I downloaded two volumes of Motion Mountain following your recommendation some time ago.
Volume 2: Relativity and Cosmology
Volume 4: The Quantum of Change
Lot's of reading there, most of which I've still to get around to!
It's worth repeating that all 5 volumes are obtainable as free downloads from here (scroll to the bottom to download individual volumes):
https://www.motionmountain.net/?gcl...L7NbClXGNicOVs9PImbJcN-SQBW8IjVBoCyXsQAvD_BwE
It's "de nada".... literally meaning "of nothing".... idiomatically meaning "you are welcome but what you are thanking me for was not a big deal for me to do, so no need to thank me"
And since you're from the UK
It's "Torremolinos", literally meaning "Torremolinos", idiomatically meaning "In the summer, it's the place with 10,000 caravans towed behind cars bearing plates from the UK"
And since you're from the UK
It's "Torremolinos", literally meaning "Torremolinos", idiomatically meaning "In the summer, it's the place with 10,000 caravans towed behind cars bearing plates from the UK"
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I didn't actually say thanks to tobydog, but I was appreciative of his scholarly intervention. 😎
Surely Tarzan's mass is the same no matter where he sits or stands?
And this is more often than not in a tree, as in the tale of him rescuing the Moon.
When the apes are frightened by an eclipse in which the Moon appears to be being devoured, Tarzan reassures them by shooting arrows at the "devourer", and as the eclipse passes is given credit by the apes for the "rescue".
Gripping stuff!
Just reading about the latest JWT discovery of the furthest galaxy yet imaged. IIUC, it was formed just 350 MYA the BB, and apparently far too big to have formed so quickly.
But, as I was reading this, I was thinking about Einstein describing one of his light bulb moments (unlike us mere mortals, he had them coming thick and fast!) which was when he looked back at the famous clock on the tower in Bern, and realised hat if he was moving away from the clock at c, the clock hands would be frozen in time i.e. they would not move as time would be standing still. Here is the clock - absolutely beautiful!
So, When we look back at those very early galaxies that to us appear to be racing away at or near lightspeed, our perception of how time is passing for them is that it is slowed down and very redshifted. But, this leads me to wonder if in fact for those galaxies receding at c, we won't catch some of them frozen or nearly frozen in time. So they in fact, may be much older than 350 million years after the BB. Which leads me to another question. Right now, we 'wind the clock back' based on the recessional velocity of far-way galaxies and and say that the universe is 13.8 billion years old. But, there is another thing to consider here. What if 13.8 billion years is just the time it takes for things to move so far apart in an expanding, accelerating universe that the most distant objects are racing away at or near c? Maybe, 13.8 billion light years isn't the time back to the BB, but some sort of physical constant, or term, that describes how long it takes for an observer in one location to see the most distant objects in the universe move at c? Since the expansion rate is accelerating, it may be that in earlier epochs, this physical constant had a different, much higher value. So as the universe ages, this time gets shorter.
But, as I was reading this, I was thinking about Einstein describing one of his light bulb moments (unlike us mere mortals, he had them coming thick and fast!) which was when he looked back at the famous clock on the tower in Bern, and realised hat if he was moving away from the clock at c, the clock hands would be frozen in time i.e. they would not move as time would be standing still. Here is the clock - absolutely beautiful!
So, When we look back at those very early galaxies that to us appear to be racing away at or near lightspeed, our perception of how time is passing for them is that it is slowed down and very redshifted. But, this leads me to wonder if in fact for those galaxies receding at c, we won't catch some of them frozen or nearly frozen in time. So they in fact, may be much older than 350 million years after the BB. Which leads me to another question. Right now, we 'wind the clock back' based on the recessional velocity of far-way galaxies and and say that the universe is 13.8 billion years old. But, there is another thing to consider here. What if 13.8 billion years is just the time it takes for things to move so far apart in an expanding, accelerating universe that the most distant objects are racing away at or near c? Maybe, 13.8 billion light years isn't the time back to the BB, but some sort of physical constant, or term, that describes how long it takes for an observer in one location to see the most distant objects in the universe move at c? Since the expansion rate is accelerating, it may be that in earlier epochs, this physical constant had a different, much higher value. So as the universe ages, this time gets shorter.
