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Ask a physicist anything. (8)

Wiccan_Child

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You flip a coin, and whatever it lands on, you get a dollar.

Thereafter every time it lands on that side in succession, you will get another dollar.

But the first time it lands on the other side, the game is over.

What is the expectation value of your winnings?
The odds of winning $N is (1/2)[sup]N[/sup], and the expectation value is the sum of each winning amount multiplied by its odds. So $1x0.5 + $2x0.25 + $3x0.125...

By my calculations, this converges to the value of $2, which is the amount you're most likely to win.

(Do you know of a casino where this game is played? $2 isn't much, but you're guaranteed a profit!)
 
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Elendur

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The odds of winning $N is (1/2)[sup]N[/sup], and the expectation value is the sum of each winning amount multiplied by its odds. So $1x0.5 + $2x0.25 + $3x0.125...

By my calculations, this converges to the value of $2, which is the amount you're most likely to win.

(Do you know of a casino where this game is played? $2 isn't much, but you're guaranteed a profit!)
Shouldn't it be $1x1 + $1x0.5 + $1x0.25 + $1x0.125...?

I.e. the geometric sum of (1/2)^n, which converges to 1, plus the initial one dollar?

(which makes the result the same, 2$, but the calculations a bit different)
 
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Wiccan_Child

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Shouldn't it be $1x1 + $1x0.5 + $1x0.25 + $1x0.125...?

I.e. the geometric sum of (1/2)^n, which converges to 1, plus the initial one dollar?

(which makes the result the same, 2$, but the calculations a bit different)
It's the same result, but if I understand your series correctly, we simply came at it from different angles.

In your series, you added up the odds of each addition to the pot - the first $1 is guaranteed, the second $2 is only 50% likely, etc. Hence, $1 x (1/2)[sup]n-1[/sup].

In my series, I added up the odds of each final total - there's a 50% chance of ending with $1, 25% of ending with $2, etc. Hence, $n x (1/2)[sup]n[/sup].

Both correctly model the system, though, just from different perspectives. I hadn't considered your way of doing it, I like it! :thumbsup:
 
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Wiccan_Child

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I have - as a thanks for your efforts - $20 to donate to either a climate change charity, or typhoon relief. Which is it to be?
Hmm...

Donating to climate change charity enables large but distant benefits, while typhoon relief charity enables immediate but limited benefits. What's the ratio of value of real current suffering to hypothetical future suffering? 1:1? 10:1? 1:10? Do we aid a few people now, or many people in the future?
 
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essentialsaltes

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(Do you know of a casino where this game is played? $2 isn't much, but you're guaranteed a profit!)

It's even more important to find this casino if they play it by the rules where your payout doubles with each successive match. since the payout grows at the same rate the odds shrink, the series diverges

$1x0.5 + $2x0.25 + $4x0.125 + $8x0.0625......

=$0.50 + $0.50 + $0.50 + $0.50 ....

The payout is infinite.

Then the headscratcher is, "How much should the casino charge you to play this game?" Or better yet, "How much would you be willing to pay to play?"

If they charge any finite amount, your expectation value is bigger than that, so it sounds like a good deal. And naive mathematicians lose all their money.

More sophisticated statisticians note that they don't have either an infinite amount of money, or an infinite amount of time to keep playing this game until they get one of the big payouts.
 
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Justatruthseeker

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It's even more important to find this casino if they play it by the rules where your payout doubles with each successive match. since the payout grows at the same rate the odds shrink, the series diverges

$1x0.5 + $2x0.25 + $4x0.125 + $8x0.0625......

=$0.50 + $0.50 + $0.50 + $0.50 ....

The payout is infinite.

Then the headscratcher is, "How much should the casino charge you to play this game?" Or better yet, "How much would you be willing to pay to play?"

If they charge any finite amount, your expectation value is bigger than that, so it sounds like a good deal. And naive mathematicians lose all their money.

More sophisticated statisticians note that they don't have either an infinite amount of money, or an infinite amount of time to keep playing this game until they get one of the big payouts.


