einstein and gravity probe b - institute of mathematical ...shyam/web/talks/gravprobebtalk.pdf ·...
TRANSCRIPT
![Page 1: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/1.jpg)
Einstein and Gravity Probe B
Ghanashyam Date
The Institute of Mathematical Sciences, Chennai
![Page 2: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/2.jpg)
Organization of the Talk
I What is Gravity Probe B?
I General Theory of Relativity
I Predictions for Spinning test bodies
I The Experiment
I Remarks
![Page 3: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/3.jpg)
Organization of the Talk
I What is Gravity Probe B?
I General Theory of Relativity
I Predictions for Spinning test bodies
I The Experiment
I Remarks
![Page 4: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/4.jpg)
Organization of the Talk
I What is Gravity Probe B?
I General Theory of Relativity
I Predictions for Spinning test bodies
I The Experiment
I Remarks
![Page 5: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/5.jpg)
Organization of the Talk
I What is Gravity Probe B?
I General Theory of Relativity
I Predictions for Spinning test bodies
I The Experiment
I Remarks
![Page 6: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/6.jpg)
Organization of the Talk
I What is Gravity Probe B?
I General Theory of Relativity
I Predictions for Spinning test bodies
I The Experiment
I Remarks
![Page 7: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/7.jpg)
Organization of the Talk
I What is Gravity Probe B?
I General Theory of Relativity
I Predictions for Spinning test bodies
I The Experiment
I Remarks
![Page 8: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/8.jpg)
What is Gravity Probe B Experiment?http://einstein.stanford.edu/
GP-B is a Highly Sophisticated Experiment designed to test
two of the predictions of Einstein’s General Theory of
Relativity:
A freely falling gyroscope will suffer two types of precession
induced by the earth’s gravitational field -
the geodetic precession due to ’curvature’ and
the frame dragging precession due to spinning of the earth.
![Page 9: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/9.jpg)
What is Gravity Probe B Experiment?http://einstein.stanford.edu/
GP-B is a Highly Sophisticated Experiment designed to test
two of the predictions of Einstein’s General Theory of
Relativity:
A freely falling gyroscope will suffer two types of precession
induced by the earth’s gravitational field -
the geodetic precession due to ’curvature’ and
the frame dragging precession due to spinning of the earth.
![Page 10: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/10.jpg)
What is Gravity Probe B Experiment?http://einstein.stanford.edu/
GP-B is a Highly Sophisticated Experiment designed to test
two of the predictions of Einstein’s General Theory of
Relativity:
A freely falling gyroscope will suffer two types of precession
induced by the earth’s gravitational field -
the geodetic precession due to ’curvature’ and
the frame dragging precession due to spinning of the earth.
![Page 11: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/11.jpg)
What is Gravity Probe B Experiment?http://einstein.stanford.edu/
GP-B is a Highly Sophisticated Experiment designed to test
two of the predictions of Einstein’s General Theory of
Relativity:
A freely falling gyroscope will suffer two types of precession
induced by the earth’s gravitational field -
the geodetic precession due to ’curvature’ and
the frame dragging precession due to spinning of the earth.
![Page 12: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/12.jpg)
What is Gravity Probe B Experiment?http://einstein.stanford.edu/
GP-B is a Highly Sophisticated Experiment designed to test
two of the predictions of Einstein’s General Theory of
Relativity:
A freely falling gyroscope will suffer two types of precession
induced by the earth’s gravitational field -
the geodetic precession due to ’curvature’ and
the frame dragging precession due to spinning of the earth.
![Page 13: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/13.jpg)
What is the General Theory of Relativity?
It a synthesis of three different notions/facts, namely,
1. The notion of space and time,
2. Riemannian geometry and
3. Gravity.
![Page 14: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/14.jpg)
What is the General Theory of Relativity?
It a synthesis of three different notions/facts, namely,
1. The notion of space and time,
2. Riemannian geometry and
3. Gravity.
![Page 15: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/15.jpg)
What is the General Theory of Relativity?
It a synthesis of three different notions/facts, namely,
1. The notion of space and time,
2. Riemannian geometry and
3. Gravity.
![Page 16: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/16.jpg)
What is the General Theory of Relativity?
It a synthesis of three different notions/facts, namely,
1. The notion of space and time,
2. Riemannian geometry and
3. Gravity.
![Page 17: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/17.jpg)
What is the General Theory of Relativity?
