Scientists use an atomic clock to measure the height of a mountain(latimes.com)
latimes.com
Scientists use an atomic clock to measure the height of a mountain
http://www.latimes.com/science/sciencenow/la-sci-sn-atomic-clock-gravity-20180212-story.html
48 comments
The innovation here is that they made the atomic clock portable enough to be taken to the top of a mountain, so that they could use the difference in clock speed to calculate the height by general relativity.
They made a strontium clock portable enough to take to the top of a mountain. There are already "chip scale" cesium clocks (see https://www.microsemi.com/product-directory/clocks-frequency...) 1x10^-11 (cesium) vs 1x10^-18 for strontium clocks [1]. The additional 7 orders of magnitude get you to measuring time dilation at very very small scales (which it what they tried to do in the article).
[1] https://www.nist.gov/news-events/news/2015/04/getting-better...
[1] https://www.nist.gov/news-events/news/2015/04/getting-better...
Yes, I think GE's "time machine" employs four cesium clocks in a compact configuration. With six, you can reduce the worldline divergence considerably.
The atomic wrist watch has been around for a while though.
http://www.leapsecond.com/pages/atomic-bill/
http://www.leapsecond.com/pages/atomic-bill/
This is an optical lattice clock which is a newer and more accurate technology than the older cesium beam type of that "wristwatch".
A portable vacuum chamber. Pretty cool. So unlike a pressure altimeter, this could also work in space. And unlike a radar altimeter, you don't need a direct line of sight to the surface.
Although seems like the measurement could also be distorted by large masses like the moon...
Although seems like the measurement could also be distorted by large masses like the moon...
Unfortunately, you still need some communication with the outside world. To the local observer, the clock will run at the same rate no matter where you are. Only distant clocks will change.
I wonder if you could measure the difference between two clocks on the same craft at a distance from one another. The strength of gravity follows the inverse square law so presumably the difference in time between the two points will change also.
That should work, and give you the local gravitational gradient, which would correspond to the altitude. You might want four arranged in a tetrahedron so you don’t need to orient them.
[deleted]
That's not really a big deal. Most GPS satellites have two in them and they're not all that huge.
GPS uses cesium (deviation, per Wikipedia: 10^−13), rather than strontium (deviation: 10^-17).
Cesium clocks were already pretty small back in the 1970s (https://en.wikipedia.org/wiki/File:President_Pi%C3%B1era_rec...), but if you're going to measure relativity-caused differences between the top and bottom of a mountain, I suspect the extra accuracy is important.
Cesium clocks were already pretty small back in the 1970s (https://en.wikipedia.org/wiki/File:President_Pi%C3%B1era_rec...), but if you're going to measure relativity-caused differences between the top and bottom of a mountain, I suspect the extra accuracy is important.
I'm not sure how this experiment works as the strength of gravity is variable across the planet anyway. Accounting for this could be tricky.
https://news.nationalgeographic.com/news/2011/04/110406-new-...
https://news.nationalgeographic.com/news/2011/04/110406-new-...
That’s across the entire surface of the plant. It isn’t going to vary much at all across the distance relative to the base and peak where they made their measurements would it?
It'd require careful calculation between the base of the mountain and the top, plus some other location for calibration purposes.
The article said that this was the first time atomic clocks were taken out of the lab to do this experiment, but definitely not the first time:
http://www.leapsecond.com/great2005/tour/
My favorite part is where he notes that with kids everything has to be exactly equal: "three kids, three clocks, three sodas." My somewhat fuzzy arguments with my brothers make me think he is a wise man.
http://www.leapsecond.com/great2005/tour/
My favorite part is where he notes that with kids everything has to be exactly equal: "three kids, three clocks, three sodas." My somewhat fuzzy arguments with my brothers make me think he is a wise man.
Quantum_clock#More_accurate_experimental_clocks:
https://en.wikipedia.org/wiki/Quantum_clock#More_accurate_ex...
> In 2015 JILA evaluated the absolute frequency uncertainty of their latest strontium-87 optical lattice clock at 2.1 × 10−18, which corresponds to a measurable gravitational time dilation for an elevation change of 2 cm (0.79 in) on planet Earth that according to JILA/NIST Fellow Jun Ye is "getting really close to being useful for relativistic geodesy".
AFAIU, this type of geodesy isn't possible with 'normal' time structs. Are nanoseconds enough?
