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

VISUALIZING: The "Twin Paradox"

In Einstein's ground-breaking 1905 paper, On the Electrodynamics of Moving Bodies, he provides the basis for the well-known "Twin Paradox" (where one twin takes a space journey at high speeds, and finds, upon returning home, that he or she has AGED less than the stay-at-home sibling):

If one of two synchronous clocks at A is moved in a closed curve with constant velocity until it returns to A, the journey lasting t seconds, then by the clock which has remained at rest the traveled clock on its arrival at A will be [1 - α] seconds slow*.

To VISUALIZE Einstein's thought experiment, let "A" be a location on Earth, where the stay-at-home twin resides, and the "closed curve" be the path followed by the traveling twin, moving at 87% of the speed of light (v/c = β = 0.8660) where α = 0.5 as depicted below.

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

As depicted, Blair and Aden are 20 years old when Aden takes off on a long space journey at ultra-high speed while Blair remains home. Aden's journey, at an average speed of 87% the speed of light, extends out to the vicinity of a Neutron Star (or a Black Hole) where Aden's spaceship "slingshots" and speeds back to Earth.

When Aden is at the half-way point, Blair has AGED 30 Earth-years and is 50 years old. However Aden, due to being in a state of high Kinetic Energy with respect to Blair, has AGED only 15 years and is only 30 years old.

By the time Aden returns from the journey, Blair has AGED an additional 30 Earth-years and has reached the ripe old age of 80. However Aden has only AGED an additional 15 years, and returns home a sprightly 50 year-old!

WHAT DOES THIS VISUALIZATION TELL US?

First of all, this is only a "thought experiment" and there are many practical limitations that make it unrealistic. None of our current spacecraft are capable of even 1% of the speed of light, much less the 87% imagined for Aden. Furthermore, even if we had such a spacecraft, and even if it carried only a clock and not a fragile human being, considering the G-forces involved,  it would take a number of years to accelerate up to 87% of the speed of light, perform the "slingshot", and decelerate to land safely on Earth.

A more realistic depiction would include those years of acceleration and deceleration and would require some portions of the journey to be faster than 87% of the speed of light so as to average 87%.

Note that the Einstein quote is from Einstein's 1905 SPECIAL RELATIVITY paper and he (wisely) specifies that the "closed curve" be at "constant velocity". It would take an additional ten years, and Einstein's 1915 GENERAL RELATIVITY paper to account for the Relativistic Effects of the acceleration and deceleration required for a practical journey. It turns out that the acceleration and deceleration of the traveling twin in the spacecraft would actually increase the difference in AGING somewhat. However, 60 years of Earth gravity, to which the stay-at-home twin would be exposed, would actually decrease the difference in AGING a bit.

On the other hand, some of the explanations of the "Twin Paradox" I found on the Internet expose what I think are misinterpretations of inertial reference frames and simultaneity.

CONFUSION IN EXPLANATIONS OF THE "TWIN PARADOX"

Symmetry and Simultaneity Run Riot !

In my explanation above, I state that, at the half-way point, Aden, on the spaceship, has aged 15 years while Blair, on Earth, has aged 30 years. Well, some would complain, if Aden has not yet done the "slingshot" turn-around, it is improper to state anything about difference in aging between Blair and Aden since both are still in their original frames of reference! They claim a symmetry where each twin sees the other as moving and the other as having a slower clock. They claim there is no such thing as simultaneity.

Different Inertial Frames

For example, right near the top of the Wikipedia explanation:

... each twin sees the other twin as moving, and so, according to an incorrect naive application of time dilation and the principle of relativity, each should paradoxically find the other to have aged more slowly. However, this scenario can be resolved within the standard framework of special relativity: the travelling twin's trajectory involves two different inertial frames, one for the outbound journey and one for the inbound journey, and so there is no symmetry between the spacetime paths of the two twins. [My emphasis]

Well, prior to the turn-around, each twin had one and only one inertial frame, not "two different inertial frames". So, does this explanation mean to say they both aged more slowly, or neither aged more slowly? Does it mean to say that, during the turn-around, the traveling twin suddenly got younger or the stay-at-home twin suddenly got older?

