Ideas and some peculiar aspects - OCA · 5 flat space & uniform time or flat spacetime inertial motions = rectilinear and uniform & no force Newton: the cause of planetary motions

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Spacetime, geometry, gravitation

Ideas and some peculiar aspects

Bertrand Chauvineau

Observatoire de la Côte d’Azur

Lagrange laboratory

Nice-Sophia Antipolis University

Nice, France

Bertrand CHAUVINEAU – Spacetime, geometry, gravitation

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Aims / contents / topics

Spacetime & geometry

Gravitation & spacetime geometry

General Relativity, the large scale Universe & L

General Relativity & ways to alternatives

Metric theories, f(R) theories, Scalar-Tensor theories


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Some keys ideas related to spacetime & to gravitation (« classical » ideas) :

Newton (1687) :

- necessity of a precise definition of the spacetime (space & time) properties

before doing physics

- Newton spacetime gravity as a « force » phenomenon

Maxwell (~1865) : electromagnetism equations versus the galilean relativity principle

The Michelson-Morley experiment (1881) & the special relativity theory (1905)

Minkowski (1908) : the relativity theory versus Minkowski’s spacetime

a drastic change in the way of thinking in theoretical physics

Einstein (1915) : gravity as a spacetime geometry effect ( General relativity)

(& later : alternatives to General Relativity)


4

Newton’s spacetime Minkowski’s spacetime

Invariant quantity between

two close eventsdt (or - dt² …)

(absolute time)

- c²dt²+dx²+dy²+dz²(relativity of time)

(fundamental) constant nonec (a priori nothing to do

with light !)

Spacetime geometry ?NO

(but space geometry -euclidean-)

YES(pseudo-euclidean)

Inertial motions(just determined by the spacetime

characteristics)

Rectilinear & uniform Rectilinear & uniform

I - Newton versus Minkowski spacetimes (in a nutschell) and gravity :

Newton’s spacetime has no spacetime geometry (unlike Minkowski’s), but its properties

are (to some extent) as precise --while different-- as the ones of Minkowski’s spacetime.


consider physical problems like simultaneity, causality, …

in both, the gravity phenomenon requires « something else » than the spacetime’s properties

(something like a « force »)

5

flat space & uniform time

or

flat spacetime

inertial motions

=

rectilinear and uniform

& no force

Newton : the cause of planetary motions is a specific force (spacetime properties unchanged)

But another possibility could be to reject the idea of a gravitational force (Riemann)

Matter generates a force field : Newton’s universal gravity theory

Matter « modifies » the spacetime properties

Motion under gravity = inertial motion in a non-newtonian spacetime

Starting from the newtonian spacetime : how changing the space geometry to recover

the Newton’s gravity theory equations ? (Riemann)

FAILS !!!


6

Starting from the minkowski spacetime :

the geodesics (extremal « length » curves inertial motions) of the spacetime of metric

are functions x(t), y(t), z(t) that satisfy, at lowest order (in U and v²), the Newtonian’s law

of motion under a gravity potential U, ie

metric) s(Newton' 2

1 22222

2

2 dzdydxdtcc

UdsN

--

Udt

xdi

i

2

2


Remark : the « time part », and only it (at lowest order) of the Minkowski spacetime is affected

Riemann could not success !!!

Obviously : the previous metric is not a « geometric gravity theory », but just a geometrical

reinterpretation of Newton’s theory !

A « geometric gravity theory » is expected to give directly the link between

the spacetime geometrical objects and its matter (ie non gravitational) content

422

2

2

,, 0 vUvUOUdt

xdds i

i

N

locally minkowskian ? (local spatial isotropy ?)

7Bertrand CHAUVINEAU – Spacetime, geometry, gravitation

The metric tensor in (very) brief

coord.) (cart. planeeuclidean 222 dydxds

) sin , cos (

) ( coord.) (pol. planeeuclidean (locally)

222

2222222

dYdXdsyxYyxX

drdrdyxdxds

) , ( spacetime Minkowski

coord.) (cart. spacetime Minkowski

222

222222

xctvxctudzdydudvds

dzdydxdtcds

-

-

) sin ( dim) (2 sphere sin 2222222 ddxdydxds

Flat spacetime/variety (metric connexion meaning) : one can find coordinates in which all

the metric tensor components are constant (or : Riemann-Christoffel curvature tensor = 0)

