massive particles) is closelyrelated to the causal character of the 4-momentum vector P. In fact,

E = −Q(Vobs,P),

where Vobs is the 4-velocity of the (co-moving) observer.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 8 / 22

Dynamics in SR III

We see thatE2 = |p|2 + m2,

so thatE =

√|p|2 + m2 ≥ |p|.

In words, P is future directed and time-like.

On the other hand,p = Ev,

so that |p| = Ev , where v = |v|.

Thus, v ≥ 1 leads to a contradiction unless v = 1 and m = 0.

This shows that no particle (massive or not!) is allowed to move faster than light.

Moreover, the positivity of energy (which always holds true for massive particles) is closelyrelated to the causal character of the 4-momentum vector P. In fact,

E = −Q(Vobs,P),

where Vobs is the 4-velocity of the (co-moving) observer.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 8 / 22

The transition to GR: The Equivalence Principle

Although extremely successful from the experimental viewpoint, SR is certainly an incompletetheory. For example, it fails to incorporate gravitational effects.

The starting point to fix this is Einstein’s happiest thought: an observer in free fall in agravitational field does not feel his own weight.

This leads to the Equivalence Principle: free fall in a uniform gravitational field is equivalent touniformly accelerated motion. In other words, gravitational effects can always be eliminatedby referring to frame that is uniformly accelerated with respect to a (local) inertial frame.

Thus, SR still works fine locally (i.e. in the presence of a uniform gravitational field).

The basic question now is to understand what happens at larger scales, where tidal effectstake place.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 9 / 22

The transition to GR: The Equivalence Principle

Although extremely successful from the experimental viewpoint, SR is certainly an incompletetheory. For example, it fails to incorporate gravitational effects.

The starting point to fix this is Einstein’s happiest thought: an observer in free fall in agravitational field does not feel his own weight.

This leads to the Equivalence Principle: free fall in a uniform gravitational field is equivalent touniformly accelerated motion. In other words, gravitational effects can always be eliminatedby referring to frame that is uniformly accelerated with respect to a (local) inertial frame.

Thus, SR still works fine locally (i.e. in the presence of a uniform gravitational field).

The basic question now is to understand what happens at larger scales, where tidal effectstake place.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 9 / 22

The transition to GR: The Equivalence Principle

Although extremely successful from the experimental viewpoint, SR is certainly an incompletetheory. For example, it fails to incorporate gravitational effects.

The starting point to fix this is Einstein’s happiest thought: an observer in free fall in agravitational field does not feel his own weight.

This leads to the Equivalence Principle: free fall in a uniform gravitational field is equivalent touniformly accelerated motion. In other words, gravitational effects can always be eliminatedby referring to frame that is uniformly accelerated with respect to a (local) inertial frame.

Thus, SR still works fine locally (i.e. in the presence of a uniform gravitational field).

The basic question now is to understand what happens at larger scales, where tidal effectstake place.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 9 / 22

The transition to GR: The Equivalence Principle

Although extremely successful from the experimental viewpoint, SR is certainly an incompletetheory. For example, it fails to incorporate gravitational effects.

The starting point to fix this is Einstein’s happiest thought: an observer in free fall in agravitational field does not feel his own weight.

This leads to the Equivalence Principle: free fall in a uniform gravitational field is equivalent touniformly accelerated motion. In other words, gravitational effects can always be eliminatedby referring to frame that is uniformly accelerated with respect to a (local) inertial frame.

Thus, SR still works fine locally (i.e. in the presence of a uniform gravitational field).

The basic question now is to understand what happens at larger scales, where tidal effectstake place.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 9 / 22

The transition to GR: The Equivalence Principle

Although extremely successful from the experimental viewpoint, SR is certainly an incompletetheory. For example, it fails to incorporate gravitational effects.

The starting point to fix this is Einstein’s happiest thought: an observer in free fall in agravitational field does not feel his own weight.

This leads to the Equivalence Principle: free fall in a uniform gravitational field is equivalent touniformly accelerated motion. In other words, gravitational effects can always be eliminatedby referring to frame that is uniformly accelerated with respect to a (local) inertial frame.

Thus, SR still works fine locally (i.e. in the presence of a uniform gravitational field).

The basic question now is to understand what happens at larger scales, where tidal effectstake place.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 9 / 22

The transition to GR: The Equivalence Principle

Although extremely successful from the experimental viewpoint, SR is certainly an incompletetheory. For example, it fails to incorporate gravitational effects.

The starting point to fix this is Einstein’s happiest thought: an observer in free fall in agravitational field does not feel his own weight.

This leads to the Equivalence Principle: free fall in a uniform gravitational field is equivalent touniformly accelerated motion. In other words, gravitational effects can always be eliminatedby referring to frame that is uniformly accelerated with respect to a (local) inertial frame.

Thus, SR still works fine locally (i.e. in the presence of a uniform gravitational field).

The basic question now is to understand what happens at larger scales, where tidal effectstake place.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 9 / 22

The conceptual framework of GR

Since tidal effects are of second order in the field, we are led to explain them by postulatingthat the geometry of space-time might be curved.

Thus, we assume that space-time now is endowed with a (not necessarily flat) Lorentzianmetric g.

The metric g says how matter moves, since free particles are constrained to trace time-likegeodesics. Hence, the metric plays the role of the gravitational potential.

But the metric itself is a dynamical entity since it should satisfy the field equations, which, inanalogy with Newtonian theory, should take the form

E = T ,

where E is a twice covariant, symmetric tensor depending on the derivatives of g up tosecond order and T is a similar tensor describing the matter-energy distribution of theuniverse.

The shape of E is determined by implementing the Lagrangian formalism in the space ofmetrics.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 10 / 22

The conceptual framework of GR

Since tidal effects are of second order in the field, we are led to explain them by postulatingthat the geometry of space-time might be curved.

Thus, we assume that space-time now is endowed with a (not necessarily flat) Lorentzianmetric g.

The metric g says how matter moves, since free particles are constrained to trace time-likegeodesics. Hence, the metric plays the role of the gravitational potential.