Sadly for me (an old engineer who cannot understand the meaning of minus two people - see the Physist/Engineer joke in my prevous posting ) YES, they do seem to think that the ages of the twins can suddenly change, based on how they do their calculations!

Gravitational Time Dilation

Further confusion in the Wikipedia explanation:

.... Explanations put forth by Albert Einstein and Max Born invoked gravitational time dilation to explain the aging as a direct effect of acceleration.

According to this Wikipedia quote, the Einstein/Born explanations invoke "gravitational time dilation to explain the aging as a direct effect of acceleration."  Well, the traveling twin certainly had to be accelerated and decelerated during launch and recovery and during the turn-around, and we learn from General Relativity that Relativistic Effects of gravity are equivalent to high-speed effects at certain levels of acceleration and speed. However, the amount of reduction in aging is proportional to the total length of time the traveling twin is at ultra-high speed, and the thought experiment could be lengthened to hundreds or millions of years, such that the acceleration/deceleration periods are an insignificant fraction of the travelling twin's journey.

Age Jump Instantly At the Turn-around

Yet further confusion in the Wikipedia explanation:

... For a moment-by-moment understanding of how the time difference between the two twins unfolds, one must understand that in special relativity there is no concept of absolute present. ...For different inertial frames there are different sets of events that are simultaneous in that frame. This relativity of simultaneity means that switching from one inertial frame to another requires an adjustment in what slice through spacetime counts as the "present". ...

... Just before turnaround, the traveling twin calculates the age of the Earth-based twin ... [but] ... Just after turnaround, if he recalculates, ... there is a jump discontinuity in the age of the Earth-based twin. ... [If the twins] regularly update each other on the status of their clocks by way of sending radio signals (which travel at light speed), then all parties will note an incremental buildup of asymmetry in time-keeping, beginning at the "turn around" point. Prior to the "turn around", each party regards the other party's clock to be recording time differently from his own, but the noted difference is symmetrical between the two parties. After the "turn around", the noted differences are not symmetrical, and the asymmetry grows incrementally until the two parties are reunited. Upon finally reuniting, this asymmetry can be seen in the actual difference showing on the two reunited clocks. [My emphasis]

OK, the twins are far apart for much of this thought experiment so radio signals between them will take years to reach their destinations. Therefore, even if the turn-around plans have been settled and the Relativistic Effects calculated before the launch, the stay-at-home twin will not know for sure whether or not they have been successful. The spacecraft may have blown up or gone off the planned course. Similarly, the traveling twin will not know the status of the stay-at-home. The Earth may have been destroyed by a meteor, etc.

But, it blows my mind that some physicists can imagine an instantaneous jump in age by any human being (much less a clock) due to a spacecraft turning around, or a calculation based on a delayed radio message.

MY (SIMPLE) EXPLANATION OF THE "TWIN PARADOX"

Yes, if two spaceships pass in the night, all they can measure is relative speed (even if one happens to be Spaceship Earth). According to all that is currently known, observers on each spaceship will measure the other as being shorter in the direction of travel than it really is (length contraction) and that the other's clock is running slow (time dilation). I got that.

As one Internet source noted, when two cars pass on a highway and each driver looks in their rear-view mirror, the other car appears to be getting smaller.  Of course, in the case of the cars, we know that neither is really getting smaller.

So, what is different in the case of the twins?

Well, for one thing, the spacecraft was loaded with fuel and the stay-at-home twin watched it blast off and accelerate. The traveling twin felt the acceleration to ultra-high speed. Due to that expenditure of fuel, the spacecraft was raised to a higher level of Kinetic Energy than it had when it was sitting on the launch pad.

Throughout its journey, the spacecraft continued at high speed relative to the Earth (assuming that any frictional losses of energy were made up by further expenditure of fuel).

I maintain that the Relativistic Effects of a slowdown of aging (clock rate) for the travelling twin compared to the stay-at-home twin is due to a relatively higher level of Kinetic Energy. (When we get to General Relativity later in this Blog series, we will learn that high levels of Potential Energy due to the acceleration of gravity have similar Relativistic Effects.)

Ira Glickstein

* Note: I've substituted "1 - α" for the equivalent, but more complex equation in Einstein's original paper, where α is the Square Root portion of the Lorentz Transformation ( ) as described in my previous Blog posting.