"[Python-Dev] PEP 564: Add new time functions with nanosecond resolution" https://mail.python.org/pipermail/python-dev/2017-October/14...
> In 2015 JILA evaluated the absolute frequency uncertainty of their latest strontium-87 optical lattice clock at 2.1 × 10−18, which corresponds to a measurable gravitational time dilation for an elevation change of 2 cm (0.79 in) on planet Earth that according to JILA/NIST Fellow Jun Ye is "getting really close to being useful for relativistic geodesy".
AFAIU, this type of geodesy isn't possible with 'normal' time structs. Are nanoseconds enough?
"[Python-Dev] PEP 564: Add new time functions with nanosecond resolution" https://mail.python.org/pipermail/python-dev/2017-October/14...
The thread you linked makes a good point: there isn't really any reason to care about the actual time, only the relative time. These sorts of clocks just use 256- or 512-bit counters; it's not like they're having overflow issues.
I am confused, I always thought that moving faster makes time go slower but the article says this:
"For example, a clock on top of a tall mountain — far from the center of the Earth — will move a tiny bit faster than a clock at the base of that mountain, where the gravity is stronger. It's not a mechanical error. Time itself actually passes faster at the top of the mountain."
From what I can understand from the Wikipedia article is that time does go slower the faster you move: "Special relativity indicates that, for an observer in an inertial frame of reference, a clock that is moving relative to him will be measured to tick slower than a clock that is at rest in his frame of reference."
I did a little search and half the sites I saw said one thing and the other half said the opposite.
Would anyone mind explaining like I'm five? Or maybe share a trustworthy sources I could read on the subject?
Thanks.
"For example, a clock on top of a tall mountain — far from the center of the Earth — will move a tiny bit faster than a clock at the base of that mountain, where the gravity is stronger. It's not a mechanical error. Time itself actually passes faster at the top of the mountain."
From what I can understand from the Wikipedia article is that time does go slower the faster you move: "Special relativity indicates that, for an observer in an inertial frame of reference, a clock that is moving relative to him will be measured to tick slower than a clock that is at rest in his frame of reference."
I did a little search and half the sites I saw said one thing and the other half said the opposite.
Would anyone mind explaining like I'm five? Or maybe share a trustworthy sources I could read on the subject?
Thanks.
They're both correct. There are two different effects.
The classic example of this is the GPS satellites. Compared to the ground, an onboard clock of a GPS satellite is
- less gravity, time goes faster.
- More speed, time goes slower.
The classic example of this is the GPS satellites. Compared to the ground, an onboard clock of a GPS satellite is
- 7 microseconds/day slower because it moves at higher velocity.
- 45 microseconds/day faster because it experiences less gravity.
[paraphrasing]
Acceleration is a kind of speed. The more you're accelerating in a given direction, the faster you're going - even if there's something stopping you. Gravity makes you accelerate - even if there's ground stopping you from moving "down". So the higher the gravity, the higher the acceleration, so the higher your (instantaneous) speed, so time goes slower for you relative to other things not moving/accelerating as much.
If you're really close to a black hole, with LOTS of gravity, time will pass really slowly for you relative to most everything else - even if something is preventing you from actually falling in.
At the top of a mountain, gravity is lower. You're farther away from most of the Earth. Less gravity (however minute) means less acceleration means less (instantaneous) speed means time traveling relatively faster (or, well, not as slow as further downhill).
In space, far far away from everything, and so long as you're kinda averaging a speed of 0 relative to everything else, time ticks by the fastest (relative to most everything else).
So if you build two clocks that are practically perfectly in sync, and you move one to the bottom of a mountain, and one to the top, and then bring them back together, you can measure the tiny difference between them caused by one (the one from the top of the mountain) experiencing time faster than the other (from the bottom).
Acceleration is a kind of speed. The more you're accelerating in a given direction, the faster you're going - even if there's something stopping you. Gravity makes you accelerate - even if there's ground stopping you from moving "down". So the higher the gravity, the higher the acceleration, so the higher your (instantaneous) speed, so time goes slower for you relative to other things not moving/accelerating as much.
If you're really close to a black hole, with LOTS of gravity, time will pass really slowly for you relative to most everything else - even if something is preventing you from actually falling in.
At the top of a mountain, gravity is lower. You're farther away from most of the Earth. Less gravity (however minute) means less acceleration means less (instantaneous) speed means time traveling relatively faster (or, well, not as slow as further downhill).