But the metric itself is a dynamical entity since it should satisfy the field equations, which, inanalogy with Newtonian theory, should take the form

E = T ,

where E is a twice covariant, symmetric tensor depending on the derivatives of g up tosecond order and T is a similar tensor describing the matter-energy distribution of theuniverse.

The shape of E is determined by implementing the Lagrangian formalism in the space ofmetrics.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 10 / 22

The conceptual framework of GR

Since tidal effects are of second order in the field, we are led to explain them by postulatingthat the geometry of space-time might be curved.

Thus, we assume that space-time now is endowed with a (not necessarily flat) Lorentzianmetric g.

The metric g says how matter moves, since free particles are constrained to trace time-likegeodesics. Hence, the metric plays the role of the gravitational potential.

But the metric itself is a dynamical entity since it should satisfy the field equations, which, inanalogy with Newtonian theory, should take the form

E = T ,

where E is a twice covariant, symmetric tensor depending on the derivatives of g up tosecond order and T is a similar tensor describing the matter-energy distribution of theuniverse.

The shape of E is determined by implementing the Lagrangian formalism in the space ofmetrics.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 10 / 22

The conceptual framework of GR

Since tidal effects are of second order in the field, we are led to explain them by postulatingthat the geometry of space-time might be curved.

Thus, we assume that space-time now is endowed with a (not necessarily flat) Lorentzianmetric g.

The metric g says how matter moves, since free particles are constrained to trace time-likegeodesics. Hence, the metric plays the role of the gravitational potential.

But the metric itself is a dynamical entity since it should satisfy the field equations, which, inanalogy with Newtonian theory, should take the form

E = T ,

where E is a twice covariant, symmetric tensor depending on the derivatives of g up tosecond order and T is a similar tensor describing the matter-energy distribution of theuniverse.

The shape of E is determined by implementing the Lagrangian formalism in the space ofmetrics.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 10 / 22

The conceptual framework of GR

Since tidal effects are of second order in the field, we are led to explain them by postulatingthat the geometry of space-time might be curved.

Thus, we assume that space-time now is endowed with a (not necessarily flat) Lorentzianmetric g.

The metric g says how matter moves, since free particles are constrained to trace time-likegeodesics. Hence, the metric plays the role of the gravitational potential.

But the metric itself is a dynamical entity since it should satisfy the field equations, which, inanalogy with Newtonian theory, should take the form

E = T ,

where E is a twice covariant, symmetric tensor depending on the derivatives of g up tosecond order and T is a similar tensor describing the matter-energy distribution of theuniverse.

The shape of E is determined by implementing the Lagrangian formalism in the space ofmetrics.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 10 / 22

The conceptual framework of GR

Since tidal effects are of second order in the field, we are led to explain them by postulatingthat the geometry of space-time might be curved.

Thus, we assume that space-time now is endowed with a (not necessarily flat) Lorentzianmetric g.

The metric g says how matter moves, since free particles are constrained to trace time-likegeodesics. Hence, the metric plays the role of the gravitational potential.

But the metric itself is a dynamical entity since it should satisfy the field equations, which, inanalogy with Newtonian theory, should take the form

E = T ,

where E is a twice covariant, symmetric tensor depending on the derivatives of g up tosecond order and T is a similar tensor describing the matter-energy distribution of theuniverse.

The shape of E is determined by implementing the Lagrangian formalism in the space ofmetrics.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 10 / 22

The conceptual framework of GR

Since tidal effects are of second order in the field, we are led to explain them by postulatingthat the geometry of space-time might be curved.

Thus, we assume that space-time now is endowed with a (not necessarily flat) Lorentzianmetric g.

The metric g says how matter moves, since free particles are constrained to trace time-likegeodesics. Hence, the metric plays the role of the gravitational potential.

But the metric itself is a dynamical entity since it should satisfy the field equations, which, inanalogy with Newtonian theory, should take the form

E = T ,

where E is a twice covariant, symmetric tensor depending on the derivatives of g up tosecond order and T is a similar tensor describing the matter-energy distribution of theuniverse.

The shape of E is determined by implementing the Lagrangian formalism in the space ofmetrics.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 10 / 22

The Lagrangian formalism in GR

Following Hilbert, we take

A(g) =∫

MRg√−g, g = det g,

as the gravitational action. Here, R denotes scalar curvature.

If δg is a variation for the metric, we have

δR = div(div δg − d tr δg)− 〈Ric, δg〉

andδ√−g =

12〈g, δg〉

√−g.

This gives

δA = −∫

M〈E , δg〉

√−g +

∫∂M

(div δg − d tr δg)(ν)√−g∂M ,

whereE = Ric−

R2

g

is the Einstein tensor.

Notice that the field equations E = T imply divgT = 0.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 11 / 22

The Lagrangian formalism in GR

Following Hilbert, we take

A(g) =∫

MRg√−g, g = det g,

as the gravitational action. Here, R denotes scalar curvature.

If δg is a variation for the metric, we have

δR = div(div δg − d tr δg)− 〈Ric, δg〉

andδ√−g =

12〈g, δg〉

√−g.

This gives

δA = −∫

M〈E , δg〉

√−g +

∫∂M

(div δg − d tr δg)(ν)√−g∂M ,

whereE = Ric−

R2

g

is the Einstein tensor.

Notice that the field equations E = T imply divgT = 0.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 11 / 22

The Lagrangian formalism in GR

Following Hilbert, we take

A(g) =∫

MRg√−g, g = det g,

as the gravitational action. Here, R denotes scalar curvature.

If δg is a variation for the metric, we have

δR = div(div δg − d tr δg)− 〈Ric, δg〉

andδ√−g =

12〈g, δg〉

√−g.

This gives

δA = −∫

M〈E , δg〉

√−g +

∫∂M

(div δg − d tr δg)(ν)√−g∂M ,

whereE = Ric−

R2

g

is the Einstein tensor.

Notice that the field equations E = T imply divgT = 0.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 11 / 22

The Lagrangian formalism in GR

Following Hilbert, we take

A(g) =∫

MRg√−g, g = det g,

as the gravitational action. Here, R denotes scalar curvature.

