Dr. Cristian Bahrim - sethi
Dr. Cristian Bahrim
WHAT IS GRAVITY?
Newton claims that every mass exerts a gravitational attraction on every other mass, no matter how far away.
Einstein considers that gravity is geometry. → the gravity arises from the curvature of space-time. A mass causes space-time to curve, and the curvature of space-time determines the paths of freely moving masses.
Dr. Cristian Bahrim
Dr. Cristian Bahrim
GEOMETRIC EFFECTS DUE TO THE CURVATURE OF SPACE-TIME
Gravitational lensing can create multiple images of a distant object.
Dr. Cristian Bahrim
GRAVITATIONAL LENSING
Dr. Cristian Bahrim
The first experimental measurement was done by Arthur Eddington (1919): during a complete solar eclipse he marked the position of certain stars and compared with the position measured during nights. He found a deviation in the position of the stars near Sun in agreement with Einstein’s predictions. FOR A RAY JUST GRAZING THE EDGE OF THE SUN THE DEFLECTION IS 1.98’’± 0.18’’ (SECONDS OF ARC).
Dr. Cristian Bahrim
EINSTEIN’S SUCCESSFUL EXPLANATION REGARDING THE DEVIATION OF LIGHT BY A MASSIVE STELLAR OBJECT The fact that a photon follows a curved path in a gravitational field can also be explained using the concept of gravitational mass associated to a photon. ♦
♦ Einstein’s equation for
the deflection of light due to the gravity gives:
θ =π +
4GM Rc 2
G = 6.67 *10 −11 Nm 2 /kg 2 c = 2.998 * 108 m/s M = mass of the stellar object R = radius of the stellar object Predicted Measured
: θ = 1.74" : θ = 1.98"
± .18"
Dr. Cristian Bahrim
General Relativity and Time Einstein suggests that time runs slower in a gravitational field or for an accelerated body.
A THOUGHT EXPERIMENT. a rocket is moving with an acceleration of g. An observer A at the nose of the rocket will release a photon which will be collected by an observer B located at the tail.
Dr. Cristian Bahrim
THE TIME RUNS DIFFERENTLY FOR ACCELERATED BODIES
Observer A (nose)
1 2 1 gt and zb = gt 2 2 2 Emission : t = 0 (1st pulse) → ∆t a (2nd pulse)
Coordinates :
za = H +
Reception : t1 (1st pulse) → ∆t b (2nd pulse) Kinematics :
1st pulse : za (t ) − zb (t1 ) = c(t1 − t ) 1 @ t = 0 → H − gt12 = ct1 (1) 2 2nd pulse : z a (∆ta ) − zb (∆t b ) = c( t1 + ∆t b − ∆t a )
H
1 1 g (∆t a ) 2 − g (t1 + ∆t b ) 2 = c( t1 + ∆t b − ∆t a ) 2 2 Expand the last equation and take times ∆t a and ∆t b small : H+
1 2 gt1 − gt1∆t b = c( t1 + ∆t b − ∆t a ) 2 Combining equations (1) and (2) we get : gt1∆t b = c∆t a − c∆t b H−
Solution :
∆t b = ∆t a (
(2)
(3)
c ) ⇒ ∆t b < ∆t a because t1 > 0. gt1 + c
Observer B (tail)
The time runs slower in the tail of the rocket than in the nose when it accelerates upward!
Dr. Cristian Bahrim
CAN GRAVITY SHIFT THE COLOR OF LIGHT? The units of time ∆tA and ∆tB can be considered as being periods of the light released by A and received by B, respectively. → The frequency (which is the reciprocal of the period), and implicitly the wavelength, of the photon traveling through the gravitational field are shifted! We can understand this effect based on the idea that we can associate an inertial mass to a photon in motion.
Dr. Cristian Bahrim
MASS OF A PHOTON By combining the Planck’s postulate regarding light with the Einstein’s relationship between energy and mass, we can define the inertial mass of a photon:
E = mc
2
E = hv hv m= 2 c
The inertial mass of a photon is defined by its energy.
Dr. Cristian Bahrim
THE PRINCIPLE OF EQUIVALENCE gravitational mass = inertial mass → an object undergoing an acceleration will behave in the same manner as it would fall in a gravitational field. Experiments testing the equality of gravitational and inertial mass compares the accelerations of bodied falling freely in a gravitational field. The most accurate test to date: accuracy of 1.5 x 10-13.