In space, far far away from everything, and so long as you're kinda averaging a speed of 0 relative to everything else, time ticks by the fastest (relative to most everything else).
So if you build two clocks that are practically perfectly in sync, and you move one to the bottom of a mountain, and one to the top, and then bring them back together, you can measure the tiny difference between them caused by one (the one from the top of the mountain) experiencing time faster than the other (from the bottom).
This is similar to the thought experiment: if you are in a spaceship with no windows, are you accelerating, or stationary on a planet with gravity; you can’t tell.
Though if you are constantly a accelerating, with the earth stopping you, what is your instantaneous speed? The faster you go, the more time dilation.
Though if you are constantly a accelerating, with the earth stopping you, what is your instantaneous speed? The faster you go, the more time dilation.
How much slower is time at the surface of the sun?
" . . . a clock on the surface of the sun will accumulate around 66.4 fewer seconds in one year [than would a distant observer's clock]."
https://en.wikipedia.org/wiki/Gravitational_time_dilation
https://en.wikipedia.org/wiki/Gravitational_time_dilation
The frame of reference in which the mountain peak is stationary is not inertial -- due to the Earth's rotation, it's actually constantly accelerating toward the center of the Earth. Hence, the quote on special relativity you posted doesn't (directly) apply.
However, time passes slower the deeper one gets into a gravity well. The center of the Earth is the center of Earth's gravity well, so the closer to it you are, the slower your clock ticks.
More info: https://en.wikipedia.org/wiki/Gravitational_time_dilation
However, time passes slower the deeper one gets into a gravity well. The center of the Earth is the center of Earth's gravity well, so the closer to it you are, the slower your clock ticks.
More info: https://en.wikipedia.org/wiki/Gravitational_time_dilation
Thank you to all those who replied.
At the top of the article you linked there's the: "This article is about time dilation due to relative gravity. For time dilation due to relative velocity, see Relative velocity time dilation."
I didn't know there were "two forms" of time dilation, knowing what to search for helps a lot.
Thanks again.
At the top of the article you linked there's the: "This article is about time dilation due to relative gravity. For time dilation due to relative velocity, see Relative velocity time dilation."
I didn't know there were "two forms" of time dilation, knowing what to search for helps a lot.
Thanks again.
On the "centre of Earth's gravity well" bit: what is the net effect of the fact that inside a sphere, the net gravitational attraction depends on the amount, distance, and vector of the gravitational mass in question, and that whilst inside a sphere at any point beneath its surface the net gravitational attraction varies.
In the case of the Earth, it actually increases to a certain depth, then falls as one approaches the centre (or would if one could; imagine, say, the neutrino version of yourself). My first thought was that the associated time dilation would follow a similar trend, though thinking this through, I'm not certain, though I'm leaning toward your answer above as not being accurate.
In the case of the Earth, it actually increases to a certain depth, then falls as one approaches the centre (or would if one could; imagine, say, the neutrino version of yourself). My first thought was that the associated time dilation would follow a similar trend, though thinking this through, I'm not certain, though I'm leaning toward your answer above as not being accurate.
https://en.wikipedia.org/wiki/Gravitational_time_dilation
Spoilers follow:
This is a theme in Pohl's Gateway, as the protagonist shoots his friends into a black hole to give himself enough velocity to escape it's pull. Later, he realizes they're still falling in, as the immense gravity of the black hole stretches time for them.
Spoilers follow:
This is a theme in Pohl's Gateway, as the protagonist shoots his friends into a black hole to give himself enough velocity to escape it's pull. Later, he realizes they're still falling in, as the immense gravity of the black hole stretches time for them.
Isn't it true that if you were falling into a black hole, you would see the end of the universe before you crossed the event horizon? Therefore from our frame of reference nothing has ever actually fallen into a black hole? Or maybe I'm thinking of reaching the singularity at the center of the black hole.
That's not true. If you are falling into a black hole, to you time will flow normally until your sphaghettified demise. To an observer at infinity, they'll never observe you going through the event horizon because no light can reach the observer from within the event horizon and because gravitational time dilation.
But doesn't the rest of the universe speed up to such an extent that billions of years pass (our frame of reference) before you cross the event horizon? It would just pass extremely quickly (from your frame of reference).
The effect they're measuring, in this case, is the weaker gravity field at the top of a mountain, less than the field intensity at the Earth's surface. So the clock ticks faster in the weaker gravity field.