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The frozen hands of the clock are due to time dilation which is a consequence of the relative motion of material bodies in the theory of special relativity.
On the other hand, the recession rate of distant galaxies is due to the expansion of spacetime which, being immaterial, is not bounded by the speed of light.
The distance beyond which objects recede from us at a rate greater than the speed of light (c) due to the expansion of the Universe is called the Hubble radius. Light from distant galaxy clusters that are receding from us at a rate greater than c can never reach us.
The Hubble radius is calculated from r = c/H and it will always be 14.4 billion light years, provided the Hubble constant (H) remains the same value in the future.
"The Hubble radius is actually smaller than the radius of the observable Universe. Because of the expansion of the universe, light from objects beyond the Hubble radius could reach us as the Universe was much smaller in the past. The observable universe is simply everything we can theoretically observe from Earth, even if it’s currently moving away from us faster than c due to the universe’s expansion." https://astroblog.cosmobc.com/hubble-volume/
"Observations indicate that the expansion of the universe is accelerating and that the Hubble constant is thought to be decreasing. Thus, sources of light outside the Hubble horizon but inside the cosmological event horizon can eventually reach us. A fairly counter-intuitive result is that photons we observe from the first ~5 billion years of the universe come from regions that are, and always have been, receding from us at superluminal speeds." https://en.wikipedia.org/wiki/Hubble_volume
On the other hand, the recession rate of distant galaxies is due to the expansion of spacetime which, being immaterial, is not bounded by the speed of light.
The distance beyond which objects recede from us at a rate greater than the speed of light (c) due to the expansion of the Universe is called the Hubble radius. Light from distant galaxy clusters that are receding from us at a rate greater than c can never reach us.
The Hubble radius is calculated from r = c/H and it will always be 14.4 billion light years, provided the Hubble constant (H) remains the same value in the future.
"The Hubble radius is actually smaller than the radius of the observable Universe. Because of the expansion of the universe, light from objects beyond the Hubble radius could reach us as the Universe was much smaller in the past. The observable universe is simply everything we can theoretically observe from Earth, even if it’s currently moving away from us faster than c due to the universe’s expansion." https://astroblog.cosmobc.com/hubble-volume/
"Observations indicate that the expansion of the universe is accelerating and that the Hubble constant is thought to be decreasing. Thus, sources of light outside the Hubble horizon but inside the cosmological event horizon can eventually reach us. A fairly counter-intuitive result is that photons we observe from the first ~5 billion years of the universe come from regions that are, and always have been, receding from us at superluminal speeds." https://en.wikipedia.org/wiki/Hubble_volume
I presume that here we are talking about co-moving space time, but, would that not include time dilation effects for objects that were moving at c?
In other words, although one is due to time dilation and the other the expansion of space time, would we not encounter similar visual effects since in both cases the relative motion between observer and object approaches c
In other words, although one is due to time dilation and the other the expansion of space time, would we not encounter similar visual effects since in both cases the relative motion between observer and object approaches c
I'm not sure what you mean by comoving spacetime.
In expanding space, distance is a dynamical quantity which changes with time and the most common way of defining distance in cosmology is comoving distance: https://en.wikipedia.org/wiki/Comoving_and_proper_distances
Anyway, I am unsure of your premise since there is no relative motion of material objects in expanding spacetime. Spacetime is somehow 'growing' between them over time.
In expanding space, distance is a dynamical quantity which changes with time and the most common way of defining distance in cosmology is comoving distance: https://en.wikipedia.org/wiki/Comoving_and_proper_distances
Anyway, I am unsure of your premise since there is no relative motion of material objects in expanding spacetime. Spacetime is somehow 'growing' between them over time.
My vague notion is that the proper way to represent Spacetime is as a 4x4 Lorentz matrix. A Quaternion in effect.
The top left element is a timelike scalar often wriiten as 1, and the 3x3 part is a vector representing Space.
I'll see if Motion Mountain helps: https://www.motionmountain.net/
In fact a 2x2 real matrix maps to the familiar complex number and angle notation of exp (i theta):
The Determinant is 1, as in cos^2 + sin^2 = 1. Determinant of 1 tells you the area doesn't change.