Dr. Cristian Bahrim
If we accept the principle of equivalence, we must also accept that light falls in a gravitational field with the same acceleration as material bodies. → in the laboratory frame the light ray will be accelerated downward with the acceleration of the laboratory. In a uniform gravitational field the light accelerates downward with the local acceleration of gravity.
Dr. Cristian Bahrim
GRAVITATIONAL SHIFT OF LIGHT Conservation of energy : Ea = Eb
Observer A
⇒ hν a + mgH = hvb
mgH ν a gH = 2 ⇒ vb ≠ ν a h c GMm Newton' s force of gravity : F = R2 dU GMm ⇒U = − Potential energy : F = dR R U GM ⇒V=Potential : V = m R 0 − mgH = − gH Change in potential : ∆V = Vb − Va = m ∆v gH GM =− 2 = >0 Relative shift in frequency : 2 va c Rc
νa
Shift in frequency : ∆v = vb −ν a =
H νb Observer B
Dr. Cristian Bahrim
Comparison for λ = 540 nm (Blue-shift) ∆ν
νa
=−
GM Rc 2
where : R and M are the Radius and Mass the stellar object
M (kg)
R (m)
∆ν ν/ν νa (%)
∆λ(nm)
λ(nm)
5.96E+24
6.378E+06
-6.93E-07
3.74E-7
539.999999
Jupiter
1.9E+27
7.15E+07
-1.97E-05
1.07E-5
539.99997
Sun
1.99E+30
6.96E+08
-2.12E-03
1.15E-3
539.9996
White Dwarf
2.57E+30
3.48E+08
-5.52E-03
2.98E-3
539.998
Neutron Star
2.59E+30
5E+03
-383.981
149.821
390.18
Black Hole
Huge
Very small
infinity
No light
?
Earth
Dr. Cristian Bahrim
DISCUSSIONS 1. Gravitational blue-shift – a photon is falling down. The blue-shift has nothing to do with the shifting that occurs due to the Doppler effect (relative motion of stars). Gravitational red-shift – a photon is leaving a star. 2. One second on the Sun lasts longer than one second on Earth! Larger the gravitational field, longer the duration of a second. Object 3. Light propagating through a gravitational field behaves as if it traverses an inhomogeneous medium having a positiondependent index of refraction n.
n1 n2
Snell’s law: n1 sin θ1 = n2 sin θ2 Where: n2 depends on the curvature of space-time.
Real
Observer
Virtual
WHAT IS GRAVITY?
Newton claims that every mass exerts a gravitational attraction on every other mass, no matter how far away.
Einstein considers that gravity is geometry. → the gravity arises from the curvature of space-time. A mass causes space-time to curve, and the curvature of space-time determines the paths of freely moving masses.
Dr. Cristian Bahrim
Dr. Cristian Bahrim
GEOMETRIC EFFECTS DUE TO THE CURVATURE OF SPACE-TIME
Gravitational lensing can create multiple images of a distant object.
Dr. Cristian Bahrim
GRAVITATIONAL LENSING
Dr. Cristian Bahrim
The first experimental measurement was done by Arthur Eddington (1919): during a complete solar eclipse he marked the position of certain stars and compared with the position measured during nights. He found a deviation in the position of the stars near Sun in agreement with Einstein’s predictions. FOR A RAY JUST GRAZING THE EDGE OF THE SUN THE DEFLECTION IS 1.98’’± 0.18’’ (SECONDS OF ARC).
Dr. Cristian Bahrim
EINSTEIN’S SUCCESSFUL EXPLANATION REGARDING THE DEVIATION OF LIGHT BY A MASSIVE STELLAR OBJECT The fact that a photon follows a curved path in a gravitational field can also be explained using the concept of gravitational mass associated to a photon. ♦
♦ Einstein’s equation for
the deflection of light due to the gravity gives:
θ =π +
4GM Rc 2
G = 6.67 *10 −11 Nm 2 /kg 2 c = 2.998 * 108 m/s M = mass of the stellar object R = radius of the stellar object Predicted Measured
: θ = 1.74" : θ = 1.98"
± .18"
Dr. Cristian Bahrim
General Relativity and Time Einstein suggests that time runs slower in a gravitational field or for an accelerated body.
A THOUGHT EXPERIMENT. a rocket is moving with an acceleration of g. An observer A at the nose of the rocket will release a photon which will be collected by an observer B located at the tail.