Blackjack. As long as you double your bet every hand when you loose, you will always win your original bet as soon as you win one hand. If you bet $1 and loose, you then bet $2. If you win you win your $2 bet back, plus $2 from the house minus your $1 original bet leaves you with $1 in winnings. Just make sure you have enough to cover several loosing hands. Even the worse player will win at least one out of every 10 hands.

Of course the drawback is you only win whatever your original bet was, so if $1 dollar you must have enough to cover doubling the bet until you win. $1, $2, $4, $8, $16, $32, etc.
 
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pgp_protector

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Blackjack. As long as you double your bet every hand when you loose, you will always win your original bet as soon as you win one hand. If you bet $1 and loose, you then bet $2. If you win you win your $2 bet back, plus $2 from the house minus your $1 original bet leaves you with $1 in winnings. Just make sure you have enough to cover several loosing hands. Even the worse player will win at least one out of every 10 hands.

Of course the drawback is you only win whatever your original bet was, so if $1 dollar you must have enough to cover doubling the bet until you win. $1, $2, $4, $8, $16, $32, etc.

You also have to be playing on a table that will handle between $1 and $512or more
 
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Wiccan_Child

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A question for everyone: what is your favourite experiment (empirical or thought) that demonstrates a complex concept in a simple way? What's your favourite, most sublime experiment that demonstrates a complex/far-reaching principle in a simple way?

As an example, I absolutely love the relatively simple thought experiment of considering a charged particle next to a live wire. By using special relativity and considering the situation from a) the POV of the particle, and b) the POV of the wire, at both times you get a force on the particle. In one perspective, it's an electrical repulsion. In another, it's magnetic. To me this is a sublime thought experiment that really, viscerally shows how the electrical and magnetic forces are one and the same.

Veritasium (whose videos I cannot get enough of) has a great video on the experiment:

How Special Relativity Makes Magnets Work - YouTube
 
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Loudmouth

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A question for everyone: what is your favourite experiment (empirical or thought) that demonstrates a complex concept in a simple way? What's your favourite, most sublime experiment that demonstrates a complex/far-reaching principle in a simple way?

In the world of physics, my favorite simple but profound experiment is Young's Double Slit experiment. It is such a simple setup, and yet has such profound implications for how the smallest of things actually work.
 
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AV1611VET

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I give you an envelope containing a certain amount of money, and you open it.

I then put into a second envelope either twice this amount or half this amount, with a fifty-fifty chance of each.

You are given the opportunity to trade envelopes.

Should you?
 
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Chesterton

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Even the worse player will win at least one out of every 10 hands.

On average, but even that's not guaranteed. You stand the outside risk of experiencing many more than ten losses, in which case even if you're a multi-billionaire, when doubling-up, there's a very real chance that you could lose every penny you own in one sitting. Do you want to potentially risk everything you own to earn one dollar? Casino owners may not be Blaise Pascals, but they're not stupid; they know what they're doing.
 
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essentialsaltes

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I give you an envelope containing a certain amount of money, and you open it.

I then put into a second envelope either twice this amount or half this amount, with a fifty-fifty chance of each.

You are given the opportunity to trade envelopes.

Should you?

You stand more to gain than to lose, so yes. Suppose the envelope has $100.

Keeping it, you have a 100% chance of getting $100.

If you switch you have 50% chance of $200, and 50% of $50. .5*$200 + .5*$50 = $125, which is better than $100.
 
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Chesterton

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A question for everyone: what is your favourite experiment (empirical or thought) that demonstrates a complex concept in a simple way? What's your favourite, most sublime experiment that demonstrates a complex/far-reaching principle in a simple way?

My favorite:

Not really an experiment, but a graphic explanation for a layman or student, as in your example:

In order to explain "spacetime", or how time is a fourth dimension, a New York professor (whose name escapes me) came up with this: You’re invited to a party at the Empire State Building. In order to attend the party you need to know four things:

1) the street (a line);
2) the intersecting street (what corner is the building on? - the intersection of the first line through a second line);
3) what floor of the building is the party on? (a vertical line); and
4) what time is the party? If you know the first three things but not the time, the event will not take place for you.