Plane (no local curvature) : euclidean plane, Minkowski spacetime, …

Local curvature (not plane) : sphere, Newton’s metric ... , 2

1 22222

2

2 dzdydxdtcc

UdsN

--

.... .... 2 11

11

10

01

00

00

,

2

dxdxxgdxdxxgdxdxxgdxdxxgds cccbac

ab

ba

metric tensor components (symmetric) should define an invertible matrix

Riemannian variety :

interval

)( ixP

)( ii dxxQ (2 close points)

-

10

01 &

10

01 1

ab

ab

ab ggg

- 22 sin0

01 &

sin0

01ab

ab gg

1000

0100

0002/1

002/10

abg

... & 0

012

ab

ab gr

g


What is c ? How interpreting it ? How naming it ?

Minkowski spacetime’s definition has nothing to do with the light/Maxwell theory !

(despite the fact it was discovered thanks to some « strange » properties of the light, first

revealed by the Michelson & Morley experiment, motivated by Maxwell’s equations, …).

c is just the modulous of a family of modulous-invariant speeds, that exists from the mere fact

that the spacetime is equipped with a ( (- + + +) pseudo-euclidean) geometry

Requiring the validity of the Maxwell equations (in their usual form, and in the usual

formulation of the theory), ie of the electromagnetic theory (that is not inherent to special

relativity, but that can be « stick » in it) in all galilean coordinate systems requires :

c 1

00

Speed of electromagnetic waves in vacuum

(as soon as Maxwell equations apply)

Speed of gravitational waves

in general relativity & ST

(in general, c related to GW)

inherent to the mere spacetime definition, as soon as it is dynamical

(at least in General Relativity & Scalar-Tensor gravity)

Refering to c as the « speed of gravity » (rather than « speed of light ») :

would it be a better motivated choice for the terminology ?

9

II – General Relativity (GR) (& motivations to go beyond ?) :

Newton : gravitational potential matter content : (Poisson equation)GU 4

eqs fieldmatter & ...8

2

1 21

4

- abababab TT

c

GRgR

GR Einstein’s equation :

Einstein (GR) : spacetime geometry matter content :

c

a

bc

ab

ab

abeaebbea

cec

ab

ce

ce

c

be

e

ac

c

ab

e

ce

c

acb

c

abcab

ggg

ggggg

RgR

R

-

--

by defined

2

1

Gravitational potential Matter content

(depends on) spacetime’s geometry Matter content (& spacetime’s geometry)

where :

Ricci curvature tensor

(Ricci) curvature scalar

metric/Christoffel connexion

contravariant vs covariant metric components


...

0

0

2

1

aba

aba

T

T

(conservation eqs)

10

GR in the weak field case :

using relevant conditions

on the coordinate system

-- abababec

ce TmTc

Ghm

2

1164

Einstein equation

(usual Dalembertian)


In vacuum 01

2

2

22

2

2

2

2

2

-

abh

tczyxc = speed of gravity

Stationarity (besides weak field) Poisson equation

back to Newton’s gravity

gravitational waves

22222

2

2 2

1 dzdydxdtcc

UdsN

-- is the lowest order (0PN) solution of GR

the spacetime’s metric is close to Minkowski 1 with abababab hhmg

1,1,1,2cdiag -

a weak (gravitational) field can not generate velocity changes of the order of c

system) coord a (

Linearized GR’s Einstein equation :

(interpreted in

relativistic terms)


GR and the Universe (try to describe the Universe as a whole) :

1917 (Einstein) : no stationary dust-filled (finite, (hyper)spherical) Universe …

Universe) theof (radiusconstant with sin

1

2222

2

2

2222 Rddr

R

r

drdtcds

-

-

… but if the field equation is completed by a cosmological term …

constant with...8

2

1 21

4L

L- ababababab TT

c

GgRgR

… thence a stationary dust-filled solution exists :

L-L

L

0000

0000

0000

000

8

2

1 satisfies

4 &

1

2

4

2

c

c

GgRgR

G

cR

E

abababEE

1920-30 Slipher, Hubble, Lemaître : Universe’s expansion

Natural behaviour of GR cosmological solutions

without (or with) L

L no longer required !!! (but may exist …)

comoving dust


- 1998-9 Perlmutter, Riess, Schmidt

- dust filled Universe

- RW cosmology

- GR

accelerated expansion

requires L !!!

What is L ? How interpreting it ?

A new fundamental constant ? Something else ?