If δg is a variation for the metric, we have

δR = div(div δg − d tr δg)− 〈Ric, δg〉

andδ√−g =

12〈g, δg〉

√−g.

This gives

δA = −∫

M〈E , δg〉

√−g +

∫∂M

(div δg − d tr δg)(ν)√−g∂M ,

whereE = Ric−

R2

g

is the Einstein tensor.

Notice that the field equations E = T imply divgT = 0.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 11 / 22

The Lagrangian formalism in GR

Following Hilbert, we take

A(g) =∫

MRg√−g, g = det g,

as the gravitational action. Here, R denotes scalar curvature.

If δg is a variation for the metric, we have

δR = div(div δg − d tr δg)− 〈Ric, δg〉

andδ√−g =

12〈g, δg〉

√−g.

This gives

δA = −∫

M〈E , δg〉

√−g +

∫∂M

(div δg − d tr δg)(ν)√−g∂M ,

whereE = Ric−

R2

g

is the Einstein tensor.

Notice that the field equations E = T imply divgT = 0.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 11 / 22

The Lagrangian formalism in GR

Following Hilbert, we take

A(g) =∫

MRg√−g, g = det g,

as the gravitational action. Here, R denotes scalar curvature.

If δg is a variation for the metric, we have

δR = div(div δg − d tr δg)− 〈Ric, δg〉

andδ√−g =

12〈g, δg〉

√−g.

This gives

δA = −∫

M〈E , δg〉

√−g +

∫∂M

(div δg − d tr δg)(ν)√−g∂M ,

whereE = Ric−

R2

g

is the Einstein tensor.

Notice that the field equations E = T imply divgT = 0.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 11 / 22

The initial value formulation of GR

TheoremLet (M, g) be a solution of Einstein equation E = T which admits a space-time hypersurface(S, h) # (M, g) with a time-like unit normal N. Then the following constraint equations hold:

T (N,N) =12

(Rh − |k |2h + K2), T (·,N) = (divgk − dK )(·), (1)

where k(·, ·) = g(D·N, ·) is the second fundamental form of S and K = trhk is the meancurvature. In particular, if (M, g) is a vacuum solution,

Rh − |k |2h + K2 = 0, divgk − dK = 0. (2)

The following result, due to Choquet-Bruhat, shows the relevance of the constraint equations.

TheoremIf (S, h, k) is an initial data set satisfying the vacuum constraint equations then there exists aspace-time (M, g) satisfying Ricg = 0, and an isometric embedding i : (S, h) ↪→ (M, g) inducing hand k as the metric and second fundamental form, respectively.

The IVF is extremely useful in Numerical GR.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 12 / 22

The initial value formulation of GR

TheoremLet (M, g) be a solution of Einstein equation E = T which admits a space-time hypersurface(S, h) # (M, g) with a time-like unit normal N. Then the following constraint equations hold:

T (N,N) =12

(Rh − |k |2h + K2), T (·,N) = (divgk − dK )(·), (1)

where k(·, ·) = g(D·N, ·) is the second fundamental form of S and K = trhk is the meancurvature. In particular, if (M, g) is a vacuum solution,

Rh − |k |2h + K2 = 0, divgk − dK = 0. (2)

The following result, due to Choquet-Bruhat, shows the relevance of the constraint equations.

TheoremIf (S, h, k) is an initial data set satisfying the vacuum constraint equations then there exists aspace-time (M, g) satisfying Ricg = 0, and an isometric embedding i : (S, h) ↪→ (M, g) inducing hand k as the metric and second fundamental form, respectively.

The IVF is extremely useful in Numerical GR.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 12 / 22

The initial value formulation of GR

TheoremLet (M, g) be a solution of Einstein equation E = T which admits a space-time hypersurface(S, h) # (M, g) with a time-like unit normal N. Then the following constraint equations hold:

T (N,N) =12

(Rh − |k |2h + K2), T (·,N) = (divgk − dK )(·), (1)

where k(·, ·) = g(D·N, ·) is the second fundamental form of S and K = trhk is the meancurvature. In particular, if (M, g) is a vacuum solution,

Rh − |k |2h + K2 = 0, divgk − dK = 0. (2)

The following result, due to Choquet-Bruhat, shows the relevance of the constraint equations.

TheoremIf (S, h, k) is an initial data set satisfying the vacuum constraint equations then there exists aspace-time (M, g) satisfying Ricg = 0, and an isometric embedding i : (S, h) ↪→ (M, g) inducing hand k as the metric and second fundamental form, respectively.

The IVF is extremely useful in Numerical GR.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 12 / 22

The initial value formulation of GR

TheoremLet (M, g) be a solution of Einstein equation E = T which admits a space-time hypersurface(S, h) # (M, g) with a time-like unit normal N. Then the following constraint equations hold:

T (N,N) =12

(Rh − |k |2h + K2), T (·,N) = (divgk − dK )(·), (1)

where k(·, ·) = g(D·N, ·) is the second fundamental form of S and K = trhk is the meancurvature. In particular, if (M, g) is a vacuum solution,

Rh − |k |2h + K2 = 0, divgk − dK = 0. (2)

The following result, due to Choquet-Bruhat, shows the relevance of the constraint equations.

TheoremIf (S, h, k) is an initial data set satisfying the vacuum constraint equations then there exists aspace-time (M, g) satisfying Ricg = 0, and an isometric embedding i : (S, h) ↪→ (M, g) inducing hand k as the metric and second fundamental form, respectively.

The IVF is extremely useful in Numerical GR.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 12 / 22

The initial value formulation of GR

TheoremLet (M, g) be a solution of Einstein equation E = T which admits a space-time hypersurface(S, h) # (M, g) with a time-like unit normal N. Then the following constraint equations hold:

T (N,N) =12

(Rh − |k |2h + K2), T (·,N) = (divgk − dK )(·), (1)

where k(·, ·) = g(D·N, ·) is the second fundamental form of S and K = trhk is the meancurvature. In particular, if (M, g) is a vacuum solution,

Rh − |k |2h + K2 = 0, divgk − dK = 0. (2)

The following result, due to Choquet-Bruhat, shows the relevance of the constraint equations.