With this explanation, as a layman, I came to understand how every "thing" is an "event" and every "event" is a "thing". A pebble lying on the ground is an event because once it wasn’t here, and someday it won’t be here again, just like a party. And that’s how I learned to understand a bit about Einsteinian "spacetime".

(Plus, maybe every particle is a bit wave, and every wave is a bit particle? I don't know.)
 
My least favorite:

When they try to demonstrate how mass warps spacetime by rolling a ball around the depressed center of a rubber sheet (as with a heavy bowling ball placed in the center of a trampoline, and a lighter ball rolled in a circle around it). This does not take into account the two other spatial dimensions and leaves me very unsatisfied as to how mass warps space in 360 degrees plus ____ (the number of degrees surrounding a sphere).
 
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Justatruthseeker

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My favorite:

Not really an experiment, but a graphic explanation for a layman or student, as in your example:

In order to explain "spacetime", or how time is a fourth dimension, a New York professor (whose name escapes me) came up with this: You’re invited to a party at the Empire State Building. In order to attend the party you need to know four things:

1) the street (a line);
2) the intersecting street (what corner is the building on? - the intersection of the first line through a second line);
3) what floor of the building is the party on? (a vertical line); and
4) what time is the party? If you know the first three things but not the time, the event will not take place for you.

With this explanation, as a layman, I came to understand how every "thing" is an "event" and every "event" is a "thing". A pebble lying on the ground is an event because once it wasn’t here, and someday it won’t be here again, just like a party. And that’s how I learned to understand a bit about Einsteinian "spacetime".

(Plus, maybe every particle is a bit wave, and every wave is a bit particle? I don't know.)
 
My least favorite:

When they try to demonstrate how mass warps spacetime by rolling a ball around the depressed center of a rubber sheet (as with a heavy bowling ball placed in the center of a trampoline, and a lighter ball rolled in a circle around it). This does not take into account the two other spatial dimensions and leaves me very unsatisfied as to how mass warps space in 360 degrees plus ____ (the number of degrees surrounding a sphere).


Nor does the rubber sheet analogy take into account that for a ball placed stationary on its edge, it would not "roll downhill" without a pre-concieved notion of an existing force beneath the sheet "pulling" it downhill. This is why the notion of the geometric interpretation of gravity as mere bent spacetime and not a force utterly fails and is a violation of cause and effect. In order for a ball placed stationary to move, a force must first cause it to move, else it would stay stationary forever.

Yet we know from experiments that two balls placed stationary near each other in space always move towards one another, once again disproving the geometric interpretation of spacetime as mere curvature of space and not a force.

As for time: time is simply a measurement of movement. This is its most direct definition. Whenever we measure time, we measure movement. We cannot measure time without measuring movement. The concept of time is dependent upon the concept of movement. Without movement, there is no time. Every clock measures movement: the vibration of a cesium atom, the swing of pendulum, the movement of a second hand, the orbit of the Earth around the Sun, or the revolution of the earth about it's axis designating days and nights.

In this way time can be thought of as a distance measurement. When we measure distance, we measure movement. We measure the change in position. When we measure time, we measure the same thing, but give it another name. Why would we do this? Why give two names and two concepts to the same thing? Distance and Time. I say, in order to compare one to the other. Time is just a second, comparative, measurement of distance. There is space and there is time, but no spacetime.
 
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Wiccan_Child

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Nor does the rubber sheet analogy take into account that for a ball placed stationary on its edge, it would not "roll downhill" without a pre-concieved notion of an existing force beneath the sheet "pulling" it downhill.
The point of the rubber sheet model is that it's a 2D model of a 3D system A heavy ball on a rubber sheet warps it down. Roll a second lighter ball in a straight line along the sheet, and its path will be bent towards the second ball - not because of gravity, but because of the warped sheet of rubber.

In 3D space, the effect is the same. Massive objects stretch space towards themselves, curving their trajectory and creating the 'force' of gravity. Einstein concluded space and time to be one object, spacetime, and so stretching space should also affect time - which it does.

