Testing the Equivalence Principle Special Lectures on Experimental Gravity Universality of free fall (Galileo) • Aristoteles view: the manner in which a body falls, does depend on its weight (at least plausible, if one does not abstract from air resistance etc.) • Galileos experiments on an inclined plane: the speed of a body is independent of its weight; all bodies, regardless of their constitution, fall with the same acceleration • This behaviour is fairly unique to gravitation; usually: the larger the acting force, the larger the acceleration. electric field vs. gravitational field electric field: gravitational field: F = q E F = m G · · Q M E G | | ∼ r2 | | ∼ r2 a q a = (m) ,a= const. | | ∼ | | ! F • For other forces, mass has only one function; as the measure of inertia. For gravity, it also fulfills a second function; as source of acceleration. Two different definitions of mass • inertial mass: • you don‘t need gravity for this.. • measures the resistance against changes of current state of motion • can be defined by using collision experiments using Newtons law: m /m = ∆v /∆v 1 2 − 2 1 Two different definitions of mass (cont‘d) • gravitational mass: • gravity leads to acceleration, too... • the force of gravity on a body is proportional to its gravitational mass: Mm F = G e r2 r Weak Equivalence Principle • these two different definitions of mass (i.e. inertial, gravitational) are equivalent; bodies of different constitution feel the same acceleration. • in classical physics, it is not entirely clear why this is the case. Early experiments were already performed by Newton, Bessel; much more accurate ones between 1906-1909 by Eötvös • in GR, gravity is explained geometrically: matter deforms space and time and all bodies are following the straightest lines in this distorted geometry ‣ no need for different mass-definitions like Newton: (a) force acting on a body depends on its gravitational mass (b) but the reaction on this force depends on the bodys inertial mass • in GR, all bodies feel the same acceleration because their motion is determined by the very same space-time around them The Eötvös-experiment • measurement using torsion balance • two test-masses of different composition with equal weight -> equal gravitational mass • a net torque will show up if the equivalence principle is violated a1 a2 9 • in term of the Eötvös-parameter η =2.0| − | 10− a + a ≈ | 1 2| The Eötvös-experiment (cont‘d) Other experiments Equivalence principle in Einsteins theory “The gravitational field has only a relative existence... Because for an observer freely falling from the roof of a house - at least in his immediate surroundings - there exists no gravitational field.“ (Einstein) • Because of the equivalence between gravitational and inertial mass, a freely falling observer won‘t feel his own weight, nor any effect of gravity • gravity can be (nearly) transformed away (at least locally) • locally, a gravitational field and a uniform accelerated frame of reference are equivalent • in any and every local Lorentz frame, anywhere and anytime in the universe, all the (nongravitational) laws of physics must take on their familiar special- relativistic forms Local Lorentz frame Applications of the Equivalence Principle I The deflection of light in a gravitational field: freely falling accelerated observer • inside view: light travels with constant speed along straight lines; laser beam leaves at same height as it enters the elevator • outside view: both observers have to agree that the light will eventually leave the elevator. Consequently, the light ray cannot remain horizontally, it has to bend -> (equivalence principle) light is deflected in the presence of gravity Applications of the Equivalence Principle II Gravitational redshift: • lightsource at the bottom of the elevator emits flashes of light with frequency ν0 • inside: receiver at the top of the elevator measures the very same frequency • outside: will measure a different frequency ν because the elevator moves away from the accelerated observer. After a time t the elevator has a velocity of v = g t = g L/c , with elevator height L · · v gL • using Dopplers equation: ν = ν 1 = ν 1 0 − c 0 − c2 ! " # $ • equivalence principle (locally, uniform acceleration cannot be distinguished from gravitational field) -> frequency of light is shifted in the presence of gravity Applications of the Equivalence Principle III Take perfect fluid energy-momentum tensor in flat space-time T µν =(ρ + p)uµuν + pηµν with uµ = (1,vi) , vi 1 and p/ρc2 1 | | ! ! Then T 00 =(ρ + p)u0u0 p ρ − ≈ T 0j = T j0 =(ρ + p)u0uj ρvj ≈ T jk =(ρ + p)ujuk + pδjk ρvjvk + pδik ≈ and T µ ν = T =0 has the components ,ν ∇ · T 00 + T 0j = ∂ρ/∂t + (ρv)=0 ,0 ,j ∇ · j0 jk j j k k j T ,0 + T ,k = ∂(ρv )/∂t + ∂(ρv v )/∂x + ∂p/∂x or ∂v/∂t +(v )v = p/ρ · ∇ −∇ Applications of the Equivalence Principle III (cont‘d) µν • This special relativistic rule, i.e. T , ν =0 holds also true in presence of gravitation; it is valid in a freely falling frame of reference • In a freely falling system, the connection coefficients (Christoffel-symbols) µ vanish, i.e. Γ νκ =0 at the origin of the freely falling system µν • In such a system at that point it is T ;ν =0 • Laws of physics are independent of the coordinate system, so in curved µν spacetime we have also T ;ν =0 Experiments I Experiments I (cont‘d) possible noise sources: • Disturbances due to variation in torques caused by gravitational field gradients. • Variable torque from a varying magnetic field acting upon magnetic contaminants. • Variable electrostatic forces. • Disturbances due to gross gas pressure effects. • Brownian motion effects. • Disturbances due to the rotation detection system. • Extraction of signal from noise. • Temperature variation effects. • Ground vibration disturbances. Experiments II testing the equivalence principle on large scales: Lunar Laser Ranging • a difference in inertial and gravitational mass of Earth and Moon will lead to corrections in their orbital motion Experiments III Satellite Test of the Equivalence Principle (STEP) joint NASA/ESA project • launch time: around 2013 • mission duration: 6 months Experiments IV MICRO-Satellite a traînee Compensee pour l'Observation du Principe d'Equivalence (MICROSCOPE).