It a synthesis of three different notions/facts, namely,
1. The notion of space and time,
2. Riemannian geometry and
3. Gravity.
![Page 18: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/18.jpg)
Space + Time − > Space-Time
Intuitively, space and time serve to catalog physical events by
giving their locations and times of occurrences. The labels are
assigned using physical systems (eg rods and clocks) and using
physical processes (eg synchronization).
Special Theory of Relativity accepts as postulate, that the
speed of light in vacuum is constant.
Instantaneous communications are out and space and time
merge into the single entity called space-time.
![Page 19: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/19.jpg)
Space + Time − > Space-Time
Intuitively, space and time serve to catalog physical events by
giving their locations and times of occurrences. The labels are
assigned using physical systems (eg rods and clocks) and using
physical processes (eg synchronization).
Special Theory of Relativity accepts as postulate, that the
speed of light in vacuum is constant.
Instantaneous communications are out and space and time
merge into the single entity called space-time.
![Page 20: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/20.jpg)
Space + Time − > Space-Time
Intuitively, space and time serve to catalog physical events by
giving their locations and times of occurrences. The labels are
assigned using physical systems (eg rods and clocks) and using
physical processes (eg synchronization).
Special Theory of Relativity accepts as postulate, that the
speed of light in vacuum is constant.
Instantaneous communications are out and space and time
merge into the single entity called space-time.
![Page 21: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/21.jpg)
Space + Time − > Space-Time
Intuitively, space and time serve to catalog physical events by
giving their locations and times of occurrences. The labels are
assigned using physical systems (eg rods and clocks) and using
physical processes (eg synchronization).
Special Theory of Relativity accepts as postulate, that the
speed of light in vacuum is constant.
Instantaneous communications are out and space and time
merge into the single entity called space-time.
![Page 22: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/22.jpg)
Riemannian Geometry
Space (or space-time) also has Geometrical properties such as
lengths of solid rods, sums of angles of triangle etc.
The geometrical properties can be encoded in terms of a
metric or a rule specifying length of a coordinate interval.
These are Riemannian geometries which are curved in general.
Pre 1905, the space is flat (Euclidean). Between 1905 and
1915, the space-time is still flat (Minkowski) but now moving
lengths contract and clocks slow down.
A rotating observer will measure the geometry to be
non-Euclidean due to the length contraction effects.
![Page 23: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/23.jpg)
Riemannian Geometry
Space (or space-time) also has Geometrical properties such as
lengths of solid rods, sums of angles of triangle etc.
The geometrical properties can be encoded in terms of a
metric or a rule specifying length of a coordinate interval.
These are Riemannian geometries which are curved in general.
Pre 1905, the space is flat (Euclidean). Between 1905 and
1915, the space-time is still flat (Minkowski) but now moving
lengths contract and clocks slow down.
A rotating observer will measure the geometry to be
non-Euclidean due to the length contraction effects.
![Page 24: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/24.jpg)
Riemannian Geometry
Space (or space-time) also has Geometrical properties such as
lengths of solid rods, sums of angles of triangle etc.
The geometrical properties can be encoded in terms of a
metric or a rule specifying length of a coordinate interval.
These are Riemannian geometries which are curved in general.
Pre 1905, the space is flat (Euclidean). Between 1905 and
1915, the space-time is still flat (Minkowski) but now moving
lengths contract and clocks slow down.
A rotating observer will measure the geometry to be
non-Euclidean due to the length contraction effects.
![Page 25: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/25.jpg)
Riemannian Geometry
Space (or space-time) also has Geometrical properties such as
lengths of solid rods, sums of angles of triangle etc.
The geometrical properties can be encoded in terms of a
metric or a rule specifying length of a coordinate interval.
These are Riemannian geometries which are curved in general.
Pre 1905, the space is flat (Euclidean). Between 1905 and
1915, the space-time is still flat (Minkowski) but now moving
lengths contract and clocks slow down.
A rotating observer will measure the geometry to be
non-Euclidean due to the length contraction effects.
![Page 26: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/26.jpg)
Riemannian Geometry
Space (or space-time) also has Geometrical properties such as
lengths of solid rods, sums of angles of triangle etc.
The geometrical properties can be encoded in terms of a
metric or a rule specifying length of a coordinate interval.
These are Riemannian geometries which are curved in general.
Pre 1905, the space is flat (Euclidean). Between 1905 and
1915, the space-time is still flat (Minkowski) but now moving
lengths contract and clocks slow down.
A rotating observer will measure the geometry to be
non-Euclidean due to the length contraction effects.