How do we know this is true? Because it correctly predicts how light bends around massive objects (classically, light shouldn't bend as much as it does). This was superbly demonstrated in the 1919 eclipse that allowed astronomers to observe that a star just behind the (now eclipsed) Sun wasn't where it should be, but was where GR predicted it should appear due to the Sun's warping of its path.

This is why the notion of the geometric interpretation of gravity as mere bent spacetime and not a force utterly fails and is a violation of cause and effect.
Nonsense. Mass stretches space, curving motion and creating the 'force' of gravity. It might be you're taking the rubber sheet analogy too literally - you know objects don't stretch space because of gravity, right?

In order for a ball placed stationary to move, a force must first cause it to move, else it would stay stationary forever.

Yet we know from experiments that two balls placed stationary near each other in space always move towards one another, once again disproving the geometric interpretation of spacetime as mere curvature of space and not a force.
Well, yeah: becuase that's not what the theory says. GR doesn't say it's a curvature of space, it says it's a curvature of spacetime. As well, don't forget that, per SR, all objects move at speed c through spacetime. If you're stationary in space, you're still moving at c through time. Start moving through space, and you experience time dilation (per SR).

No object is stationary, which is why the objects in your example accelerate to one another - their geodesics are bent towards each other, warping their trajectory through spacetime.

From the object's point of view, there is no force of gravity. It moves along its geodesic, along its locally straight line, and it experiences no inertial deviation.

As for time: time is simply a measurement of movement. This is its most direct definition. Whenever we measure time, we measure movement. We cannot measure time without measuring movement. The concept of time is dependent upon the concept of movement. Without movement, there is no time. Every clock measures movement: the vibration of a cesium atom, the swing of pendulum, the movement of a second hand, the orbit of the Earth around the Sun, or the revolution of the earth about it's axis designating days and nights.

In this way time can be thought of as a distance measurement. When we measure distance, we measure movement. We measure the change in position. When we measure time, we measure the same thing, but give it another name. Why would we do this? Why give two names and two concepts to the same thing? Distance and Time. I say, in order to compare one to the other. Time is just a second, comparative, measurement of distance. There is space and there is time, but no spacetime.
GR, and its host of passed experiments, beg to differ. Space and time are one continuum, spacetime, as demonstrated by GR (and more directly by the effect gravity has on time; qv. GPS).
 
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Wiccan_Child

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My favorite:

Not really an experiment, but a graphic explanation for a layman or student, as in your example:

In order to explain "spacetime", or how time is a fourth dimension, a New York professor (whose name escapes me) came up with this: You’re invited to a party at the Empire State Building. In order to attend the party you need to know four things:

1) the street (a line);
2) the intersecting street (what corner is the building on? - the intersection of the first line through a second line);
3) what floor of the building is the party on? (a vertical line); and
4) what time is the party? If you know the first three things but not the time, the event will not take place for you.

With this explanation, as a layman, I came to understand how every "thing" is an "event" and every "event" is a "thing". A pebble lying on the ground is an event because once it wasn’t here, and someday it won’t be here again, just like a party. And that’s how I learned to understand a bit about Einsteinian "spacetime".

(Plus, maybe every particle is a bit wave, and every wave is a bit particle? I don't know.)
That's an interesting example, I hadn't heard of that before.

 
My least favorite:

When they try to demonstrate how mass warps spacetime by rolling a ball around the depressed center of a rubber sheet (as with a heavy bowling ball placed in the center of a trampoline, and a lighter ball rolled in a circle around it). This does not take into account the two other spatial dimensions and leaves me very unsatisfied as to how mass warps space in 360 degrees plus ____ (the number of degrees surrounding a sphere).
Well, 3D angles are called 'solid' angles, are measured in steradians (just as 2D angles are measured in radians, or degrees, or gradians), and there are 4pi steradians in a sphere (just as there are 2pi radians in a circle).

The trampoline analogy is actually quite good, but a lot of people get hung up on the apparent circular logic - it seems to say gravity causes warping causes gravity. This is just the limitation of the analogy.