TheoremIf (S, h, k) is an initial data set satisfying the vacuum constraint equations then there exists aspace-time (M, g) satisfying Ricg = 0, and an isometric embedding i : (S, h) ↪→ (M, g) inducing hand k as the metric and second fundamental form, respectively.

The IVF is extremely useful in Numerical GR.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 12 / 22

The initial value formulation of GR

TheoremLet (M, g) be a solution of Einstein equation E = T which admits a space-time hypersurface(S, h) # (M, g) with a time-like unit normal N. Then the following constraint equations hold:

T (N,N) =12

(Rh − |k |2h + K2), T (·,N) = (divgk − dK )(·), (1)

where k(·, ·) = g(D·N, ·) is the second fundamental form of S and K = trhk is the meancurvature. In particular, if (M, g) is a vacuum solution,

Rh − |k |2h + K2 = 0, divgk − dK = 0. (2)

The following result, due to Choquet-Bruhat, shows the relevance of the constraint equations.

TheoremIf (S, h, k) is an initial data set satisfying the vacuum constraint equations then there exists aspace-time (M, g) satisfying Ricg = 0, and an isometric embedding i : (S, h) ↪→ (M, g) inducing hand k as the metric and second fundamental form, respectively.

The IVF is extremely useful in Numerical GR.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 12 / 22

The energy problem in GR

In GR, the Equivalence Principle prohibits the existence of any energy density.

This shows in particular that any definition of total energy should necessarily involve a globalconstruction.

The ADM formulation of GR allows us to define the total mass for isolated systems.

Thus we assume that our space-time admits a space-like slice (S, h, k) so that, at infinity,

hij = δij + O(r−1)

andkij = O(r−2).

In words, the solution becomes Minkowskian at spatial infinity.

The idea now is to implement the Hamiltonian scheme to this situation.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 13 / 22

The energy problem in GR

In GR, the Equivalence Principle prohibits the existence of any energy density.

This shows in particular that any definition of total energy should necessarily involve a globalconstruction.

The ADM formulation of GR allows us to define the total mass for isolated systems.

Thus we assume that our space-time admits a space-like slice (S, h, k) so that, at infinity,

hij = δij + O(r−1)

andkij = O(r−2).

In words, the solution becomes Minkowskian at spatial infinity.

The idea now is to implement the Hamiltonian scheme to this situation.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 13 / 22

The energy problem in GR

In GR, the Equivalence Principle prohibits the existence of any energy density.

This shows in particular that any definition of total energy should necessarily involve a globalconstruction.

The ADM formulation of GR allows us to define the total mass for isolated systems.

Thus we assume that our space-time admits a space-like slice (S, h, k) so that, at infinity,

hij = δij + O(r−1)

andkij = O(r−2).

In words, the solution becomes Minkowskian at spatial infinity.

The idea now is to implement the Hamiltonian scheme to this situation.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 13 / 22

The energy problem in GR

In GR, the Equivalence Principle prohibits the existence of any energy density.

This shows in particular that any definition of total energy should necessarily involve a globalconstruction.

The ADM formulation of GR allows us to define the total mass for isolated systems.

Thus we assume that our space-time admits a space-like slice (S, h, k) so that, at infinity,

hij = δij + O(r−1)

andkij = O(r−2).

In words, the solution becomes Minkowskian at spatial infinity.

The idea now is to implement the Hamiltonian scheme to this situation.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 13 / 22

The energy problem in GR

In GR, the Equivalence Principle prohibits the existence of any energy density.

This shows in particular that any definition of total energy should necessarily involve a globalconstruction.

The ADM formulation of GR allows us to define the total mass for isolated systems.

Thus we assume that our space-time admits a space-like slice (S, h, k) so that, at infinity,

hij = δij + O(r−1)

andkij = O(r−2).

In words, the solution becomes Minkowskian at spatial infinity.

The idea now is to implement the Hamiltonian scheme to this situation.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 13 / 22

The energy problem in GR

In GR, the Equivalence Principle prohibits the existence of any energy density.

This shows in particular that any definition of total energy should necessarily involve a globalconstruction.

The ADM formulation of GR allows us to define the total mass for isolated systems.

Thus we assume that our space-time admits a space-like slice (S, h, k) so that, at infinity,

hij = δij + O(r−1)

andkij = O(r−2).

In words, the solution becomes Minkowskian at spatial infinity.

The idea now is to implement the Hamiltonian scheme to this situation.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 13 / 22

The energy problem in GR

In GR, the Equivalence Principle prohibits the existence of any energy density.

This shows in particular that any definition of total energy should necessarily involve a globalconstruction.

The ADM formulation of GR allows us to define the total mass for isolated systems.

Thus we assume that our space-time admits a space-like slice (S, h, k) so that, at infinity,

hij = δij + O(r−1)

andkij = O(r−2).

In words, the solution becomes Minkowskian at spatial infinity.

The idea now is to implement the Hamiltonian scheme to this situation.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 13 / 22

The energy problem in GR

In GR, the Equivalence Principle prohibits the existence of any energy density.

This shows in particular that any definition of total energy should necessarily involve a globalconstruction.

The ADM formulation of GR allows us to define the total mass for isolated systems.

Thus we assume that our space-time admits a space-like slice (S, h, k) so that, at infinity,

hij = δij + O(r−1)

andkij = O(r−2).

In words, the solution becomes Minkowskian at spatial infinity.

The idea now is to implement the Hamiltonian scheme to this situation.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 13 / 22

The Hamiltonian formalism in GR I

As in the IVF, we foliate M by a one-prameter family of space-like slices St .

We assume that the slices evolve in time with speed given by

∂t = λN + Λ,

where λ is the lapse function and Λ is the shift vector. Here, N is the unit normal vector to theslices.

In these coordinates,

g = −λ2dt2 + hij (dx i + Λi dt)(dx j + Λj dt).

Thus, it is natural to take q = (h, λ,Λ) as the configuration space.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 14 / 22

The Hamiltonian formalism in GR I

As in the IVF, we foliate M by a one-prameter family of space-like slices St .

We assume that the slices evolve in time with speed given by

∂t = λN + Λ,

where λ is the lapse function and Λ is the shift vector. Here, N is the unit normal vector to theslices.