![Page 27: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/27.jpg)
Gravity
A curious fact: gravitational mass = inertial mass.
Uniform gravity is indistinguishable from uniform acceleration!
But, acceleration affects geometry, so gravity affects
geometry!!
But, matter/energy produces gravity, so matter/energy also
affects geometry!!!
GTR: Matter moves in a curved Riemannian manifold whose
geometry is determined by the matter distribution.
![Page 28: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/28.jpg)
Gravity
A curious fact: gravitational mass = inertial mass.
Uniform gravity is indistinguishable from uniform acceleration!
But, acceleration affects geometry, so gravity affects
geometry!!
But, matter/energy produces gravity, so matter/energy also
affects geometry!!!
GTR: Matter moves in a curved Riemannian manifold whose
geometry is determined by the matter distribution.
![Page 29: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/29.jpg)
Gravity
A curious fact: gravitational mass = inertial mass.
Uniform gravity is indistinguishable from uniform acceleration!
But, acceleration affects geometry, so gravity affects
geometry!!
But, matter/energy produces gravity, so matter/energy also
affects geometry!!!
GTR: Matter moves in a curved Riemannian manifold whose
geometry is determined by the matter distribution.
![Page 30: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/30.jpg)
Gravity
A curious fact: gravitational mass = inertial mass.
Uniform gravity is indistinguishable from uniform acceleration!
But, acceleration affects geometry, so gravity affects
geometry!!
But, matter/energy produces gravity, so matter/energy also
affects geometry!!!
GTR: Matter moves in a curved Riemannian manifold whose
geometry is determined by the matter distribution.
![Page 31: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/31.jpg)
Gravity
A curious fact: gravitational mass = inertial mass.
Uniform gravity is indistinguishable from uniform acceleration!
But, acceleration affects geometry, so gravity affects
geometry!!
But, matter/energy produces gravity, so matter/energy also
affects geometry!!!
GTR: Matter moves in a curved Riemannian manifold whose
geometry is determined by the matter distribution.
![Page 32: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/32.jpg)
Gravity
A curious fact: gravitational mass = inertial mass.
Uniform gravity is indistinguishable from uniform acceleration!
But, acceleration affects geometry, so gravity affects
geometry!!
But, matter/energy produces gravity, so matter/energy also
affects geometry!!!
GTR: Matter moves in a curved Riemannian manifold whose
geometry is determined by the matter distribution.
![Page 33: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/33.jpg)
Novelties of GTR
Light bends around massive bodies (confirmed);
Planetary orbits precess (confirmed);
Stars can collapse to form black holes (widely believed);
Gravitational waves (indirect evidence);
“Big Bang cosmology” (widely believed);
A small spinning body would precess due to curvature near a
massive body and an additional precession if the body is
rotating (Being tested!).
![Page 34: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/34.jpg)
Novelties of GTR
Light bends around massive bodies (confirmed);
Planetary orbits precess (confirmed);
Stars can collapse to form black holes (widely believed);
Gravitational waves (indirect evidence);
“Big Bang cosmology” (widely believed);
A small spinning body would precess due to curvature near a
massive body and an additional precession if the body is
rotating (Being tested!).
![Page 35: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/35.jpg)
Novelties of GTR
Light bends around massive bodies (confirmed);
Planetary orbits precess (confirmed);
Stars can collapse to form black holes (widely believed);
Gravitational waves (indirect evidence);
“Big Bang cosmology” (widely believed);
A small spinning body would precess due to curvature near a
massive body and an additional precession if the body is
rotating (Being tested!).
![Page 36: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/36.jpg)
Novelties of GTR
Light bends around massive bodies (confirmed);
Planetary orbits precess (confirmed);
Stars can collapse to form black holes (widely believed);
Gravitational waves (indirect evidence);
“Big Bang cosmology” (widely believed);
A small spinning body would precess due to curvature near a
massive body and an additional precession if the body is
rotating (Being tested!).
![Page 37: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/37.jpg)
Novelties of GTR
Light bends around massive bodies (confirmed);
Planetary orbits precess (confirmed);
Stars can collapse to form black holes (widely believed);
Gravitational waves (indirect evidence);
“Big Bang cosmology” (widely believed);
A small spinning body would precess due to curvature near a
massive body and an additional precession if the body is
rotating (Being tested!).