Others have already given you the gist of it.
What I find really interesting is how both types of time dilation (and most of the rest of both special and general relativity) follow from three simple observations about the world.
1. Imagine you are in a spaceship that is deep in interstellar space. Your engines are off and you feel no acceleration. There is another ship in the direction you consider to be behind your ship, moving in a constant velocity in the direction you consider to be the front of your ship. The people in the other ship also have their engines off and feel no acceleration. From their point of view, they think you are in front of them, moving backward at constant velocity.
The first observation that forms the basis of relativity is that neither you or the people in the other ship can do any physics experiment to determine which if you is "really" moving.
2. The second observations is that the speed of light is the same for all observers.
Suppose you make a clock that consists of a source the emits a very brief pulse of light, perpendicular to the direction the ship is pointing. This pulse bounces of a mirror that you have placed 0.5 light-nanosecond away from the emitter. When the pulse reflects back and reaches the emitter, the emitter emits another pulse. Every time a reflected pulse gets back to emitter counts as a tick of this clock, so it ticks 10^9 times per second.
Suppose the people on the other ship also make such a clock, and as your ships pass you synchronize your clock with theirs.
As the ships pull apart and you watch their clock and yours, you will note that because their shipping is moving, their light is not moving perpendicular to the direction the ships are pointing like yours is. Theirs has to move diagonally so it hits the mirror where the mirror will be after the light cross that 0.5 light-nanosecond gap between the emitter and mirror.
Now if this were a bouncing ball clock instead of a light clock, that would be no problem. You would see their balls as having the same perpendicular velocity as yours, plus their would have a forward velocity equal to the speed you see their ship moving, and so the diagonal velocity would combine both of those to something bigger than either (for the same reason the hypotenuse of a right triangle is longer than either side). Their ball would have farther to go than yours, but it would be going faster, so their clock would tick at the same rate as yours, and all would be as common sense requires.
But light has the same speed for all observers! So you do not see their light going faster than your light. It's going the same speed, but has farther to go because of the diagonal. So you see their clock ticking slower than your clock!
But #1 says that we can't tell who is "really" moving. That means that every clock the other ship has must slow down by the same rate their light clock slows down or they could tell their are moving because their light clock goes out of sync with their other clocks.
But when every possible way to measure time slows down at the same rate, it is hard not to say that time has slowed down.
Keep going on in this direction, reconciling the speed of light being constant for all observers and the impossibility of determining which ship is "really" moving, and you end up with pretty much all of special relativity--the time dilation, the Lorentz contraction, the Lorentz transformation, and so on.
3. The third observation is that if you turn your engines on and they provide uniform acceleration, it looks like gravity to you. If your windows are closed you can't tell if you are accelerating or if you ship is at rest on top of a planet. This holds even for non-uniform acceleration--you cannot tell if the forces you feel are due to varying acceleration or varying gravity.
Consider two people in a large cylinder rotating around its axis, so they feel a force pushing them toward the wall. Suppose they each have a clock, and the clocks are synchronized. Now suppose one of the person takes his clock, climbs a ladder up to the axis of the cylinder, hangs out there a while, and then climbs back down.
You are watching that. You see that the guy who stays on the wall is moving fast, so you see his clock moving slow. As the other guy climbs, you see his clock speeding up, because you see him slowing down as he gets nearer the axis. When he's at the axis, his clock runs the same speed as yours. Then when he climbs back "down", his clock slows down more and more, until he gets back with his buddy. Their clocks are again running at the same speed, but they are now out of sync.
From your point of view, this is all just special relativity, like we worked out with the two spaceships.
But for the two guys in the cylinder, #3 says they need to get the same result if that force toward the wall is due to gravity, not acceleration. From their point of view they are in a gravitation field pointing toward the wall, and the guy on the ladder climbed up and then back down. So we have to conclude from this that clocks run faster when you are "higher" in a gravitational field.
If you imagine now that they two guys decide to measure Pi, by measuring the circumference and radius of the cylinder, you can similarly apply special relativity from your point of view to figure out what they will measure, and find that they get a value of Pi that is too high. By #3 that means the same thing must happen in a gravitational field, and from that we (well...if we were Einstein) would be able to figure out that this means mass must warp space.
There are some books meant for the general public that go over this far better than I just did. This is getting long, so I'm going to post the book recommendations in a reply to this.