In these coordinates,

g = −λ2dt2 + hij (dx i + Λi dt)(dx j + Λj dt).

Thus, it is natural to take q = (h, λ,Λ) as the configuration space.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 14 / 22

The Hamiltonian formalism in GR I

As in the IVF, we foliate M by a one-prameter family of space-like slices St .

We assume that the slices evolve in time with speed given by

∂t = λN + Λ,

where λ is the lapse function and Λ is the shift vector. Here, N is the unit normal vector to theslices.

In these coordinates,

g = −λ2dt2 + hij (dx i + Λi dt)(dx j + Λj dt).

Thus, it is natural to take q = (h, λ,Λ) as the configuration space.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 14 / 22

The Hamiltonian formalism in GR I

As in the IVF, we foliate M by a one-prameter family of space-like slices St .

We assume that the slices evolve in time with speed given by

∂t = λN + Λ,

where λ is the lapse function and Λ is the shift vector. Here, N is the unit normal vector to theslices.

In these coordinates,

g = −λ2dt2 + hij (dx i + Λi dt)(dx j + Λj dt).

Thus, it is natural to take q = (h, λ,Λ) as the configuration space.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 14 / 22

The Hamiltonian formalism in GR I

As in the IVF, we foliate M by a one-prameter family of space-like slices St .

We assume that the slices evolve in time with speed given by

∂t = λN + Λ,

where λ is the lapse function and Λ is the shift vector. Here, N is the unit normal vector to theslices.

In these coordinates,

g = −λ2dt2 + hij (dx i + Λi dt)(dx j + Λj dt).

Thus, it is natural to take q = (h, λ,Λ) as the configuration space.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 14 / 22

The Hamiltonian formalism in GR I

As in the IVF, we foliate M by a one-prameter family of space-like slices St .

We assume that the slices evolve in time with speed given by

∂t = λN + Λ,

where λ is the lapse function and Λ is the shift vector. Here, N is the unit normal vector to theslices.

In these coordinates,

g = −λ2dt2 + hij (dx i + Λi dt)(dx j + Λj dt).

Thus, it is natural to take q = (h, λ,Λ) as the configuration space.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 14 / 22

The Hamiltonian formalism in GR II

We have seen that the gravitational Lagrangian density is

L = Rg√−g = 2(EabNaNb − RabNaNb)λ

√h,

where h = det h.

ButEabNaNb =

12

(Rh − |k |2h + K2),

RabNaNb = K 2 − |k |2h − Da(NaDcNc) + Dc(NaDaNc),

andkab =

12λ−1(ḣab −∇aΛb −∇bΛa).

Thus, the conjugate momenta are

πab =∂L∂ḣab

=√

h(kab − Khab), πa =∂L∂Λ̇a

= 0, π =∂L∂λ̇

= 0.

As expected, λ and Λ are not dynamical variables, so that q = h indeed.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 15 / 22

The Hamiltonian formalism in GR II

We have seen that the gravitational Lagrangian density is

L = Rg√−g = 2(EabNaNb − RabNaNb)λ

√h,

where h = det h.

ButEabNaNb =

12

(Rh − |k |2h + K2),

RabNaNb = K 2 − |k |2h − Da(NaDcNc) + Dc(NaDaNc),

andkab =

12λ−1(ḣab −∇aΛb −∇bΛa).

Thus, the conjugate momenta are

πab =∂L∂ḣab

=√

h(kab − Khab), πa =∂L∂Λ̇a

= 0, π =∂L∂λ̇

= 0.

As expected, λ and Λ are not dynamical variables, so that q = h indeed.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 15 / 22

The Hamiltonian formalism in GR II

We have seen that the gravitational Lagrangian density is

L = Rg√−g = 2(EabNaNb − RabNaNb)λ

√h,

where h = det h.

ButEabNaNb =

12

(Rh − |k |2h + K2),

RabNaNb = K 2 − |k |2h − Da(NaDcNc) + Dc(NaDaNc),

andkab =

12λ−1(ḣab −∇aΛb −∇bΛa).

Thus, the conjugate momenta are

πab =∂L∂ḣab

=√

h(kab − Khab), πa =∂L∂Λ̇a

= 0, π =∂L∂λ̇

= 0.

As expected, λ and Λ are not dynamical variables, so that q = h indeed.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 15 / 22

The Hamiltonian formalism in GR II

We have seen that the gravitational Lagrangian density is

L = Rg√−g = 2(EabNaNb − RabNaNb)λ

√h,

where h = det h.

ButEabNaNb =

12

(Rh − |k |2h + K2),

RabNaNb = K 2 − |k |2h − Da(NaDcNc) + Dc(NaDaNc),

andkab =

12λ−1(ḣab −∇aΛb −∇bΛa).

Thus, the conjugate momenta are

πab =∂L∂ḣab

=√

h(kab − Khab), πa =∂L∂Λ̇a

= 0, π =∂L∂λ̇

= 0.

As expected, λ and Λ are not dynamical variables, so that q = h indeed.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 15 / 22

The Hamiltonian formalism in GR II

We have seen that the gravitational Lagrangian density is

L = Rg√−g = 2(EabNaNb − RabNaNb)λ

√h,

where h = det h.

ButEabNaNb =

12

(Rh − |k |2h + K2),

RabNaNb = K 2 − |k |2h − Da(NaDcNc) + Dc(NaDaNc),

andkab =

12λ−1(ḣab −∇aΛb −∇bΛa).

Thus, the conjugate momenta are

πab =∂L∂ḣab

=√

h(kab − Khab), πa =∂L∂Λ̇a

= 0, π =∂L∂λ̇

= 0.

As expected, λ and Λ are not dynamical variables, so that q = h indeed.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 15 / 22

The Hamiltonian formalism in GR II

We have seen that the gravitational Lagrangian density is

L = Rg√−g = 2(EabNaNb − RabNaNb)λ

√h,

where h = det h.

ButEabNaNb =

12

(Rh − |k |2h + K2),

RabNaNb = K 2 − |k |2h − Da(NaDcNc) + Dc(NaDaNc),

andkab =

12λ−1(ḣab −∇aΛb −∇bΛa).

