Twin Paradox Explained
Standard textbooks do not present the concepts of Relativity well. In particular these books askew understanding in favor of stressing calculational acumen blended with an excess of anthropomorphizing. I will try not to succumb to these same faults. So, when I write, "Bob and Alice perceive the events of the Twin Paradox via arrays of rods and clocks.", I need to carefully explain (in some detail) what that sentence means.
Let's start with "Bob The Person" versus "Bob The Observer":
"Bob The Person" is Bob sitting in his stationary ship in a region of space otherwise devoid of mass.
"Bob The Observer" is something quite different, indeed. Bob The Observer is an array of rods and clocks. In a flat region of space the rods are of equal length. The most useful arrangement is to think of the rods as forming cubes locked together in a regular array. Binding them together at each corner is a clock synchronized to all other clocks in this structure. But these clocks are not like the clocks you are used to, these clocks do not go 'tick' - 'tock' - 'tick' - 'tock' - 'tick' - 'tock'. In this region of flat space all clocks read the same value, a value that does not change . . . ever. So, I hope, we are all "seeing" a "jungle gym" of synchronized (frozen) clocks at the ends of ridged (equal length) rods.
In flat space, all synchronized clocks read the same number. In regions of non-flat space, synchronized clocks do not all read the same number.
In flat space, all rigid rods are of the same length. In regions of non-flat space, rigid rods are not all of the same length.
Next Step: Imagine this "jungle gym" being reproduced an infinite number of times. Each "jungle gym" reproduction occupies the same place as the original "jungle gym" and is different in only one way. That difference is that the synchronized (frozen) clocks in the reproductions do not read the same number as the original. In the easiest to display examples the numbers for the various reproductions differ by the same amount.
So, I hope (if I used the proper words) we are all "seeing" a four dimensional "jungle gym" of mutually perpendicular ridged rods. The ridged rods separating the various reproduced clocks have a length commensurate with the aforementioned time difference. The rods representing time differences do not need to be of the same length as the rods representing space differences. However the "pictures" are so much easier to "see" if those two sets of rods are of the same length. So, I will use that convention. Another convention I will use is to make sure all rods are measured using a common dimensionality (and preferably the same units).
- A furlong is of the same dimensionality as a meter, but they are different units.
- A lightyear is of the same dimensionality as a parsec, but they are different units.
- A "meter of time" is the same as a "meter of distance", 3 meters (of time) ≈ 10 nanoseconds (of time)
- A "year of time" is the same as a "year of distance".
- A "lightyear of time" is the same as a "lightyear of distance".
In spite of what you may think Han Solo was not wrong about the units. But, "less than twelve parsecs" is not every impressive ‽
Bob The Observer operates by recording events using Time-Space labels that looks like this ⇒ ( ct , x , y , z ). So, events labeled 1, 2, and so on to i would look like:
BE1 = ( Bct1 , Bx1 , By1 , Bz1 ), BE2 = ( Bct2 , Bx2 , By2 , Bz2 ), and so on . . . , BEi = ( Bcti , Bxi , Byi , Bzi )
Alice The Observer operates by recording events using Time-Space labels that looks like this ⇒ ( ct , x , y , z ). So, events labeled 1, 2, and so on to i would look like:
AE1 = ( Act1 , Ax1 , Ay1 , Az1 ), AE2 = ( Act2 , Ax2 , Ay2 , Az2 ), and so on . . . , AEi = ( Acti , Axi , Ayi , Azi )
As Alice The Person moves relative to Bob The Person, Alice The Observer is an array of rods and clocks. This array is similar to the array described above. The difference is that as a function of velocity this array is distorted relative to the array described above. The usual convention is to make BE0 = ( 0 , 0 , 0 , 0 ) coincide with AE0 = ( 0 , 0 , 0 , 0 ), and rotate the arrays so that the third and fourth axes are perpendicular to the motions of interest. This action allows us to suppress those other two dimensions and write:
Bob The Observer records events with labels that look like this ⇒ ( ct , x ). So, events labeled 1, 2, 3, and so on to i would look like:
BE1 = ( Bct1 , Bx1 ), BE2 = ( Bct2 , Bx2 ), BE3 = ( Bct3 , Bx3 ), and so on . . . , BEi = ( Bcti , Bxi )
Alice The Observer records events with labels that look like this ⇒ ( ct , x ). So, events labeled 1, 2, 3, and so on to i would look like:
AE1 = ( Act1 , Ax1 ), AE2 = ( Act2 , Ax2 ), AE3 = ( Act3 , Ax3 ), and so on . . . , AEi = ( Acti , Axi )
The values each Observer assigns to a particular event are governed by the distortion of Alice's array relative to Bob's array as displayed in the next two TimeSpace diagrams.