Instead of thinking about the heavy object warping the rubber, instead think about taking a marble and rolling it along a straight line across the rubber. What will it do? The marble follows a 'local' straight line, called a geodesic, but on a large-scale this geodesic is itself bent towards the heavy object. So it analogy is meant to show that the marble's straight line is deflected and curved simply because the sheet on which it sits is itself curved. The analogy uses gravity to warp the sheet, but that's just a convenience (we could also pin it to the floor). In the real world case of planets and stars, the Sun warps spacetime, which creates the fictitious force of gravity. But the Sun doesn't warp spacetime because of gravity, and geodesics don't curve towards the Sun because of gravity - they curve because that's the straightest line from their point of view. Why the Sun warps spacetime at all is another question altogether.
 
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Justatruthseeker

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The point of the rubber sheet model is that it's a 2D model of a 3D system A heavy ball on a rubber sheet warps it down. Roll a second lighter ball in a straight line along the sheet, and its path will be bent towards the second ball - not because of gravity, but because of the warped sheet of rubber.

And once again, the ball warps the rubber sheet because you once again think of a pre-conceived force beneath the sheet pulling it downwards. The rubber sheet analogy is the worst analogy one could ever use to describe space. Your spacetime is composed of nothing, since space is not an ether. So basically you want something to warp nothing, then nothing to tell something how to move. Not only that, you want this nothing to stretch and expand as well, and not just to expand but to accelerate as it expands.

So why do you treat it as an ether, then refuse to consider ether theories?



In 3D space, the effect is the same. Massive objects stretch space towards themselves, curving their trajectory and creating the 'force' of gravity. Einstein concluded space and time to be one object, spacetime, and so stretching space should also affect time - which it does.
No it doesn't. Nearness to a charged object affects the vibrational rate at which the cesium atom vibrates. The atom is controlled by the electromagnetic force.

Electromagnetism - Wikipedia, the free encyclopedia

"The electromagnetic force is the interaction responsible for almost all the phenomena encountered in daily life, with the exception of gravity. Ordinary matter takes its form as a result of intermolecular forces between individual molecules in matter. Electrons are bound by electromagnetic wave mechanics into orbitals around atomic nuclei to form atoms, which are the building blocks of molecules. This governs the processes involved in chemistry, which arise from interactions between the electrons of neighboring atoms, which are in turn determined by the interaction between electromagnetic force and the momentum of the electrons...."

So the energy content of the atom determines the rate at which the atom oscillates, which is why acceleration also affects the rate at which clocks tick. When acceleration ends, the extra energy buildup is disipitated and the clock resumes normal operation. Unless of course you want me to believe that acceleration warps spacetime as well as objects of mass, and this warping during acceleration causes the cesium atom to vary in oscillation rates?

"The theoretical implications of electromagnetism, in particular the establishment of the speed of light based on properties of the "medium" of propagation (permeability and permittivity), led to the development of special relativity by Albert Einstein in 1905."

So based on the medium of propagation of the electromagnetic force, E developed SR. But yet there is no ether according to mainstream, no medium through which the force propagates. Space is composed of nothing, yet nothing warps, expands and accelerates. yah, if you say so.

It seems in modern astronomy nothing is responsible for quite a lot of things.


How do we know this is true? Because it correctly predicts how light bends around massive objects (classically, light shouldn't bend as much as it does). This was superbly demonstrated in the 1919 eclipse that allowed astronomers to observe that a star just behind the (now eclipsed) Sun wasn't where it should be, but was where GR predicted it should appear due to the Sun's warping of its path.
No, once again it is the medium through which the EM force propagates, it's permeability and permittivity that affects the electromagnetic force. This is why light bends in water (a medium) and why light bends around massive objects (the suns plasma atmosphere) and has nothing to do with gravity. Light is constant, does not slow or speed up near a gravitational source, but does slow when propagating through a medium. As it is also refracted, deflected or absorbed.

Refraction - Wikipedia, the free encyclopedia

"Refraction is the change in direction of a wave due to a change in its transmission medium."

So as light passes through the solar atmosphere it is refracted, not bent by your imaginary warped nothing. Of this you have direct experimental results, that confirm it is the sun's plasma atmosphere that refracts the light, not your mythical warped space.