Thus, the conjugate momenta are

πab =∂L∂ḣab

=√

h(kab − Khab), πa =∂L∂Λ̇a

= 0, π =∂L∂λ̇

= 0.

As expected, λ and Λ are not dynamical variables, so that q = h indeed.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 15 / 22

The Hamiltonian formalism in GR III

The Hamiltonian density is

H = πab ḣab − L

=√

h

(λ

[−Rh +

|π|2hh−

Π2

2h

]+ Λa

[−2∇b

(πab√h

)])+

+2∇a

(Λbπ

ab√h

),

where Π = trhπ, and the full Hamiltonian is

H =∫

StH.

Variation of H with respect to λ and Λ yield

C := −Rh +|π|2hh−

Π2

2h= 0, Ca := −2∇b

(πab√h

)= 0,

which are the initial value constraints we have met earlier.

These should be appended to the Hamiltonian equations

ḣab =δHδπab

, π̇ab = −δHδhab

,

in order to obtain a constrained Hamiltonian formulation for GR.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 16 / 22

The Hamiltonian formalism in GR IIIThe Hamiltonian density is

H = πab ḣab − L

=√

h

(λ

[−Rh +

|π|2hh−

Π2

2h

]+ Λa

[−2∇b

(πab√h

)])+

+2∇a

(Λbπ

ab√h

),

where Π = trhπ, and the full Hamiltonian is

H =∫

StH.

Variation of H with respect to λ and Λ yield

C := −Rh +|π|2hh−

Π2

2h= 0, Ca := −2∇b

(πab√h

)= 0,

which are the initial value constraints we have met earlier.

These should be appended to the Hamiltonian equations

ḣab =δHδπab

, π̇ab = −δHδhab

,

in order to obtain a constrained Hamiltonian formulation for GR.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 16 / 22

The Hamiltonian formalism in GR IIIThe Hamiltonian density is

H = πab ḣab − L

=√

h

(λ

[−Rh +

|π|2hh−

Π2

2h

]+ Λa

[−2∇b

(πab√h

)])+

+2∇a

(Λbπ

ab√h

),

where Π = trhπ, and the full Hamiltonian is

H =∫

StH.

Variation of H with respect to λ and Λ yield

C := −Rh +|π|2hh−

Π2

2h= 0, Ca := −2∇b

(πab√h

)= 0,

which are the initial value constraints we have met earlier.

These should be appended to the Hamiltonian equations

ḣab =δHδπab

, π̇ab = −δHδhab

,

in order to obtain a constrained Hamiltonian formulation for GR.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 16 / 22

The Hamiltonian formalism in GR IIIThe Hamiltonian density is

H = πab ḣab − L

=√

h

(λ

[−Rh +

|π|2hh−

Π2

2h

]+ Λa

[−2∇b

(πab√h

)])+

+2∇a

(Λbπ

ab√h

),

where Π = trhπ, and the full Hamiltonian is

H =∫

StH.

Variation of H with respect to λ and Λ yield

C := −Rh +|π|2hh−

Π2

2h= 0, Ca := −2∇b

(πab√h

)= 0,

which are the initial value constraints we have met earlier.

These should be appended to the Hamiltonian equations

ḣab =δHδπab

, π̇ab = −δHδhab

,

in order to obtain a constrained Hamiltonian formulation for GR.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 16 / 22

The Hamiltonian formalism in GR IIIThe Hamiltonian density is

H = πab ḣab − L

=√

h

(λ

[−Rh +

|π|2hh−

Π2

2h

]+ Λa

[−2∇b

(πab√h

)])+

+2∇a

(Λbπ

ab√h

),

where Π = trhπ, and the full Hamiltonian is

H =∫

StH.

Variation of H with respect to λ and Λ yield

C := −Rh +|π|2hh−

Π2

2h= 0, Ca := −2∇b

(πab√h

)= 0,

which are the initial value constraints we have met earlier.

These should be appended to the Hamiltonian equations

ḣab =δHδπab

, π̇ab = −δHδhab

,

in order to obtain a constrained Hamiltonian formulation for GR.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 16 / 22

The ADM 4-momentum vector

The total energy of the system at a given time is obtained by evaluating the full Hamiltonianon the corresponding slice.

Carefully keeping track of boundary terms we get

H(t) =∫

St

(λC + ΛaCa

)√ht + (Eλ− paΛa),

where (λ,Λ) = limr→+∞(λ,Λ),

E =1

16πlim

r→+∞

∫Σr

(hab,b − hbb,a)νadΣr ,

andpa =

18π

limr→+∞

∫Σr

(kab − Khab)νbdΣr ,

where Σr ⊂ S is a large coordinate sphere.

Thus, the total energy, as measured by an observer at spatial infinity moving with 4-velocity(λ,Λ), is

H = −Q((

λ

Λ

),

(Ep

)).

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 17 / 22

The ADM 4-momentum vector

The total energy of the system at a given time is obtained by evaluating the full Hamiltonianon the corresponding slice.

Carefully keeping track of boundary terms we get

H(t) =∫

St

(λC + ΛaCa

)√ht + (Eλ− paΛa),

where (λ,Λ) = limr→+∞(λ,Λ),

E =1

16πlim

r→+∞

∫Σr

(hab,b − hbb,a)νadΣr ,

andpa =

18π

limr→+∞

∫Σr

(kab − Khab)νbdΣr ,

where Σr ⊂ S is a large coordinate sphere.

Thus, the total energy, as measured by an observer at spatial infinity moving with 4-velocity(λ,Λ), is

H = −Q((

λ

Λ

),

(Ep

)).

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 17 / 22

The ADM 4-momentum vector

The total energy of the system at a given time is obtained by evaluating the full Hamiltonianon the corresponding slice.

Carefully keeping track of boundary terms we get

H(t) =∫

St

(λC + ΛaCa

)√ht + (Eλ− paΛa),

where (λ,Λ) = limr→+∞(λ,Λ),

E =1

16πlim

r→+∞

∫Σr

(hab,b − hbb,a)νadΣr ,

andpa =

18π

limr→+∞

∫Σr

(kab − Khab)νbdΣr ,

where Σr ⊂ S is a large coordinate sphere.