Bob is stationary at Bxi = 0 LY. The events he experiences are characterized by the labels BEi = ( Bcti , 0.0 LY). That is, you may think of him as moving along the vertical axis in this diagram. Alice is moving toward the positive direction. The events she experiences are characterized by the labels AEi = ( Acti , 0.0 LY). That is, you may think of her as moving along the ct' axis in this diagram.
The tick marks on all the axes have 2 LY spacings. So notice that Alice ages by eight years on the outbound trip and travels zero distance with respect to her frame of reference. But, from Bob's point of view Alice travels six LY distance in ten LY of time. The red line shows Alice's return path.
In the diagram below (from Alice's point of view) Alice ages by eight years while Bob moves to the left a bit less than five LY (actually 4.8 LY) as he ages by a bit more than six years (actually 6.4 years). The horizontal red line shows Bob suddenly moving from the 4.8 LY mark to the needed 6.0 LY mark as Alice comes to a stop. Then Bob suddenly moves from the 6.0 LY mark to the 4.8 LY mark before traveling toward Alice at 60% the speed of light.
The second diagram is clearly not physically reasonable and any conclusions it might suggest are suspect. Hence, these two diagrams make it clear that Alice will be younger than Bob when she arrives back at Bob's position.
The above scenario is not real convenient for the graphical representation of the full round trip including communicating via radio signals. So I will proceed with some graphs from an example I used in a class a few years ago. In that example I set β = 0.8. As a result, the length contraction factor is γ-1 = 0.6, the time dilation factor is γ = 1.6666, and we get 3.0 as the doppler factor.
The twins were 20 year old at the start. Alice flew a ship to a docking station 8 LY from Bob's position at 80% the speed of light. At two year intervals both Alice and Bob sent birthday messages toward each other.
Notice how they send signals with a frequency of one every two years. As they travel away from the other they receive signals with a frequency of one every six years, As they travel toward the other they receive signals with a frequency of three every two years. That's the Doppler Effect in action.
Notice the weirdness in the graph below. As Alice briefly comes to a stop "out there" Bob seems to change position by more than three LY in a very little time. Wow, that's a big jump for someone that is stationary, right ‽
In addition the radio signals just make no sense.
I have a Better Scenario:
All this is much easier to analyze if Alice accelerates smoothly up to a top speed, then accelerates smoothly to a stop. The acceleration continues to drive her ship to a top speed moving toward Bob, and finally accelerates to a stop next to Bob.
Comments
Isn't the problem with original formulation of the TP
1) the implicit (impossible) assumption that Alice accelerates from 0 to 0.8beta "instantaneously" and even reverses direction to -0.8beta.
2) there must be periods of acceleration from 0 to 0.8b, deceleration from 0.8 to 0, acceleration to -0.8b, and deceleration form -0.8b to 0.
During these periods of acceleration and deceleration, A must feel a "force" (unless she be "massless", i.e. a photon) which means that A and B are no longer in "mutually" inertial frames.
Thus, A and B are in different frames of reference. The most common representation of this as you showed is to "tilt" one wrt the other, which is not the most intuitive way of approaching this. It remains to proved that the difference in "aging", if any, is independent of the 4 acceleration/deceleration parameters.
(Let's ignore the (small) accelerations associated with the circular orbits of the Earth around the Sun and the rotation of the Galaxy for the argument.)
Good post.
There are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns – the ones we don't know we don't know.
"No" is the answer to this question
The "trick" was to realize that Bob remains in one frame of reference during the entire process and that Alice switched frames somewhere between the event of leaving Bob's location and arriving back at Bob's location. Even though this "frame jumping" is an acceleration. If you want to take the anthropomorphizing seriously: Just set it up so two Alices are traveling on two oppositely moving frames of references. If you time it right they will pass each other at the outer point and as they pass they will be able to compare "local" clocks and report these same numbers. Easy Peasy, and no accelerations to tidy up.