![Page 38: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/38.jpg)
Novelties of GTR
Light bends around massive bodies (confirmed);
Planetary orbits precess (confirmed);
Stars can collapse to form black holes (widely believed);
Gravitational waves (indirect evidence);
“Big Bang cosmology” (widely believed);
A small spinning body would precess due to curvature near a
massive body and an additional precession if the body is
rotating (Being tested!).
![Page 39: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/39.jpg)
Novelties of GTR
Light bends around massive bodies (confirmed);
Planetary orbits precess (confirmed);
Stars can collapse to form black holes (widely believed);
Gravitational waves (indirect evidence);
“Big Bang cosmology” (widely believed);
A small spinning body would precess due to curvature near a
massive body and an additional precession if the body is
rotating (Being tested!).
![Page 40: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/40.jpg)
Spinning Bodies - Newton’s Bucket
If a half filled bucket is set spinning, the water starts spinning
too and its surface becomes curved. What is this rotation
relative to?
It can’t be the relative rotation between water and the bucket
walls. Newton favored Absolute space, others favored Distant
stars! (Mach, 1893)
If the distant stars rotated and not the bucket, would the
surface be curved?
May be notions of inertia, dynamics be such that with either
view, the surface should be curved – Mach
![Page 41: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/41.jpg)
Spinning Bodies - Newton’s Bucket
If a half filled bucket is set spinning, the water starts spinning
too and its surface becomes curved. What is this rotation
relative to?
It can’t be the relative rotation between water and the bucket
walls. Newton favored Absolute space, others favored Distant
stars! (Mach, 1893)
If the distant stars rotated and not the bucket, would the
surface be curved?
May be notions of inertia, dynamics be such that with either
view, the surface should be curved – Mach
![Page 42: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/42.jpg)
Spinning Bodies - Newton’s Bucket
If a half filled bucket is set spinning, the water starts spinning
too and its surface becomes curved. What is this rotation
relative to?
It can’t be the relative rotation between water and the bucket
walls. Newton favored Absolute space, others favored Distant
stars! (Mach, 1893)
If the distant stars rotated and not the bucket, would the
surface be curved?
May be notions of inertia, dynamics be such that with either
view, the surface should be curved – Mach
![Page 43: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/43.jpg)
Spinning Bodies - Newton’s Bucket
If a half filled bucket is set spinning, the water starts spinning
too and its surface becomes curved. What is this rotation
relative to?
It can’t be the relative rotation between water and the bucket
walls. Newton favored Absolute space, others favored Distant
stars! (Mach, 1893)
If the distant stars rotated and not the bucket, would the
surface be curved?
May be notions of inertia, dynamics be such that with either
view, the surface should be curved – Mach
![Page 44: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/44.jpg)
Spinning Bodies - Newton’s Bucket
If a half filled bucket is set spinning, the water starts spinning
too and its surface becomes curved. What is this rotation
relative to?
It can’t be the relative rotation between water and the bucket
walls. Newton favored Absolute space, others favored Distant
stars! (Mach, 1893)
If the distant stars rotated and not the bucket, would the
surface be curved?
May be notions of inertia, dynamics be such that with either
view, the surface should be curved – Mach
![Page 45: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/45.jpg)
In GTR, this is possible since a rotating universe will have a
different geometry which will curve the water surface
[Brill-Cohen (1966), Pfister-Braun (1985)].
This entails space-time being not only curved, but also
dragged by a rotating body.
It is hard to rotate the whole universe or go very close to a
rotating black hole, but we can just watch a spinning
gyroscope falling freely around the rotating earth.
This is just what Gravity Probe B Experiment is.
![Page 46: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/46.jpg)
In GTR, this is possible since a rotating universe will have a
different geometry which will curve the water surface
[Brill-Cohen (1966), Pfister-Braun (1985)].
This entails space-time being not only curved, but also
dragged by a rotating body.
It is hard to rotate the whole universe or go very close to a
rotating black hole, but we can just watch a spinning
gyroscope falling freely around the rotating earth.
This is just what Gravity Probe B Experiment is.
![Page 47: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/47.jpg)
In GTR, this is possible since a rotating universe will have a
different geometry which will curve the water surface
[Brill-Cohen (1966), Pfister-Braun (1985)].
This entails space-time being not only curved, but also
dragged by a rotating body.
It is hard to rotate the whole universe or go very close to a
rotating black hole, but we can just watch a spinning
gyroscope falling freely around the rotating earth.
This is just what Gravity Probe B Experiment is.