What I find really interesting is how both types of time dilation (and most of the rest of both special and general relativity) follow from three simple observations about the world.
1. Imagine you are in a spaceship that is deep in interstellar space. Your engines are off and you feel no acceleration. There is another ship in the direction you consider to be behind your ship, moving in a constant velocity in the direction you consider to be the front of your ship. The people in the other ship also have their engines off and feel no acceleration. From their point of view, they think you are in front of them, moving backward at constant velocity.
The first observation that forms the basis of relativity is that neither you or the people in the other ship can do any physics experiment to determine which if you is "really" moving.
2. The second observations is that the speed of light is the same for all observers.
Suppose you make a clock that consists of a source the emits a very brief pulse of light, perpendicular to the direction the ship is pointing. This pulse bounces of a mirror that you have placed 0.5 light-nanosecond away from the emitter. When the pulse reflects back and reaches the emitter, the emitter emits another pulse. Every time a reflected pulse gets back to emitter counts as a tick of this clock, so it ticks 10^9 times per second.
Suppose the people on the other ship also make such a clock, and as your ships pass you synchronize your clock with theirs.
As the ships pull apart and you watch their clock and yours, you will note that because their shipping is moving, their light is not moving perpendicular to the direction the ships are pointing like yours is. Theirs has to move diagonally so it hits the mirror where the mirror will be after the light cross that 0.5 light-nanosecond gap between the emitter and mirror.
Now if this were a bouncing ball clock instead of a light clock, that would be no problem. You would see their balls as having the same perpendicular velocity as yours, plus their would have a forward velocity equal to the speed you see their ship moving, and so the diagonal velocity would combine both of those to something bigger than either (for the same reason the hypotenuse of a right triangle is longer than either side). Their ball would have farther to go than yours, but it would be going faster, so their clock would tick at the same rate as yours, and all would be as common sense requires.
But light has the same speed for all observers! So you do not see their light going faster than your light. It's going the same speed, but has farther to go because of the diagonal. So you see their clock ticking slower than your clock!
But #1 says that we can't tell who is "really" moving. That means that every clock the other ship has must slow down by the same rate their light clock slows down or they could tell their are moving because their light clock goes out of sync with their other clocks.
But when every possible way to measure time slows down at the same rate, it is hard not to say that time has slowed down.
Keep going on in this direction, reconciling the speed of light being constant for all observers and the impossibility of determining which ship is "really" moving, and you end up with pretty much all of special relativity--the time dilation, the Lorentz contraction, the Lorentz transformation, and so on.
3. The third observation is that if you turn your engines on and they provide uniform acceleration, it looks like gravity to you. If your windows are closed you can't tell if you are accelerating or if you ship is at rest on top of a planet. This holds even for non-uniform acceleration--you cannot tell if the forces you feel are due to varying acceleration or varying gravity.
Consider two people in a large cylinder rotating around its axis, so they feel a force pushing them toward the wall. Suppose they each have a clock, and the clocks are synchronized. Now suppose one of the person takes his clock, climbs a ladder up to the axis of the cylinder, hangs out there a while, and then climbs back down.
You are watching that. You see that the guy who stays on the wall is moving fast, so you see his clock moving slow. As the other guy climbs, you see his clock speeding up, because you see him slowing down as he gets nearer the axis. When he's at the axis, his clock runs the same speed as yours. Then when he climbs back "down", his clock slows down more and more, until he gets back with his buddy. Their clocks are again running at the same speed, but they are now out of sync.
From your point of view, this is all just special relativity, like we worked out with the two spaceships.
But for the two guys in the cylinder, #3 says they need to get the same result if that force toward the wall is due to gravity, not acceleration. From their point of view they are in a gravitation field pointing toward the wall, and the guy on the ladder climbed up and then back down. So we have to conclude from this that clocks run faster when you are "higher" in a gravitational field.
If you imagine now that they two guys decide to measure Pi, by measuring the circumference and radius of the cylinder, you can similarly apply special relativity from your point of view to figure out what they will measure, and find that they get a value of Pi that is too high. By #3 that means the same thing must happen in a gravitational field, and from that we (well...if we were Einstein) would be able to figure out that this means mass must warp space.
There are some books meant for the general public that go over this far better than I just did. This is getting long, so I'm going to post the book recommendations in a reply to this.
The first book is one Einstein himself wrote to explain relativity to a general audience. There are some good versions of this on Project Gutenberg. Links below.