Remember this conundrum was usually in one of the final chapters of every Calc-Physics textbook. The notion was that the students would rely upon the basic properties of Special Relativity and not even realize they could (if they were careful) construct the transformation equations for accelerations. The challenge was that most of our fellow students were not able to "think outside the box", like some of us. Not boasting, just stating the obvious.
Deforming one frame wrt the other: "which is not the most intuitive way of approaching this." I suppose this is my goto approach because I think "geometrically", rather than "mathematically".
Actually the difference in aging is very real and depends sensitively on the acceleration/deceleration schedule. Calculating the general result is as straight forward as it is tedious. I like the presentation in the last few pages of Louis N. Hand and Janet D. Finch.
That's cheating.
It's not a paradox unless both objects are treated similarly. You can't simply anchor one reference frame and say THAT'S the one we use to determine the effects of time dilation on the entire system, especially when the other node will observe precisely the same effect in reverse when looking at its opposite number.
From A's time perspective, it is B who is accelerating away from her, and B's clock which is slowing down. Anchor A instead of B you get the negative of the answer of B to A, but that's not right either.
Instead, in our empty, free floating, binary system where no perspective is favored, everything cancels. There is no time dilation difference and A & B will have each aged the same as the other upon return.
The current working assumption appears to be that our Shroedinger's Cat system is still alive. But what if we all suspect it's not, and the real problem is we just can't bring ourselves to open the box?
It's a bit more subtle than that.
But, sometimes the effect is the same as simply anchoring one reference frame. Ah, but it is subtle . . . I have one lecture on how to find the "preferred" (please do notice the quotation marks) frame of reference. The whole lecture is about the Interval Between Events. It is a bit subtle and results in so many intractable problems becoming trivial. The thing is, Reality cares not for our rods and clocks, the only thing Reality cares about is the Reality of Intervals.
TimeSpace diagrams are just Feynman Diagrams in a classical setting. The rules are different but using them makes much stuff of physics rather intuitive.
Well, I must call it a night, and on the morrow prep for another trip to Medford.
The problem with your categorical final statement is
that we know, quite certainly, that it is incorrect. Time dilation as described by PR really happens, and we can and do observe it happening. If we really did this experiment, but sent Alice a longer distance at a higher speed, she could return having scarcely aged, but find Bob a moldering corpse.
Acceleration changes everything, though for almost everybody, including me, it's almost impossible to fully understand exactly why.
The earth is a multibillion-year-old sphere.
The Nazis killed millions of Jews.
On 9/11/01 a Boeing 757 (AA77) flew into the Pentagon.
AGCC is happening.
If you cannot accept these facts, I cannot fake an interest in any of your opinions.
Acceleration is observed both ways...
in exactly the same way.
Because one party 'feels' acceleration has absolutely nothing to do each party's observations of the other's time dilation.
That is simply not true given the parameters of the hypothetical.
Were Bob on the earth and Alice accelerated away then you would be correct, because the earth's time rate anchors Bob, but both parties out in empty space is a completely different scenario.
The current working assumption appears to be that our Shroedinger's Cat system is still alive. But what if we all suspect it's not, and the real problem is we just can't bring ourselves to open the box?
i'm sorry, but you are just flat-out wrong,
and we know from experimental evidence that you are wrong. it has nothing to do with what alice "feels" -- which is the dangerous thing about expressing this conundrum in terms of human "observers". alice's acceleration with respect to bob is just as real regardless of whether he "anchored" to earth's gravitational field.
The earth is a multibillion-year-old sphere.
The Nazis killed millions of Jews.
On 9/11/01 a Boeing 757 (AA77) flew into the Pentagon.
AGCC is happening.
If you cannot accept these facts, I cannot fake an interest in any of your opinions.
No I'm not.
but if you'd like to show me you experimental evidence id be happy to take a look.
In this hypo Alice's observation of Bob's acceleration is the same as Bob's observation of Alice's acceleration.
If you don't understand that, you don't understand the paradox.
Also, acceleration may affect an object's entropy, but it doesn't affect time dilation. Velocity and mass are the only relevant factors.