![Page 48: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/48.jpg)
Basic Equation for Spin precession
S
J
h
Geodetic
Frame dragging
d~S
dt= 〈~Ω〉 × ~S ,
〈~Ω〉 =G
2r 3
[~JE − h(h · ~JE )
]+
3(GME )3/2
2r 5/2h
![Page 49: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/49.jpg)
Basic Equation for Spin precession
S
J
h
Geodetic
Frame dragging
d~S
dt= 〈~Ω〉 × ~S ,
〈~Ω〉 =G
2r 3
[~JE − h(h · ~JE )
]+
3(GME )3/2
2r 5/2h
![Page 50: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/50.jpg)
Basic details and Estimates
I Polar circular orbit with r ∼ 400 miles;
I Duration of experiment ∼ 12 – 14 months;
I Estimated Geodetic precession rate ∼ 6.6 arcsec/yr;
I Estimated Frame-dragging precession rate ∼ 40.9
milliarcsec/yr;
![Page 51: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/51.jpg)
Experimental Requirements
Basically, all sources which could induce precession must do so
with amount less than the estimate and since the change is so
minute, the detection of the changes must be extremely
precise.
1. Gyroscopes must be stable and drag-free;
2. Method to read the spin direction without disturbance;
3. Stable reference to a Guide Star;
4. Trust worthy Guide Star;
5. Isolating Relativity Effects;
6. Calibration Scheme.
![Page 52: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/52.jpg)
Experimental Requirements
Basically, all sources which could induce precession must do so
with amount less than the estimate and since the change is so
minute, the detection of the changes must be extremely
precise.
1. Gyroscopes must be stable and drag-free;
2. Method to read the spin direction without disturbance;
3. Stable reference to a Guide Star;
4. Trust worthy Guide Star;
5. Isolating Relativity Effects;
6. Calibration Scheme.
![Page 53: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/53.jpg)
Experimental Requirements
Basically, all sources which could induce precession must do so
with amount less than the estimate and since the change is so
minute, the detection of the changes must be extremely
precise.
1. Gyroscopes must be stable and drag-free;
2. Method to read the spin direction without disturbance;
3. Stable reference to a Guide Star;
4. Trust worthy Guide Star;
5. Isolating Relativity Effects;
6. Calibration Scheme.
![Page 54: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/54.jpg)
Experimental Requirements
Basically, all sources which could induce precession must do so
with amount less than the estimate and since the change is so
minute, the detection of the changes must be extremely
precise.
1. Gyroscopes must be stable and drag-free;
2. Method to read the spin direction without disturbance;
3. Stable reference to a Guide Star;
4. Trust worthy Guide Star;
5. Isolating Relativity Effects;
6. Calibration Scheme.
![Page 55: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/55.jpg)
The arrangement
![Page 56: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/56.jpg)
The Gyroscope
I Pure quartz ball of diameter 3.81 cm, spherical within few
atomic layers (< 3 10−7 inches), homogeneous within
(10−6), electrical dipole moment vanishingly small;
I Coated with Niobium, suspended electrically in its
housing with a clearance of 10−3 inches, remains centered
within 10−6 inches. Spin rate up to 10000 RPM;
![Page 57: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/57.jpg)
The Gyroscope
I Pure quartz ball of diameter 3.81 cm, spherical within few
atomic layers (< 3 10−7 inches), homogeneous within
(10−6), electrical dipole moment vanishingly small;
I Coated with Niobium, suspended electrically in its
housing with a clearance of 10−3 inches, remains centered
within 10−6 inches. Spin rate up to 10000 RPM;
![Page 58: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/58.jpg)
The Gyroscope
I Pure quartz ball of diameter 3.81 cm, spherical within few
atomic layers (< 3 10−7 inches), homogeneous within
(10−6), electrical dipole moment vanishingly small;
I Coated with Niobium, suspended electrically in its
housing with a clearance of 10−3 inches, remains centered
within 10−6 inches. Spin rate up to 10000 RPM;
![Page 59: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/59.jpg)
I Detecting spin direction: London moment induced in the
rotating super-conductor. Read by a SQUID with
sensitivity of 10−14 gauss. Cut earth’s magnetic field by
10−13 by enclosing in lead bags;
I Spin-up is achieved by evaporating super-fluid Helium
passing around the gyroscope.
![Page 60: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/60.jpg)
I Detecting spin direction: London moment induced in the
rotating super-conductor. Read by a SQUID with
sensitivity of 10−14 gauss. Cut earth’s magnetic field by
10−13 by enclosing in lead bags;
I Spin-up is achieved by evaporating super-fluid Helium
passing around the gyroscope.