Dr. Cristian Bahrim
THE TIME RUNS DIFFERENTLY FOR ACCELERATED BODIES
Observer A (nose)
1 2 1 gt and zb = gt 2 2 2 Emission : t = 0 (1st pulse) → ∆t a (2nd pulse)
Coordinates :
za = H +
Reception : t1 (1st pulse) → ∆t b (2nd pulse) Kinematics :
1st pulse : za (t ) − zb (t1 ) = c(t1 − t ) 1 @ t = 0 → H − gt12 = ct1 (1) 2 2nd pulse : z a (∆ta ) − zb (∆t b ) = c( t1 + ∆t b − ∆t a )
H
1 1 g (∆t a ) 2 − g (t1 + ∆t b ) 2 = c( t1 + ∆t b − ∆t a ) 2 2 Expand the last equation and take times ∆t a and ∆t b small : H+
1 2 gt1 − gt1∆t b = c( t1 + ∆t b − ∆t a ) 2 Combining equations (1) and (2) we get : gt1∆t b = c∆t a − c∆t b H−
Solution :
∆t b = ∆t a (
(2)
(3)
c ) ⇒ ∆t b < ∆t a because t1 > 0. gt1 + c
Observer B (tail)
The time runs slower in the tail of the rocket than in the nose when it accelerates upward!
Dr. Cristian Bahrim
CAN GRAVITY SHIFT THE COLOR OF LIGHT? The units of time ∆tA and ∆tB can be considered as being periods of the light released by A and received by B, respectively. → The frequency (which is the reciprocal of the period), and implicitly the wavelength, of the photon traveling through the gravitational field are shifted! We can understand this effect based on the idea that we can associate an inertial mass to a photon in motion.
Dr. Cristian Bahrim
MASS OF A PHOTON By combining the Planck’s postulate regarding light with the Einstein’s relationship between energy and mass, we can define the inertial mass of a photon:
E = mc
2
E = hv hv m= 2 c
The inertial mass of a photon is defined by its energy.
Dr. Cristian Bahrim
THE PRINCIPLE OF EQUIVALENCE gravitational mass = inertial mass → an object undergoing an acceleration will behave in the same manner as it would fall in a gravitational field. Experiments testing the equality of gravitational and inertial mass compares the accelerations of bodied falling freely in a gravitational field. The most accurate test to date: accuracy of 1.5 x 10-13.
Dr. Cristian Bahrim
If we accept the principle of equivalence, we must also accept that light falls in a gravitational field with the same acceleration as material bodies. → in the laboratory frame the light ray will be accelerated downward with the acceleration of the laboratory. In a uniform gravitational field the light accelerates downward with the local acceleration of gravity.
Dr. Cristian Bahrim
GRAVITATIONAL SHIFT OF LIGHT Conservation of energy : Ea = Eb
Observer A
⇒ hν a + mgH = hvb
mgH ν a gH = 2 ⇒ vb ≠ ν a h c GMm Newton' s force of gravity : F = R2 dU GMm ⇒U = − Potential energy : F = dR R U GM ⇒V=Potential : V = m R 0 − mgH = − gH Change in potential : ∆V = Vb − Va = m ∆v gH GM =− 2 = >0 Relative shift in frequency : 2 va c Rc
νa
Shift in frequency : ∆v = vb −ν a =
H νb Observer B
Dr. Cristian Bahrim
Comparison for λ = 540 nm (Blue-shift) ∆ν
νa
=−
GM Rc 2
where : R and M are the Radius and Mass the stellar object
M (kg)
R (m)
∆ν ν/ν νa (%)
∆λ(nm)
λ(nm)
5.96E+24
6.378E+06
-6.93E-07
3.74E-7
539.999999
Jupiter
1.9E+27
7.15E+07
-1.97E-05
1.07E-5
539.99997
Sun
1.99E+30
6.96E+08
-2.12E-03
1.15E-3
539.9996
White Dwarf
2.57E+30
3.48E+08
-5.52E-03
2.98E-3
539.998
Neutron Star
2.59E+30
5E+03
-383.981
149.821
390.18
Black Hole
Huge
Very small
infinity
No light
?
Earth
Dr. Cristian Bahrim
DISCUSSIONS 1. Gravitational blue-shift – a photon is falling down. The blue-shift has nothing to do with the shifting that occurs due to the Doppler effect (relative motion of stars). Gravitational red-shift – a photon is leaving a star. 2. One second on the Sun lasts longer than one second on Earth! Larger the gravitational field, longer the duration of a second. Object 3. Light propagating through a gravitational field behaves as if it traverses an inhomogeneous medium having a positiondependent index of refraction n.
n1 n2
Snell’s law: n1 sin θ1 = n2 sin θ2 Where: n2 depends on the curvature of space-time.
Real
Observer
Virtual