The current working assumption appears to be that our Shroedinger's Cat system is still alive. But what if we all suspect it's not, and the real problem is we just can't bring ourselves to open the box?
again -- talking about people observing people is
the wrong way to go.
if i send two ships, A and B, out into the far reaches of space (per PR's description), and if i put an electronic clock driven by a quartz crystal onto unmanned ship A, and i put a comparable clock onto unmanned ship B, and then send unmanned ship A out onto a journey relative to B's position (per PR's description), and unmanned ship A eventually returns per PR's description, and the two ships dock, and then fly themselves back to me, or indeed to anywhere in the universe, and someone -- anyone, with any history -- eventually climbs on board the two ships and compares clock A to clock B, clock B will signify that an several extra years have passed versus clock A. neither clock "observes" acceleration, either for itself or for the other. one clock, however, experiences acceleration with respect to their initially shared frame of reference, and the other does not.
i'm afraid you are the one who appears not to understand the paradox; or at least, if you do understand the paradox, your understanding is at odds with the one held by, as far as i know, every physicist who has an opinion on this. the original formulation of the paradox was framed in terms of two ships moving at relativistic velocity with respect to each other, and as they pass, each measures the other's clock to be ticking more slowly than its own; yet, somehow, when the ships resolve their differences in velocity and in space, somehow, at the very least, one of the ships measurements must indicate that the clocks are either synchronized, or that its own clock is running behind the other. contrary to your insistence about the equivalence of all frames of reference, two notable "edge-cases" have straightforward results:
case 0: as the ships pass, they both begin immediately to decelerate at exactly the same rate; when they have decelerated such that they are no longer moving with respect to each other, they accelerate back towards each other at precisely the same rate, but only until they are halfway back to their original rendezvous point, at which point they again begin to decelerate, so that they come to stop with respect to each other exactly at the original rendezvous point; at this point the crews of both ships will agree that the exact same amount of time has transpired since they originally passed each other.
case 1: when the ships reach that distant point at which they are no longer moving with respect to one another, only ship A accelerates/decelerates back, while ship B kills its engines. when the ships now rendezvous (in a different position relative to their original rendezvous), both ship's crews will agree that ship A's clock has ticked fewer times than has ship B's, since the moment of their original rendezvous.
i understand that you adamantly believe this is not the case, but i do not know why you believe it not to be so, and the closest you've come to explaining or justifying your adamance is to continue stating that acceleration just doesn't matter -- which i again note, is not the position put forward by Them What Decides What Physics Says.
to the limited extent that I understand it, Einstein couches his own discussion of this effect in terms of gravity, asserting that relatively speaking, anything that is accelerating is in its own self-created gravity field, which induces a gravity-time-dilation effect etc etc. more generally, the assertion is that when a thing accelerates it changes its inertial frame of reference, and this alteration in inertial frame of reference is what gives rise to the different experience of elapsed time relative to a thing that has not changed its inertial frame of reference.
The earth is a multibillion-year-old sphere.
The Nazis killed millions of Jews.
On 9/11/01 a Boeing 757 (AA77) flew into the Pentagon.
AGCC is happening.
If you cannot accept these facts, I cannot fake an interest in any of your opinions.
You are imposing a third party observer
onto the hypothetical that does not exist.
The original question involved what both A & B observe about each other. Not what you or any one else observes. And it doesn't matter what a third party observe anyway because the third party runs at its own time rate that distorts its own observation of the other two and so only serves to confuse the true relationship between A & B.
In that hypothetical, PR sepcifically stated:
Bottom line: this a binary hypothetical. I'm sorry you refuse to accept that.
Where do you get the notion that acceleration (not velocity) is a component necessary for calculating time dilation? That simply isn't true.
The current working assumption appears to be that our Shroedinger's Cat system is still alive. But what if we all suspect it's not, and the real problem is we just can't bring ourselves to open the box?
Clearly we are not going to agree.
I am 100% certain that you are 100% wrong, and that I would have as much trouble finding a physicist to agree with you as I would have finding a climate scientist to agree with an AGCC denier; and you are 100% certain that you are right (and I presume, that 99% of the physicists would agree with you). I'm also 100% certain that you aren't any smarter than I am, so I've no reason to prefer your certainty to mine; and your latest 2 responses tell me that we've reached a point where we are no longer even usefully communicating, but just talking past each other.