![Page 61: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/61.jpg)
Guide Star
I Ideally should be a quasar – too faint;
I Star with known motion relative to a quasar, preferably a
binary with a radio source;
I convenient position in the sky – IM Pegasi;
![Page 62: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/62.jpg)
Guide Star
I Ideally should be a quasar – too faint;
I Star with known motion relative to a quasar, preferably a
binary with a radio source;
I convenient position in the sky – IM Pegasi;
![Page 63: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/63.jpg)
Guide Star
I Ideally should be a quasar – too faint;
I Star with known motion relative to a quasar, preferably a
binary with a radio source;
I convenient position in the sky – IM Pegasi;
![Page 64: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/64.jpg)
Guide Star
I Ideally should be a quasar – too faint;
I Star with known motion relative to a quasar, preferably a
binary with a radio source;
I convenient position in the sky – IM Pegasi;
![Page 65: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/65.jpg)
Telescope
I Should detect changes of 10−4 arcsec;
I Telescope should remain exactly pointed at the center of
the star’s image despite diffraction – split image;
I 5.6 inch diameter, 4 inch length, made up from quartz
elements with optical contact.
![Page 66: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/66.jpg)
Telescope
I Should detect changes of 10−4 arcsec;
I Telescope should remain exactly pointed at the center of
the star’s image despite diffraction – split image;
I 5.6 inch diameter, 4 inch length, made up from quartz
elements with optical contact.
![Page 67: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/67.jpg)
Telescope
I Should detect changes of 10−4 arcsec;
I Telescope should remain exactly pointed at the center of
the star’s image despite diffraction – split image;
I 5.6 inch diameter, 4 inch length, made up from quartz
elements with optical contact.
![Page 68: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/68.jpg)
Telescope
I Should detect changes of 10−4 arcsec;
I Telescope should remain exactly pointed at the center of
the star’s image despite diffraction – split image;
I 5.6 inch diameter, 4 inch length, made up from quartz
elements with optical contact.
![Page 69: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/69.jpg)
Near ZerosTemperature 1.80 K
Pressure < 10−11 Torr
Gravitational Acceleration < 10−10 g
material homogeneity < 3 10−6
mechanical sphericity < 3 10−7 inch
electrical sphericity < 5 10−7
Magnetic field < 10−6 Gauss
![Page 70: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/70.jpg)
Remarks
This experiment, conceived over 40 years ago, illustrates a
number of points:
I Even for a universally believed theory, there is a
qualitatively distinct prediction, which is yet to be
confirmed and is being tested;
I The predicted deviations are tiny making the tests very
much harder calling for tremendous sophistication;
I Yet, it is worthwhile to devise experiments, even if only to
test very esoteric ideas, since they spur new technological
developments.
![Page 71: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/71.jpg)
Remarks
This experiment, conceived over 40 years ago, illustrates a
number of points:
I Even for a universally believed theory, there is a
qualitatively distinct prediction, which is yet to be
confirmed and is being tested;
I The predicted deviations are tiny making the tests very
much harder calling for tremendous sophistication;
I Yet, it is worthwhile to devise experiments, even if only to
test very esoteric ideas, since they spur new technological
developments.
![Page 72: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/72.jpg)
Remarks
This experiment, conceived over 40 years ago, illustrates a
number of points:
I Even for a universally believed theory, there is a
qualitatively distinct prediction, which is yet to be
confirmed and is being tested;
I The predicted deviations are tiny making the tests very
much harder calling for tremendous sophistication;
I Yet, it is worthwhile to devise experiments, even if only to
test very esoteric ideas, since they spur new technological
developments.
![Page 73: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/73.jpg)
Remarks
This experiment, conceived over 40 years ago, illustrates a
number of points:
I Even for a universally believed theory, there is a
qualitatively distinct prediction, which is yet to be
confirmed and is being tested;
I The predicted deviations are tiny making the tests very
much harder calling for tremendous sophistication;
I Yet, it is worthwhile to devise experiments, even if only to
test very esoteric ideas, since they spur new technological
developments.
![Page 74: Einstein and Gravity Probe B - Institute of Mathematical ...shyam/Web/Talks/GravProbeBTalk.pdf · Einstein and Gravity Probe B Ghanashyam Date The Institute of Mathematical Sciences,](https://reader034.vdocuments.net/reader034/viewer/2022052014/602b0d0ef08888661220dd13/html5/thumbnails/74.jpg)
Thank You