An appealing point :

~~~

882

1 8

2

1--L- PTTRgRTgRgR

the cosmological term is to a perfect fluid with the vacuum eq of state !!! (QFT)

As a (QFT) field, the vacuum should gravitate the presence of L is natural !!!

… BUT … !!!!!! 10~ 120

observed

QFT

L

L

-- 2222

2

22222 sin

1 ddr

kr

drtadtcds

asymptotically …

?

??

?

?

???

?


The « equivalence » :

~~~

882

1 8

2

1--L- PTTRgRTgRgR

other options : change neither theory nor matter, but allow for

- voids (local inhomogeneities effect) ;

- remove large scale symmetries ;

- …

suggests :

don’t change the Universe’s

matter content, but the gravity

theory alternative gravity theoriesreminiscent from

Mercury’s perihelion’s shift problem

suggests :

don’t change the gravity

theory, but the Universe’s matter

content dark energy theoriesreminiscent from

Uranus’ orbit’s anomalies problem

The main topic of the following … (just some alternatives)

… could be worth looking for an alternative « story » !!!

Going beyond GR ?


GR lagrangian :

… it suggests some possible ways for alternatives

...;;216

22114

-

-L-- abNGabNGGR gLggLgRg

G

c

abNGGR gLgRgG

c; 2

16

4

-L--

gravity sector « matter » sector

Could the Newton’s constant

be upgraded as a scalar field ?* More intricate dependence

in the metric ?

* Metric as the alone

geometrical field ?

Does the matter necessarily

couple with the metric only ?

- pure geometric theories (metric &/or independent connection)

pure metric theories

- scalar(s)-tensor theories

- others (scalar-vector-tensor, bimetric, massive gravity, …)

abT2

abT1

The metric’s connexion

is just one peculiar connexion (that nevertheless possesses some fair properties)


III – Purely metric gravity theories (MGT) :

Let us identify 2 basic geometric concepts :

- the metric g : defines intervals (generalizes the « length » concept)

- the connection : defines (local) parallelism

ba

ab dxdxgds 2

dxxQ xP

iA

iii AAB

vector

at P

vector

at Q• displacement dx

• connection

abeaebbea

cec

abggggg -

2

1

But if these objects are considered as independent (metric-affine approach)

- varying wrt the metric an equation (the « Einstein’s equation » of the theory)

- varying wrt an equation that links the connexion to the metric

GR’s action g) !!!metric-affine

formalism

cba

bc

a dxAA

:with

-


From now on, we do the following choices :

- MGT : the gravitational sector of the theory depends on the metric only

- the connexion is a priori the metric’s one g) (second order formalism)

abNG

abcd

abcd

ab

abba

ab

ba

ab

metric gLgRRRRRgRRgRFgG

c;,...,,,,

16

4

--

The most general MGT’s lagrangian one could imagine :

Put all the metric (only) dependent terms you can imagine …

where F is required to be a scalar (invariant in all coordinate transforms)

ensures the « covariance » of the theory (relativity principle)

The simplest choice (for a non trivial theory) : F(…) = R GR !!!

GR is the simplest MGT ! (the following step being GR with L )

« covariant » derivative

(includes curvature effects)


What is the Einstein’s equation of an MGT expected to look like ?

Preliminaries : some generalities on lagrangian systems

1rst order lagrangian ', qqLFq

L

q

L

dt

d FF

'

Euler-Lagrange

second order eq :

','' qqfq

''

q

LqLE F

FF

-The associated « energy » reads and one shows that this energy

may be bound, in which case the theory is « stable » (Ostrogradski stability)

2cd order lagrangian '',', qqqLS

''''

''''

q

Lq

q

L

dt

d

q

LqLE SSS

SS

-

-

q

L

q

L

dt

d

q

L

dt

d SSS

-

''

' 2

2

fourth order eq :

''','',','''' qqqqfq

The associated « energy » reads

and one shows that this energy is generically unbound the theory is « unstable »

(Ostrogradski instability) R. Woodard (2007)

nondegeneracy


MGT lagrangians depend on R, thence on the metric’s second derivatives

expect

- the corresponding Einstein equation is of fourth order ;

- Ostrogradski instability of the theory !!!

BUT :

If : F(…) = R (or R+cst ) : the second derivatives terms surface terms

A strong argument supporting the f(R) theories among the whole MGT family

In the other cases : fourth order Einstein’s equation ...

… but NO Ostrogradski instability if F(…) = f(R) and in this case only !!!