Thus, the total energy, as measured by an observer at spatial infinity moving with 4-velocity(λ,Λ), is

H = −Q((

λ

Λ

),

(Ep

)).

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 17 / 22

The ADM 4-momentum vector

The total energy of the system at a given time is obtained by evaluating the full Hamiltonianon the corresponding slice.

Carefully keeping track of boundary terms we get

H(t) =∫

St

(λC + ΛaCa

)√ht + (Eλ− paΛa),

where (λ,Λ) = limr→+∞(λ,Λ),

E =1

16πlim

r→+∞

∫Σr

(hab,b − hbb,a)νadΣr ,

andpa =

18π

limr→+∞

∫Σr

(kab − Khab)νbdΣr ,

where Σr ⊂ S is a large coordinate sphere.

Thus, the total energy, as measured by an observer at spatial infinity moving with 4-velocity(λ,Λ), is

H = −Q((

λ

Λ

),

(Ep

)).

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 17 / 22

The ADM 4-momentum vector

The total energy of the system at a given time is obtained by evaluating the full Hamiltonianon the corresponding slice.

Carefully keeping track of boundary terms we get

H(t) =∫

St

(λC + ΛaCa

)√ht + (Eλ− paΛa),

where (λ,Λ) = limr→+∞(λ,Λ),

E =1

16πlim

r→+∞

∫Σr

(hab,b − hbb,a)νadΣr ,

andpa =

18π

limr→+∞

∫Σr

(kab − Khab)νbdΣr ,

where Σr ⊂ S is a large coordinate sphere.

Thus, the total energy, as measured by an observer at spatial infinity moving with 4-velocity(λ,Λ), is

H = −Q((

λ

Λ

),

(Ep

)).

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 17 / 22

The positive mass theorem

The theory is consistent only if the observer at infinity sees the ADM 4-momentum as a futuredirected time-like vector. Equivalently, H > 0.

This important question motivates the following outstanding result, due to Schoen-Yau andWitten.

TheoremLet (S, h) ↪→ (M, g) as above. Define the the energy and momentum densities along S by

µ =12

(Rh − |k |2h + K2), J i = ∇j hij −∇i K ,

and assume that the dominant energy condition holds: µ ≥ |J i |. Then the ADM 4-momentumvector is time-like and future directed unless it vanishes and (S, h) arises from a slice of Minkowskispace.

Thus, if we defined the total mass of the system by

m =√

E2 − |p|2,

then m ≥ 0 and m = 0 only if (S, h) arises from a slice of Minkowski space.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 18 / 22

The positive mass theorem

The theory is consistent only if the observer at infinity sees the ADM 4-momentum as a futuredirected time-like vector. Equivalently, H > 0.

This important question motivates the following outstanding result, due to Schoen-Yau andWitten.

TheoremLet (S, h) ↪→ (M, g) as above. Define the the energy and momentum densities along S by

µ =12

(Rh − |k |2h + K2), J i = ∇j hij −∇i K ,

and assume that the dominant energy condition holds: µ ≥ |J i |. Then the ADM 4-momentumvector is time-like and future directed unless it vanishes and (S, h) arises from a slice of Minkowskispace.

Thus, if we defined the total mass of the system by

m =√

E2 − |p|2,

then m ≥ 0 and m = 0 only if (S, h) arises from a slice of Minkowski space.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 18 / 22

The positive mass theorem

The theory is consistent only if the observer at infinity sees the ADM 4-momentum as a futuredirected time-like vector. Equivalently, H > 0.

This important question motivates the following outstanding result, due to Schoen-Yau andWitten.

TheoremLet (S, h) ↪→ (M, g) as above. Define the the energy and momentum densities along S by

µ =12

(Rh − |k |2h + K2), J i = ∇j hij −∇i K ,

and assume that the dominant energy condition holds: µ ≥ |J i |. Then the ADM 4-momentumvector is time-like and future directed unless it vanishes and (S, h) arises from a slice of Minkowskispace.

Thus, if we defined the total mass of the system by

m =√

E2 − |p|2,

then m ≥ 0 and m = 0 only if (S, h) arises from a slice of Minkowski space.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 18 / 22

The positive mass theorem

The theory is consistent only if the observer at infinity sees the ADM 4-momentum as a futuredirected time-like vector. Equivalently, H > 0.

This important question motivates the following outstanding result, due to Schoen-Yau andWitten.

TheoremLet (S, h) ↪→ (M, g) as above. Define the the energy and momentum densities along S by

µ =12

(Rh − |k |2h + K2), J i = ∇j hij −∇i K ,

and assume that the dominant energy condition holds: µ ≥ |J i |. Then the ADM 4-momentumvector is time-like and future directed unless it vanishes and (S, h) arises from a slice of Minkowskispace.

Thus, if we defined the total mass of the system by

m =√

E2 − |p|2,

then m ≥ 0 and m = 0 only if (S, h) arises from a slice of Minkowski space.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 18 / 22

The positive mass theorem

The theory is consistent only if the observer at infinity sees the ADM 4-momentum as a futuredirected time-like vector. Equivalently, H > 0.

This important question motivates the following outstanding result, due to Schoen-Yau andWitten.

TheoremLet (S, h) ↪→ (M, g) as above. Define the the energy and momentum densities along S by

µ =12

(Rh − |k |2h + K2), J i = ∇j hij −∇i K ,

and assume that the dominant energy condition holds: µ ≥ |J i |. Then the ADM 4-momentumvector is time-like and future directed unless it vanishes and (S, h) arises from a slice of Minkowskispace.

Thus, if we defined the total mass of the system by

m =√

E2 − |p|2,

then m ≥ 0 and m = 0 only if (S, h) arises from a slice of Minkowski space.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 18 / 22

The time-symmetric (or Riemannian) case

If k = 0 along the slice then p = 0. Moreover, the energy condition becomes Rh ≥ 0. In thiscase, it is natural to take λ = 1 and call E the ADM mass of (S, h), denoted m(S,h).

We then obtain the following special but important case.