So let's drop it.
The earth is a multibillion-year-old sphere.
The Nazis killed millions of Jews.
On 9/11/01 a Boeing 757 (AA77) flew into the Pentagon.
AGCC is happening.
If you cannot accept these facts, I cannot fake an interest in any of your opinions.
Also...
The whole point of the exercise it that there are no other gravity generating bodies around. IT's just A & B in deep space. As I said before, I would agree with you if B were on the earth, but without this anchoring (inertial) frame of reference, your analysis falls apart because the original reference frames of A & B do not change with either distance or velocity.
The current working assumption appears to be that our Shroedinger's Cat system is still alive. But what if we all suspect it's not, and the real problem is we just can't bring ourselves to open the box?
Finally...
I find your incessant appeals to authority an annoying and unnecessary distraction. If you would like to provide authority for your assertions then by all mean give me links and I will look at them, but you gain no argumentative high ground by simply stating that unnamed scientists agree with you.
I'd also point out that this whole conversation started with the recognition that current physical theories of the universe have been shown in many ways to be substantially flawed, and that much of what we have long assumed about the 'Laws of Physics' have been shown time and again to simply not work when removed from the terrestrial sphere.
So if you would like to continue discussing this, please stop lecturing me for not agreeing with your less than perfect understanding of scientific dogma. Because frankly, I don't have time for ideologues masquerading as truth seekers.
The current working assumption appears to be that our Shroedinger's Cat system is still alive. But what if we all suspect it's not, and the real problem is we just can't bring ourselves to open the box?
And I don't have time for people who tell me I don't understand
things that I actually understand quite well.
The earth is a multibillion-year-old sphere.
The Nazis killed millions of Jews.
On 9/11/01 a Boeing 757 (AA77) flew into the Pentagon.
AGCC is happening.
If you cannot accept these facts, I cannot fake an interest in any of your opinions.
I was going to say, "Exactly so" to this:
This is exactly how I was taught to analyze the TP problem back in a second-year SR class, in 1965 by LEH Trainor (q.v.). [See above comment].
Still, it needs to be proved that one gets the same result for any value of the four a/d parameters. Don't assume a(out) = d(out) = a(in) = d(in).
There are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns – the ones we don't know we don't know.
Should I include the equations . . .
I have a completed example that I planned to HTMLize, for this forum. I have shown all the work within the LaTeX file of the example and it would be straight forward but tedious to make all four accelerations different. The real pain in the ass bit is to seamlessly join the four hyperbolic curves.
Would you like a copy of my TwinParadox.pdf, it is just a "refined" presentation based upon Hand & Finch. Actually I think it is rather well done, if I do say so myself.
Certainly I would like a copy.
Thanks!
There are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns – the ones we don't know we don't know.
Here's another delicious one.
Again, sorry for the anthropmorphisation.
I'll use, PR, as my example
PR is in a space ship travelling at a sufficiently high (constant) velocity to make special relativistic effects manifest.
He is approaching an exoplanet of radius R at a transverse distance of, say, 4R.
Describe what he sees during (a) approach, (b) passage, and (c) departure.
An observer, O on said exoplanet is clearly following PR's approach, etc. via superhuman vision or other means. Describe O's observation.
There are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns – the ones we don't know we don't know.
This look like a fun exercise.
How about we make it a superdense ball of (primarily) Neutrons and title it "Dragon's Egg Revisited ". Bob (Robert L. Forward, for the non-cognoscenti) didn't make the flyby fast enough to include those interesting effects, but then his focus was on the effects of intense gravitational fields and tiny sentient beings.
Great idea!
I was only thinking of the SR effects (in fact my "junior thesis" was precisely on my above question, for which I received a grade that I was not unhappy with).
Adding GR considerations to this adds several levels of complexities. Why not increase the masses even further and describe the motion and behaviour of one moving black hole relative to a stationary one? Suppose the two black holes have masses and entropies (equivalent to the area by the Hawking-Beckenstein equation) M1, S1 and M2, S2.
Let's let BH 1 move at a speed [beta] relative to BH2 at a distance of closest approach d.