- GR Einstein’s eq is of 2cd order

- no Ostrogradskian instability

GR

F(R

)

related to the prior hypothesis g)

R. Woodard (2007)

First order effective

lagrangian


equationsmatter usual the&

8 ' '

2

1'

4 ab

c

cabbaabab Tc

GfgffgRf

--Field equations :

generate fourth order terms

(as soon as f is not affine)

-- 2222

2

22222 sin

1 ddr

kr

drtadtcds

A result on RW cosmology in this f(R) framework :

One can choose (reconstruct) a function f(R) in such a way that a(t)

fits any prior Universe’s history !!!

f(R) theories would be of weak interest if limited to this physical problem

(but also : Solar System, stellar’s structure, cosmological perturbations, …)

R. Woodard (2007)


IV – (Usual) scalar-tensor (ST) theories :

abNGba

ab gLgUgRgc

;216

4

-

---

Motivations : - some attempts to quantize gravity (or unify with other interactions)

Brans-Dicke (BD) like theory (BD = ST with constant)

- in some cases, close to GR in some sense (see later)

OK solar Systems’ tests

Field equations :

c

cabba

c

cabbaab

ababab ggc

TUgRgR -

--

1

2

18

2

124

equationsmatter usual the&

8223

4T

cd

dUU

d

d a

a

a

a

-

ST may be locally interpreted as a gravity theory with varying

effective Newton’s constant (in a Cavendish-like experiment) :

1

32

42

effG

minimal matter-gravity coupling

Metric only (choice)


- 2cd order field equations

- no Ostrogradskian instability, as soon as > -3/2Remarks :

f(R) versus ST :

''2

1' :gravity fgfTfgRfRf c

cabbaababab --

Rd

UdTUU

gTgURR

a

a

c

cabbaababab

-

---

2or '23

22

1 :gravity BD-0

The ressemblance of the (Einstein’s) equations is suggestive !

can go further ?

Let us define, from f(R) :

) ' ( '

2 & ' 1 -- fRRf

RfRURf

The f(R) theory can be seen as a peculiar BD (thence ST) theory

f(R) Ostrogradskian stability, this correspondance requiring 0 > -3/2

P. Teyssandier, P. Tourrenc (1983)

(excludes RG)


constant) (with 2

1 0 finite UTUgRgR abababab -

2323

'2

- TUU

ST versus GR :

The ST Einstein’s equation in the constant scalar case :

ie GR gravity. But the scalar eq yields

Consider (for simplicity) the case without potential. The scalar equation then requires :

) 0 (if 023

TT

Thence, the convergence of ST to GR generically requires

BUT :

c

cabba

c

cabbaab

abab ggT

RgR -

--

1

2

1

2

12

!!! 0 OK 0

limit = GR, but with an extra matter term : massless scalar field

originating in the

BD scalar vanishing part !(turns out to be = 0 in stationary cases…)


Physical relevance : ask a physical question :

Consider a flat RW dust-filled universe, with the observational constraint that H0 is known.

What can be said on its age in the frameworks of (1) GR ; (2) (infinite )-BD ?

GR’s answer :

03

2

HTGR

BD’s answer :

34

12,

34

3/2111

00 HHT BD

Thence the (infinite )-BD’s answer :

00

lim3

2,

3

1

HHTBD

different answers (infinite )-BD differs from GR

… but ]1/(3H0),2/(3H0)] corresponds to the GR’s answer got

for a dust + massless scalar filled flat RW universe (in accordance with …) BC (2007)


Rmks : - the residual scalar field is zero considering stationary solutions/problems

- the same conclusions essentially work, in some sense, for general ST

Experiments/propagation of light (Cassini) : ST pass Solar System tests if > 40 000

Some large scale (cosmology, …) studies are grounded on ST theories effects that are

by far more important than expected from solar system constraints

Conciliating the two ? A possibility could be the so called chameleon mechanism

(in brief)The scalar has an effective « mass » increasing with local density

1mass range -- large range in galactic, cosmological, … mediums

- weak range (then no effect) in planetary, solar systems, … mediums

Yukawa like (spatial) damping

J. Khoury, A. Weltman (2004)

C. Will (2014)


V – Scalar-tensor theories with an external scalar (EST) :

abNGba

ab gLgUgRgc

;216

4

-

---

The ST theories’ lagrangian

Vary wrt metric gab Einstein’s equation

Vary wrt scalar field Scalar equation (after combining with Einstein’s equation)