Here's how Einstein described this book:
> The present book is intended, as far as possible, to give an exact insight into the theory of Relativity to those readers who, from a general scientific and philosophical point of view, are interested in the theory, but who are not conversant with the mathematical apparatus of theoretical physics. The work presumes a standard of education corresponding to that of a university matriculation examination , and, despite the shortness of the book, a fair amount of patience and force of will on the part of the reader. The author has spared himself no pains in his endeavour to present the main ideas in the simplest and most intelligible form, and on the whole, in the sequence and connection in which they actually originated. In the interest of clearness, it appeared to me inevitable that I should repeat myself frequently, without paying the slightest attention to the elegance of the presentation. I adhered scrupulously to the precept of that brilliant theoretical physicist L. Boltzmann, according to whom matters of elegance ought to be left to the tailor and to the cobbler. I make no pretence of having withheld from the reader difficulties which are inherent to the subject. On the other hand, I have purposely treated the empirical physical foundations of the theory in a "step-motherly" fashion, so that readers unfamiliar with physics may not feel like the wanderer who was unable to see the forest for the trees. May the book bring some one a few happy hours of suggestive thought!
The book is called "Relativity: The Special and the General Theory".
Here is a copy of the 3rd edition in PDF and TeX made via OCR of the physical book [1].
Here is a copy that is available in HTML, MS Word, and TeX. I'm not sure what edition this is [2].
There's also a Kindle version of the 3rd edition on Amazon for $0.99 that is good. Books with math are often terrible on Kindle due to publishers sometimes doing the equations as small image files that are hard to read and ugly if you zoom them. This one, though, is specifically touted as being "with readable equations", and they are right.
Unless you actually want to read on a Kindle there is no advantage that I can see that it has over either of the Gutenberg copies I listed above. If you do want to read on a Kindle and are willing to cough up $0.99, here is the link [3].
Another book that goes over special and general relativity, in a way similar to what I gave in the prior comment (much of mine was ripped off from this, with my contribution just the probably introduction of errors) is Brian Greene's "The Elegant Universe" [4]. I grabbed this when it was free on Amazon Prime Reading a few months ago, and just recently started it. I'm only about 15% of the way through, but it has been quite good so far. The relativity material is only in the first couple of chapters, though, so if you aren't interested in the rest of the material it would probably not be worth it.
[1] http://www.gutenberg.org/ebooks/36114
[2] http://www.gutenberg.org/ebooks/5001
[3] https://www.amazon.com/gp/product/B004M8S53U
[4] https://www.amazon.com/Elegant-Universe-Superstrings-Dimensi...
Here's how Einstein described this book:
> The present book is intended, as far as possible, to give an exact insight into the theory of Relativity to those readers who, from a general scientific and philosophical point of view, are interested in the theory, but who are not conversant with the mathematical apparatus of theoretical physics. The work presumes a standard of education corresponding to that of a university matriculation examination , and, despite the shortness of the book, a fair amount of patience and force of will on the part of the reader. The author has spared himself no pains in his endeavour to present the main ideas in the simplest and most intelligible form, and on the whole, in the sequence and connection in which they actually originated. In the interest of clearness, it appeared to me inevitable that I should repeat myself frequently, without paying the slightest attention to the elegance of the presentation. I adhered scrupulously to the precept of that brilliant theoretical physicist L. Boltzmann, according to whom matters of elegance ought to be left to the tailor and to the cobbler. I make no pretence of having withheld from the reader difficulties which are inherent to the subject. On the other hand, I have purposely treated the empirical physical foundations of the theory in a "step-motherly" fashion, so that readers unfamiliar with physics may not feel like the wanderer who was unable to see the forest for the trees. May the book bring some one a few happy hours of suggestive thought!
The book is called "Relativity: The Special and the General Theory".
Here is a copy of the 3rd edition in PDF and TeX made via OCR of the physical book [1].
Here is a copy that is available in HTML, MS Word, and TeX. I'm not sure what edition this is [2].
There's also a Kindle version of the 3rd edition on Amazon for $0.99 that is good. Books with math are often terrible on Kindle due to publishers sometimes doing the equations as small image files that are hard to read and ugly if you zoom them. This one, though, is specifically touted as being "with readable equations", and they are right.
Unless you actually want to read on a Kindle there is no advantage that I can see that it has over either of the Gutenberg copies I listed above. If you do want to read on a Kindle and are willing to cough up $0.99, here is the link [3].