Where it always gets interesting is if you extend it so each element can be complex in itself. This leads to spinors AFAIK, and circles become hypertbolas, leading to hyperbolic functions.
This is Dirac and Fermi spin stuff in Quantum Mechanics. You get this relation between trig functions and hyperbolic functions. I shall ponder it.
Latest Astro news! I have bought a better Nikon £92 lens. f1.8 which is happy at 4 seconds to avoid trailing worse than 1 arc minute. The Moon is 30 arc minutes for comparison. This hugely exceeds the stock zoom which is f5 at 35mm.
Here we have planet Uranus at mag 5.7 in the exteme bottom right below the Pleiades about 1cm in.
https://theskylive.com/planetarium?obj=uranus
Here we have a genuine star double Epsilon Lyrae at 3 arc minutes, takes about 100,000 years to rotate and is 150 LY away:
https://en.wikipedia.org/wiki/Epsilon_Lyrae
I was up early today, and I missed the slightly unexciting partial moon eclipse, but nabbed the planet despite a lot of glare from the full moon and usual light pollution. This lens is excellent.
I think ISO 800 and 4 seconds at a sharper f2.8 might be best for these Portsmouth skies on my squat tripod. I also use a 10s time delay to avoid bumping the camera.
😎
The top left element is a timelike scalar often wriiten as 1, and the 3x3 part is a vector representing Space.
I'll see if Motion Mountain helps: https://www.motionmountain.net/
In fact a 2x2 real matrix maps to the familiar complex number and angle notation of exp (i theta):
The Determinant is 1, as in cos^2 + sin^2 = 1. Determinant of 1 tells you the area doesn't change.
Where it always gets interesting is if you extend it so each element can be complex in itself. This leads to spinors AFAIK, and circles become hypertbolas, leading to hyperbolic functions.
This is Dirac and Fermi spin stuff in Quantum Mechanics. You get this relation between trig functions and hyperbolic functions. I shall ponder it.
Latest Astro news! I have bought a better Nikon £92 lens. f1.8 which is happy at 4 seconds to avoid trailing worse than 1 arc minute. The Moon is 30 arc minutes for comparison. This hugely exceeds the stock zoom which is f5 at 35mm.
Here we have planet Uranus at mag 5.7 in the exteme bottom right below the Pleiades about 1cm in.
https://theskylive.com/planetarium?obj=uranus
Here we have a genuine star double Epsilon Lyrae at 3 arc minutes, takes about 100,000 years to rotate and is 150 LY away:
https://en.wikipedia.org/wiki/Epsilon_Lyrae
I was up early today, and I missed the slightly unexciting partial moon eclipse, but nabbed the planet despite a lot of glare from the full moon and usual light pollution. This lens is excellent.
I think ISO 800 and 4 seconds at a sharper f2.8 might be best for these Portsmouth skies on my squat tripod. I also use a 10s time delay to avoid bumping the camera.
😎
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Let me clarify that - it should have read 'comoving distance'. Objects further away are moving faster than closer objects and we can see that because of the redshift.
Here is the wiki entry for comoving space time
https://en.wikipedia.org/wiki/Comoving_and_proper_distances#:~:text=The comoving time coordinate is,tells when an event occurs.
In comoving spacetime, the distance between objects moving apart does not change since the ruler or metric used to measure it is also changing by the same factor. Here is the wiki summary
In standard cosmology, comoving distance and proper distance (or physical distance) are two closely related distance measures used by cosmologists to define distances between objects. Comoving distance factors out the expansion of the universe, giving a distance that does not change in time due to the expansion of space (though this may change due to other, local factors, such as the motion of a galaxy within a cluster).[1] Proper distance roughly corresponds to where a distant object would be at a specific moment of cosmological time, which can change over time due to the expansion of the universe. Comoving distance and proper distance are defined to be equal at the present time. At other times, the Universe's expansion results in the proper distance changing, while the comoving distance remains constant.
In an expanding cosmos, in flat space time every metre is stretching as it expands. That would imply that if we compared the proper distance c would have moved in 1 second 500 million years after the BB to how far it travels today, there would be a difference. Is c a function of the age of the universe then?
My observation/question about the clock and galaxies that are moving away from us at C is: would they disappear or be frozen in time to just before they disappeared?