Nonsense. Mass stretches space, curving motion and creating the 'force' of gravity. It might be you're taking the rubber sheet analogy too literally - you know objects don't stretch space because of gravity, right?
Indeed, I know objects don't stretch space at all, because space is composed of nothing and there is nothing to stretch or to be bent.


Well, yeah: becuase that's not what the theory says. GR doesn't say it's a curvature of space, it says it's a curvature of spacetime. As well, don't forget that, per SR, all objects move at speed c through spacetime. If you're stationary in space, you're still moving at c through time. Start moving through space, and you experience time dilation (per SR).

No object is stationary, which is why the objects in your example accelerate to one another - their geodesics are bent towards each other, warping their trajectory through spacetime.

From the object's point of view, there is no force of gravity. It moves along its geodesic, along its locally straight line, and it experiences no inertial deviation.
No, moving charged objects curve in space as they near other charged objects because of the magnetic force. Of which once again we have a wealth of experimental evidence.

Charged Particle in a Magnetic Field

This is also why planets orbit the sun in the sun's magnetic field, and why moons orbit planets in their magnetic field, and why the sun orbits the galaxy in its magnetic field. All rotational affects are caused by electric currents, they do not begin to spin by chance from the collapse of interstellar dust as gravity pulls the material inward, Gravity is a spherical force, so the collapse should be equal from all directions, counteracting spin.

Birkeland current - Wikipedia, the free encyclopedia

300px-Magnetic_rope.svg.png





GR, and its host of passed experiments, beg to differ. Space and time are one continuum, spacetime, as demonstrated by GR (and more directly by the effect gravity has on time; qv. GPS).
GR is simply a generalization of SR and SR was developed from the electromagnetic force (both Lorentz and Maxwell equations) and the effects of permeability and permittivity. SR is "On the Electrodynamics of Moving Bodies", being a generalization it can in no instance violate SR. That you fail to account for the Electrodynamics in GR is not E's fault, even he was never satisfied with GR.

Einstein's Pathway

"By his own later judgment, Einstein did not, in the end, find a theory that fully satisfied Mach's Principle. The immediate benefit of his new principle of equivalence, however, was that it let Einstein learn a lot about gravitation. For the principle delivered to Einstein one special case of a gravitational field that, he believed, conformed with relativity theory and in which all bodies truly fell alike. Einstein's program of research on gravity in the five years following 1907 was simply to examine the properties of this one special case and to try to generalize them to recover a full theory. His early hope was that the generalization of the principle of relativity would somehow emerge in the course of those investigations."

So your reliance on E's theory of GR, that he himself was not satisfied with is quite irrelevant. He hoped the generalization would emerge in later years, instead you threw in more Fairie Dust, expending and bending nothing. Dark matter, Dark Energy. All undetectable entities in your attempt to fudge the math of GR into a semblance of what we observe outside the solar system.

Don't get me wrong, GR is a good approximation of bound matter or matter in close confines where the electric and magnetic fields are balanced. But in the depths of space it fails utterly, and so Dark Matter, Dark Energy and bent nothing was invented in an attempt to fusge the math to fit observations in a universe dominated by unbound matter and matter NOT in close confines. Plasma, a distinct state of matter that does not behave like solids liquids and gasses (bound matter or matter in close confines where the EM forces are balanced). So once one leaves the vicinity of the solar system or center of galaxies, the math no longer works anymore and must be fudged by supernatural gap fillers.

Because the gravitational laws do not apply to unbound Plasma in space, the electromagnetic laws do. This is why Plasma does not behave like solids, liquids or gasses and is considered a distinct state of matter.

Plasma (physics) - Wikipedia, the free encyclopedia

"The presence of a non-negligible number of charge carriers makes the plasma electrically conductive so that it responds strongly to electromagnetic fields. Plasma, therefore, has properties quite unlike those of solids, liquids, or gases and is considered a distinct state of matter."

Yet you treat it no differently in your math than those solids, liquids and gasses you say it behave quite unlike. Then you wonder why after you ignore what 99% of the universe is, you must add 96% Fairie Dust to make the math work. Word games and semantics is all you are left with.
 
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