- 3263 - SPACETIME - Space-Time Warp of Earth’s Gravity? Einstein’s Theory of Relativity predicted that gravity would warp the geometry of space-time just like acceleration would. If a satellite orbits the Earth it would follow the least energy path through this modified space-time. This would in turn cause the plane of the satellite’s orbit to wobble, or ‘precess‘. The effect of precession is very small, one part in a few trillion. This review tries to learn how the calculations were made and what the measurement results have been to prove Einstein is right.
---------------- 3263 - SPACETIME - Space-Time Warp of Earth’s Gravity?
- The idea of the above measurement was to put a precision gyroscope in orbit and measure its spin axis pointing direction over time. The “precession wobble angle” was expected to drift over time as the satellite orbited through the space-time distortions created by the massive, spinning Earth.
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- Think of the Earth’s mass as a bowling ball on a rubber sheet. The dimple in the center of the sheet represents warped space-time around a heavy mass. If the Earth were stationary that would be it. But, the Earth is spinning causing the dimple in the sheet to twist in the direction of rotation.
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- In Isaac Newton’s classical physic equations gravity depends on mass of the body alone and does not depend on rotation. Albert Einstein’s relativity physics calculates that a rotating mass drags the framework of space and time around with it. The effect is small around the Earth, but, around a Blackhole that is billions of times more massive than our Sun, the effect is mind blowing.
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- To test this theory of frame dragging the “Gravity-Probe B” satellite was launched April, 2004. It contained 4 precision gyroscopes made using the most perfect rotating spheres ever produced by mankind. The surface of the spheres we so perfect they did not vary more than the width of 40 atom diameters.
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- The satellite spin axis was pointed at a distant star as a fixed reference point. As the rotating Earth twists space-time the orbiting gyroscopes axis direction would drift. This angle of precession would be measured over a year’s period and compared to Einstein’s calculations which predicted a drift of:
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---------------------------- 0.041 arc seconds per year.
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- After measuring the gyroscopes precession angle drift for 50 weeks the results were:
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------------------------- 0.038 arc seconds per year
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- The Gravity Probe B experiment results proved Einstein’s calculations to within 17% uncertainty. The team at Stanford University were hoping to get results with better than 1% uncertainty. The measurement was very difficult. Measuring to within 0.0005 arc seconds is equivalent to viewing the edge of a sheet of paper 100 miles away.
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- For those not so faint of heart let’s try learning how the calculations were made: The gyroscopes would wobble as they orbited the mass of the Earth and their precession angle would change as the space-time around the Earth was changed. The angle of the precession would be directly proportional to the mass and rotational velocity of the Earth.
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- The bigger the mass and the faster the spin the greater the angle. The angle would also be inversely proportional to the radius of orbit. The further away the gyroscope was the smaller the angle of precession. The angle is more sensitive to distance away and it is inversely proportional to the radius cubed.
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-------------------------- Precession Angle is proportional to “p / r^3”
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------------------------- “p” = Angular Momentum of the Earth
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------------------------- “r” = radius of the satellite orbit.
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- Any time a function has proportionality it can be turned into an equivalency by using the proper “Constant of Proportionality“.
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-------------------------- Precession Angle = Constant * p / r^3
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- According to Einstein the proper Constant was:
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-------------------------- Precession Angle = (G / 2 * c^2) * p / r^3
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------------------------- G = the Gravitational Constant
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------------------------- G = 6.67 * 10^-11 meters ^3 / ( kilogram * seconds^2)
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------------------------- c = the speed of light
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------------------------- c = 300,000,000 meters per second
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------------------------ c^2 = 9 * 10^16 m^2 / sec^2
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------------------------ Constant = 6.67*10^-11 / 18 * 10^16 m / kg
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------------------------ Constant = 0.37 * 10^-5 m / kg
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- Therefore, ---------- Precession Angle = 0.37 *10^-5 * p / r^3
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- The Angular Momentum of the Earth, “p”, is a function of the Inertial Mass, “m”, and the velocity of rotation, “w”. The rotational velocity is one rotation of 2*pi radians in 24 hours or , 86,400 seconds.
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-------------------- w = 2 * pi / 8.64*10^4 radians / second
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- To calculate Inertia we have to make a major assumption, in my case. The Inertia ,“I”, of a perfect uniform sphere:
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-------------------- I = 2/5 * m * r^2
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-------------------- m = mass of the Earth = 5.97 * 10^24 kilograms
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-------------------- r = radius of the Earth = 6.38 * 10^6 meters
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-------------------- r^2 = 40.7 * 10^12 m^2
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- Therefore ,---- I = 2/5 * 5.97 * 10^24 * 40.7 * 10^12
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------------------- I = 97 *10^36 kg * m^2
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- Next, Angular Momentum = Inertial mass * radial velocity
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-------------------- p = I * w
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-------------------- p = 97 *10^36 * 2 * pi / 8.64*10^4
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-------------------- p = 7.069 *10^33 kg * m^2 / sec
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- The Earth is not a perfect sphere so after some complex modifications a better estimate for Earth’s Angular Momentum is:
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------------------ p = 5.861^10^33 kg * m^2 / sec
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- Now, putting all of these calculations back in to the original equation for the angle of precession:
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-------------------------- Precession Angle = Constant * p / r^3
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------- Angle = (0.37 * 10^-5 m / kg) * (5.861^10^33 kg * m^2 / sec) / (7.02*10^6 m)^3
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- In this case the radius of the satellite must be its elevation of 399 miles plus the radius of the Earth:
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----------------------- r = 6,378,000 + 624,000 = 7.02*10^6 meters
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-------------------------- Precession Angle = 6.28 * 10^-15 radians / second
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-------------------------- Precession Angle = 3.6 * 10^-13 degrees / second
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- There are 31,000,000 seconds in a year
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-------------------------- Precession Angle = 0.04018 arc seconds / year
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- Einstein’s calculation was for 0.041 arc seconds / year. The Gravity Probe B’s measurement was 0.039 arc seconds + or - 0.007 arc seconds. So the measurement uncertainty was 17%, from 0.032 to 0.046 arc seconds with the measurement errors.
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- You can see how difficult the measurement was. And, I can imagine how difficult it was for anyone to follow the math involved.
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- The concept you have just proven is that gravity and acceleration are the same thing and have the same effect on physics, space and time.
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- When you are in a space ship without windows you can not tell the difference if you are standing still in a gravitational field, or, if you are in free space and accelerating. Both pull you to the floor and all the physics works the same in both situations.
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- When you accelerate near the speed of light space contracts and time slows down. The same thing happens when you are in the gravitational field of a large mass, space contracts and time slows, space-time warps.
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- Not a significant change occurs when orbiting the space-time around the mass of the Earth. But, it is much more significant around a Neutron Star, or a Blackhole.
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------------------------ Earth’s precession angle = 0.041 arc seconds per year
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------------------------ Neutron Star precession angle = 3,960 arc seconds / year
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- The effect of the precession of the central mass is called the De Sitter Effect.
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- The effect of the mass’ rotation is called the Lens-Thirring Effect.
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- The common name is frame - dragging of space-time.
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- September 3, 2021 SPACETIME - Space-Time Warp of Earth’s Gravity? 3263
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--------------------- --- Friday, September 3, 2021 ---------------------------
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