Vary wrt matter fields H Matter field equations

However, considering that : - physical consideration sometimes lead to scalar fields that are

imposed in the theory (external, ie not varied in the lagrangian)

- resorting to external fields is sometimes required in physics

(unimodular gravity, bimetric/massive gravity, …)

… it may be worth to take a close look at the consequences if the scalar field

is not varied at the action (lagrangian) level …

conservation equations (energy, Euler, …)

S. Weinberg (1989) ; C. de Rham et al (2011) ; C. Böhmer, N. Tamanini (2013)

M. Reuter, H. Weyer (2004)


Digression : the « role » of the geometric identities

0 & 0 & ...8

2

1 2121

4

- ab

aab

aabababab TTTT

c

GRgR

GR with 2 independent matter fields 1 & 2

got varying the action w.r.t. the metric

Take the divergence and get (thanks to some geometrical identities) :

got thanks to

variation w.r.t. 1

got thanks to

variation w.r.t. 2

021

abab

a TT

(nothing new)

The same theory (ie same action), but with 2 not to be varied (external/non-dynamical field)

!!! all s that'and 0 & ...8

2

1 121

4

- ab

aabababab TTT

c

GRgR

got varying the action w.r.t. the metricgot thanks to

variation w.r.t. 1

Take the divergence and get :

021

abab

a TT

!!!back is 02

aba T

Not a new equation, but made explicit thanks to the geometrical identities

(showing that Einstein + conservation eqs are the same in both theories)


Vary wrt metric gab Einstein’s equation

Vary wrt scalar field Scalar equation

Vary wrt matter fields H Matter field equations

not varied the scalar equation is lost

Is the resulting theory « less constrained » ?

One could be tempted to claim :

NO, because geometrical identities & conservation equations ensure that the scalar

equation is back

02

1 : id Bianchi contracted

0 : id Ricci

-

abab

a

bca

RgR

g

Let us check !!! Two points : (a) conservation equations, (b) use geometrical identities

(a) Conservation equations :

not a trivial task, but OK here since : - the matter action is a scalar

- all matter fields in the matter sector

- does not enter the matter sector

- no external field in the matter sector

0 ab

aT


(b) Use geometrical identities : it yields

0

23

'2'8 4

----

-

b

a

aa

a

UUTc

theoriesST (usual)in 00 if

solutions like-GR toleads

- usual ST solutions

- (but also) GR solutions

- (and even !!!) some mixes of the two !!!

The EST theory equations admit :

2222222 RW ) 0 (Flat dzdydxtadtcdsk -

An (unexpected) cosmological solution :

( = cst & U = 0)- EST

equations in the dust filled case :

0'

'''2

'

2

''''2

) ' ( ''

3'

28

'3

3

22

22

--

-

-

a

a

a

a

a

a

a

dt

dXX

a

a

a

a


Let us remark the form of the induced equation : 032

8' ''

33

-

aa

3/2Ata

C

-

-

-

-

--

--

qq

pp

ttttBa

tttt0

34

3/211

34

3/2131

sq

sp

... & ...0 A

B

C

continuity of a, a’, and ’ (and )

A possible solution a GR phase followed by a Brans-Dicke one (with > -4/3)(or the converse … inflationary-like scenarios ?)

0 1 2 3 4 5 60

1

2

3

4

t

GR scalar

BD scalarIllustration with numerical values :

- = 0

- matching time = 1

- A = 1

- C = 2

discontinuity of second derivatives,

but continuity of /''/''2 aa

BC, D. Rodrigues, J. Fabris (2015)


The EST gravity allows the coexistence in a same spacetime

of (exact) GR regions & (exact) ST regions

some (new) kind of sceening mechanism for ST theories ?

Some questions : - external scalar : does it mean it should be fixed a priori ?

not the case here, but happens in some theories with ext elements

- deterministic status of EST ? In the previous cosmological example :

- the matching time (GR BD) is arbitrary

- the « jump GR ST or ST GR » is not ensured to occur

more data provided for the Cauchy problem ?

- …

To be explored/in progress : - spherical symmetry : static, LTB-like, …

screening mechanism ?

- perturbed solutions in vacuum

gravitational waves ? Emission mechanisms ?

- perturbing about RW solution

revisiting cosmological perturbations ? « matching » ?

- other kinds of external fields ?

… thank you for your attention !!!


Documents

Ideas and some peculiar aspects - OCA · 5 flat space & uniform time or flat spacetime inertial motions = rectilinear and uniform & no force Newton: the cause of planetary motions