TheoremIF (S, h) is a complete, asymptotically flat Riemannian manifold satisfying Rh ≥ 0 everywherethen m(S,h) ≥ 0. Moreover, the equality holds if and only if (S, h) is isometric to (R3, δ).

Notice that the statement makes sense in any dimension n ≥ 3. This is the famous PositiveMass Conjecture, which is known to be true for n ≤ 7 (Schoen-Yau) and any dimension if S isspin (Witten).

Surprisingly enough, this result has extremely important applications in Geometric Analysis,in connection with the study of the space of solutions of the Yamabe problem.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 19 / 22

The time-symmetric (or Riemannian) case

If k = 0 along the slice then p = 0. Moreover, the energy condition becomes Rh ≥ 0. In thiscase, it is natural to take λ = 1 and call E the ADM mass of (S, h), denoted m(S,h).

We then obtain the following special but important case.

TheoremIF (S, h) is a complete, asymptotically flat Riemannian manifold satisfying Rh ≥ 0 everywherethen m(S,h) ≥ 0. Moreover, the equality holds if and only if (S, h) is isometric to (R3, δ).

Notice that the statement makes sense in any dimension n ≥ 3. This is the famous PositiveMass Conjecture, which is known to be true for n ≤ 7 (Schoen-Yau) and any dimension if S isspin (Witten).

Surprisingly enough, this result has extremely important applications in Geometric Analysis,in connection with the study of the space of solutions of the Yamabe problem.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 19 / 22

The time-symmetric (or Riemannian) case

If k = 0 along the slice then p = 0. Moreover, the energy condition becomes Rh ≥ 0. In thiscase, it is natural to take λ = 1 and call E the ADM mass of (S, h), denoted m(S,h).

We then obtain the following special but important case.

TheoremIF (S, h) is a complete, asymptotically flat Riemannian manifold satisfying Rh ≥ 0 everywherethen m(S,h) ≥ 0. Moreover, the equality holds if and only if (S, h) is isometric to (R3, δ).

Notice that the statement makes sense in any dimension n ≥ 3. This is the famous PositiveMass Conjecture, which is known to be true for n ≤ 7 (Schoen-Yau) and any dimension if S isspin (Witten).

Surprisingly enough, this result has extremely important applications in Geometric Analysis,in connection with the study of the space of solutions of the Yamabe problem.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 19 / 22

The time-symmetric (or Riemannian) case

If k = 0 along the slice then p = 0. Moreover, the energy condition becomes Rh ≥ 0. In thiscase, it is natural to take λ = 1 and call E the ADM mass of (S, h), denoted m(S,h).

We then obtain the following special but important case.

TheoremIF (S, h) is a complete, asymptotically flat Riemannian manifold satisfying Rh ≥ 0 everywherethen m(S,h) ≥ 0. Moreover, the equality holds if and only if (S, h) is isometric to (R3, δ).

Notice that the statement makes sense in any dimension n ≥ 3. This is the famous PositiveMass Conjecture, which is known to be true for n ≤ 7 (Schoen-Yau) and any dimension if S isspin (Witten).

Surprisingly enough, this result has extremely important applications in Geometric Analysis,in connection with the study of the space of solutions of the Yamabe problem.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 19 / 22

The time-symmetric (or Riemannian) case

If k = 0 along the slice then p = 0. Moreover, the energy condition becomes Rh ≥ 0. In thiscase, it is natural to take λ = 1 and call E the ADM mass of (S, h), denoted m(S,h).

We then obtain the following special but important case.

TheoremIF (S, h) is a complete, asymptotically flat Riemannian manifold satisfying Rh ≥ 0 everywherethen m(S,h) ≥ 0. Moreover, the equality holds if and only if (S, h) is isometric to (R3, δ).

Notice that the statement makes sense in any dimension n ≥ 3. This is the famous PositiveMass Conjecture, which is known to be true for n ≤ 7 (Schoen-Yau) and any dimension if S isspin (Witten).

Surprisingly enough, this result has extremely important applications in Geometric Analysis,in connection with the study of the space of solutions of the Yamabe problem.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 19 / 22

The time-symmetric (or Riemannian) case

If k = 0 along the slice then p = 0. Moreover, the energy condition becomes Rh ≥ 0. In thiscase, it is natural to take λ = 1 and call E the ADM mass of (S, h), denoted m(S,h).

We then obtain the following special but important case.

TheoremIF (S, h) is a complete, asymptotically flat Riemannian manifold satisfying Rh ≥ 0 everywherethen m(S,h) ≥ 0. Moreover, the equality holds if and only if (S, h) is isometric to (R3, δ).

Notice that the statement makes sense in any dimension n ≥ 3. This is the famous PositiveMass Conjecture, which is known to be true for n ≤ 7 (Schoen-Yau) and any dimension if S isspin (Witten).

Surprisingly enough, this result has extremely important applications in Geometric Analysis,in connection with the study of the space of solutions of the Yamabe problem.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 19 / 22

The Schwarzschild solution

Naturally, Minkowski space-time

ds2 = −dt2 + r2dΘ2, r > 0,Θ ∈ S2,

is a solution to Einstein field equations in vacuum (T = 0).

Another solution was obtained by Schwarzschild in 1916:

ds2m = −(

1−2mr

)dt2 +

dr2

1− 2mr+ r2dΘ2, m > 0.

A direct computation shows that the parameter m is precisely the ADM mass of the solution.

In principle, this solution is only defined for r > rm := 2m, the Schwarzschild radius.

Only much later people realized that the singularity r = rm is apparent, that is, it can beremoved by means of a suitable coordinate change.

On the other hand, the central singularity r = 0 é essential. In particular, the geometry (and,hence, the physics) completely looses its meaning in sufficiently small neighborhoods of thesingularity.

From the modern viewpoint, Schwarzschild solution (and its various generalizations) modelthe final stage of the gravitational collapse of massive bodies (formation of black holes). Inthis context, the null hypersurface r = rm is the corresponding event horizon.

Levi Lopes de Lima (DM–UFC) Mass in General Relativity Gelosp 2013 - USP/SP 20 / 22

The Schwarzschild solutionNatur