An obvious question (among many) then arises: Under what conditions will the two black holes coalesce?
Thanks, RiP, this is great fun!
There are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns – the ones we don't know we don't know.
(No subject)
There are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns – the ones we don't know we don't know.
Damn, you had me, sorta, until the formulas
Well, actually, I had a flashback to Java class when I read the word "array", and that put me into gag/tearing up mode. After a regroup, and a reread, I was fine until the formulas. Sigh. . . I have wonderful wishes for you and your granddaughter, though!
Cheers!
I wholeheartedly agree . . .
The equations can be a bit off-putting, that's why I prefer to present this via graphical representations. With graphics (IRL) I can point at the various features and convey some really sophisticated concepts with high fidelity. This is, of course, best done when the presentation mode is a conversation rather than a lecture. Unfortunately, the latter works in this medium, the former, not so much.
Thanks to you-all, I get to practice what I have shunned throughout my years of teaching.
Yes, that was how I parsed it as well.
Just to show you that I do not know more about physics than I think I know, the first time I read the problem, I wondered when and where Bob and Alice met Ted and Carol. (Never saw the film, but the title burrowed into my mind for some unknown reason.)
And, as usual I will try to leave you singing:
[video: https://www.youtube.com/watch?v=M4D9kT7Phvs]
(If anyone feel this kind of post trolls these brilliant science threads, please let me know by post or pm, so I know to stop. Disrupting them is not my intention.)
Yea, I don't get the formulas either!
Which I think is what you're saying by your diversion. I started a calculus class once without taking Algebra II first--I was always good at math and thought I could wing it-boy was I wrong. Dropped it after the first exam when I realized how lost I was.
My performance was so poor . . .
I was not allowed (in high school and junior college) to take some classes in a timely fashion so I said, "Oh, what the hell!" and just skipped the middle two terms of the calculus sequence at the four year college. This resulted in a few interesting interactions with my Advanced Calculus teacher (ex-marine drill sergeant) . . . on occasion, hilarious. My record was consistent, a little over fifty percent pass rate on exams.
Aw, I think I got a 40%...the
Aw, I think I got a 40%...the end of the world for an A student...
Many "A" students hate me, sigh.
Did the same.
The class lost me on day one. Something about the f of x. f of x? Who cares about the f of x, I thought this was a math class? I went to a couple more classes, then spent most of that class time in the cafeteria, waiting for political science or gym to start.
the little things you can do are more valuable than the giant things you can't! - @thanatokephaloides. On Twitter @wink1radio. (-2.1) All about building progressive media.
You're way ahead of me.
I barely read a quarter of the problem before my eyelashes started to hurt.
Trolling . . . Hardly
I am fascinated by obscure artistic (and pun-ish) references to illustrate concepts and connections. Also, I like the trip down memory lane. You might need to check out the dates to get that last one.
Say no more.
[video:https://www.youtube.com/watch?v=Pe3yPvY8Z10]
I know much about how the brain works . . .
at the microscopic level, but how we can have such sharp (well sort of sharp) recall is astounding. Encoding, decoding, and modifying memory is almost magic. I used to be a violinist and stuff like this triggers an incredible number of memories. That voice will never get old . . . Thanks for the memories.
Thanks! Good to know I am welcome in the class clown corner.
There is no paradox
The fallacy is assuming that everything in the universe ages at the same rate, i.e. clocks are unaffected by velocity and acceleration. Which is to say the Newtonian Universe. But that was only a low speed low acceleration approximation to reality. Physics has had to reject concepts many times like Phlogiston and the Luminerous Ether, both of which were attempts to reconcile experiment with pre-conceived pre-Science expectations.
I've seen lots of changes. What doesn't change is people. Same old hairless apes.
The paradox is that relativity "appears" to assert that when
they meet again, Alice will be younger than Bob, and Bob will be younger than Alice. Far from assuming that everything in the universe ages at the same rate, the paradox is explicitly built on the understanding that such is not the case.
The earth is a multibillion-year-old sphere.
The Nazis killed millions of Jews.
On 9/11/01 a Boeing 757 (AA77) flew into the Pentagon.
AGCC is happening.
If you cannot accept these facts, I cannot fake an interest in any of your opinions.