VISUALIZING: General Relativity

It took some ten years for Einstein to publish an extension of his (Special) Theory of Relativity from the relatively [pun intended] simple case of constant velocity inertial reference frames to the more General case of accelerating frames of reference. His General Theory of Relativity was published in 1915.
LINKS TO RELATED POSTINGS AND RESOURCES
VISUALIZING Relativity - PowerPoint Show
VISUALIZING Relativity - Excel Spreadsheet
VISUALIZING for Science and Technology - Blog Posting
VISUALIZING Einstein's "Miracle Year" - Blog Posting
VISUALIZING My Insight Into Lorentz Gamma - Blog Posting
VISUALIZING the "Twin Paradox" - Blog Posting

Again, Einstein utilized his uncanny ability to VISUALIZE a complex situation and gain a unique insight. He recognized that:

• Gravity is equivalent to Acceleration, and 
• Massive bodies cause SpaceTime to CURVE in their vicinity.

He VISUALIZED a scientist, confined to a sealed box with instruments, and tasked to determine by measurements, if the box was "at rest" on the surface of the Earth, and therefore subject to Earth Gravity or in a spacecraft far from any massive object, and being accelerated at 9.8 m/s2 (32.2 ft/s2), which is the acceleration of gravity on Earth. Einstein concluded that the scientist could not make that determination.

[This is not strictly true. Given very sensitive accelerometers at head and foot level, not available in Einstein's time, the scientist would note a small difference if "at rest" on Earth because head and foot are different distances from the center of the Earth and Gravity varies as the square of the distance from the center of mass. In an accelerating spacecraft far from massive bodies, the acceleration at foot and head level would be equal.]

HOW DOES GENERAL RELATIVITY RELATE TO SPECIAL RELATIVITY?

In my research for this project, I happened upon a fact that is not prominently mentioned by many Internet expositions of Relativity. Namely that:

 the Relativistic Effects of Gravity 

in the vicinity of a massive body 

are exactly equal to 

the Relativistic Effects in a spacecraft 

(in deep space far from any massive body) 

moving at the Escape Velocity 

corresponding to that level of Gravity!

Escape Velocity from the Earth Surface is 11.2 km/s (about 25,000 MPH). It is defined as the launch speed required for a spacecraft, pointing straight up, such that it will not fall back to Earth (ignoring air friction and rotation of the Earth).

The formula for Escape Velocity from the vicinity of a massive body is the square root of 2GM/r, where G is the universal gravitational constant (6.67×10⁻¹¹ m³ kg⁻¹ s⁻²), M is the mass of the body, and r is the radius from the center of the body to the spacecraft at launch.

From this equation you should be able to deduce that Escape Velocity is less if the spacecraft is flown to a position that is high above the Earth Surface, and launched there, increasing r. That is one reason for multi-stage rockets. The final stage does not fire until far from the Surface. Less obvious is that a horizontal launch requires less speed than a vertical launch. Thus, the spacecraft is usually placed into high orbit prior to the final acceleration to escape.

EQUIVALENCE OF KINETIC AND POTENTIAL ENERGY

When you throw a ball straight up into the air, at some initial vertical speed, it continuously slows until it reaches the point where its speed is zero, and then it falls, continuously increasing downward speed, until it returns to your glove. If we ignore air friction, the ball will strike your glove at the same speed as your initial throw.

This is a perfect illustration of the exchange of Kinetic Energy for Potential Energy.

Your initial throw imparts a given vertical speed to the ball. From that speed, you can compute the Kinetic Energy. As the ball rises and slows due to the force of Gravity, the Kinetic Energy is converted to Potential Energy (ignoring loss to air friction). At the highest point, the ball has zero Kinetic Energy, and maximum Potential Energy. By conservation of Energy, the Potential Energy at the peak is exactly equal to the initial Kinetic Energy of the throw. As the ball falls, the process is reversed, with the Potential Energy being converted to Kinetic Energy.

Please note that we are speaking here of the Kinetic and Potential Energy referenced to your glove height. If you happened to be near a deep hole in the ground, such as a well, you could drop the ball and it would speed as it fell, because your glove is at a higher Potential Energy level than the bottom of the well.

If the hole extended all the way through the Earth, the ball would speed, gaining Kinetic Energy (converted from the Potential Energy) until it passed the center of the Earth, where the Potential Energy would be zero, having all been converted to Kinetic Energy. The ball would continue to the other side of the Earth, trading Kinetic for Potential Energy (again ignoring air friction and assuming the ball does not touch the sides of the hole, etc.)

The first equation in the graphic is the equation you probably learned in your physics class for computing Kinetic Energy, using Newtonian physics. This equation is "close enough" for virtually all practical engineering applications on Earth. (m is the mass of the ball, and v is the initial velocity.)

The second equation is based on Einsteinian Physics, and must be used to obtain absolutely accurate results for Kinetic Energy at speeds that are a significant fraction of the speed of light. (m is the mass of the ball, and v is accounted for by ϒ. See previous posting in the Blog series for the definition of ϒ the Greek letter Gamma).

The third equation is based on the equivalence of Kinetic and Potential Energy. It solves for Potential Energy, using Einsteinian Physics, given knowledge of  G, the universal gravitational constant (6.67×10⁻¹¹ m³ kg⁻¹ s⁻²), m is the mass of the body, and r is the radius from the center of the body to the ball (or spacecraft).

Ira Glickstein