Another book that goes over special and general relativity, in a way similar to what I gave in the prior comment (much of mine was ripped off from this, with my contribution just the probably introduction of errors) is Brian Greene's "The Elegant Universe" [4]. I grabbed this when it was free on Amazon Prime Reading a few months ago, and just recently started it. I'm only about 15% of the way through, but it has been quite good so far. The relativity material is only in the first couple of chapters, though, so if you aren't interested in the rest of the material it would probably not be worth it.
[1] http://www.gutenberg.org/ebooks/36114
[2] http://www.gutenberg.org/ebooks/5001
[3] https://www.amazon.com/gp/product/B004M8S53U
[4] https://www.amazon.com/Elegant-Universe-Superstrings-Dimensi...
A similar experiment (but to only detect the effect of altitude, not to measure it) was done about 12 years ago, described here: http://www.leapsecond.com/great2005/
I wouldn't think this would be exceptional as a way to measure altitude.
However it is exceptional as a way to measure gravity differences arising from stuff inside of the Earth. See https://academo.org/demos/gravity-map/ for a picture of how that varies over the Earth.
(Though you wouldn't be measuring the strength of gravitational pull directly. You would be measuring the gravitational potential. The strength of gravity is the gradient of the potential field.)
However it is exceptional as a way to measure gravity differences arising from stuff inside of the Earth. See https://academo.org/demos/gravity-map/ for a picture of how that varies over the Earth.
(Though you wouldn't be measuring the strength of gravitational pull directly. You would be measuring the gravitational potential. The strength of gravity is the gradient of the potential field.)
Why is the Indonesia area so red on that map?
I get that other red spots have tall mountains... but why Indonesia?
I get that other red spots have tall mountains... but why Indonesia?
Continental crust is less dense than marine crust. So just
because the Himalayas are tall, they also displace denser
mantle below, so they don’t contribute much more to
gravitational pull.
The Andes and Indonesia are strong subduction zones. Besides
the crust on the surface, there’s another layer of (dense
oceanic) crust below adding to the pull.
-- Darren #44 comment at http://blogs.discovermagazine.com/badastronomy/2011/03/31/the-earths-lumpy-gravity/
Harabaglia, Paolo & Doglioni, Carlo. (1998). Topography and
gravity across subduction zones. Geophysical Research
Letters - GEOPHYS RES LETT. 25. 703-706. 10.1029/98GL00137.How the hell did they measure clock speed if the passage of time itself is different?
"Dear tahw, what time and date is it over there? Please write back. I shall write again next month. Sincerely, phyzome"
...
"Dear tahw, over the past year, I have noticed something interesting in our correspondences -- more hours have passed on your end than mine, on average. I believe our clocks are ticking at different rate. Yours, phyzome"
(except measuring number of ticks, as communicated over fiber optics, and possibly with other tricks to cancel out communication lag)
...
"Dear tahw, over the past year, I have noticed something interesting in our correspondences -- more hours have passed on your end than mine, on average. I believe our clocks are ticking at different rate. Yours, phyzome"
(except measuring number of ticks, as communicated over fiber optics, and possibly with other tricks to cancel out communication lag)
Relative to a twin clock at sea level.
"It didn't work as nicely as we hoped, but we learned a lot and it's a start," Lisdat said. "Sometimes you just have to begin, and then you can figure out how to improve."
Good sentiment to live/work by.
Good sentiment to live/work by.
> Good sentiment to live/work by.
Definitely. I've found in my personal life that just getting started on something can be the hardest part about any given task (whether something physical, or implementing an idea as code or something like that). But once you get rolling on it, things tend to come along easier.
You might not reach the end, things might fail (sometimes spectacularly), or things will change midstream and you might not end up where you had planned. Regardless, though, the journey can end up being better than the original goal.
...you just have to begin.
Definitely. I've found in my personal life that just getting started on something can be the hardest part about any given task (whether something physical, or implementing an idea as code or something like that). But once you get rolling on it, things tend to come along easier.
You might not reach the end, things might fail (sometimes spectacularly), or things will change midstream and you might not end up where you had planned. Regardless, though, the journey can end up being better than the original goal.
...you just have to begin.
Is it possible that they measured the distance not height to the reference clock? It takes time for the signal to travel to the reference clock. But I’m not sure about the exact geometry.