In the last article of the series, I narrated the story of how the ancients managed to amass some crucial info about the nature and geometry of light. We have also seen how thinkers like Empedocles and Aryabhata I shared their intellectual revelations with the world, which helped us understand that the same laws of terrestrial geometry would also be applicable to a system of beams (or rays) of light. However, as we have also seen, the debate regarding the speed of light continued. Most believed that light is an all-pervading presence, and not any finite element (to use the term in a broader sense). However, there were others who challenged this belief.
This simple question, asked so innocently, had to be answered in a larger-than-life and profound way. This was because, unlike sound which only travels at around 331 m/s in air at STP, the speed of light is unchanging across all media, and it is also the speed-limit of the universe.
The debate continues
However, during the renaissance, this started to change. Great thinkers and scientists like Kepler and Descartes believed that the speed of light is infinite, while others such as Galileo challenged this notion. Galileo is known to have conducted a series to experiments to find out the exact value of the speed of light.
In one such experiments, Galileo travelled to a ground at night, with a covered lamp. His companion remained on the other side of the vast field. Now, the arrangement was that, as Galileo removes the cover (thus exposing the lamp), light beams will travel in all directions. As soon as his companion sees the light of the first lamp, he would repeat the process and expose his own covered lamp. Galileo would get to see the light beams coming from the second lamp.
The time interval between Galileo exposing the first lamp, and him seeing the light from the second lamp, would actually be noted as precisely and accurately as possible. Also, the shortest straight-line distance between Galileo and his companion was recorded. Say, this is d.
Now, Galileo’s premise was simple: light would take time t to reach his companion from his position, and another time t to reach him from his companion’s position. This would amount to a total time of t + t = 2t. Now, using the very definition of speed (let us stick to the scalar quantity here, to avoid confusion), speed = distance / time.
Naturally, the speed of light S = 2 x ( d / 2t )
The time interval between Galileo exposing the first lamp, and him seeing the light from the second lamp, would actually be noted as precisely and accurately as possible. Also, the shortest straight-line distance between Galileo and his companion was recorded. Say, this is d.
Now, Galileo’s premise was simple: light would take time t to reach his companion from his position, and another time t to reach him from his companion’s position. This would amount to a total time of t + t = 2t. Now, using the very definition of speed (let us stick to the scalar quantity here, to avoid confusion), speed = distance / time.
Naturally, the speed of light S = 2 x ( d / 2t )
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| Galileo's simple but elegant experiment |
Longitudes and Latitudes: Geography in the service of physics
Such an event occured back in the 17th century. Back in the golden era of European explorers, it was crucial for captains and navigators to correctly estimate the positions of their ships. After all, there was no GPS back then. They had to rely on astronomical phenomena, trigonometry and an accurate clock for that.
As any person with a basic knowledge of geography would know, to estimate our location on earth, we would need the knowledge of two parameters, the latitude and the longitude. Latitude could easily be estimated when you were on the northern hemisphere, since the angle made by the north star (Polaris) with the horizontal on earth (connecting the place and the north pole) is the value of the latitude. In the southern hemisphere, things became a little bit more difficult, since there’s no star directly overhead the south pole, but using the polaris itself, that is possible with some extra calculations.
However, the real problem came with the longitudes. We know that the earth rotates 360 degrees to complete one term of rotation, and this causes the phenomena of day and night. Now, in this rotation takes place exactly in 24 hours, as we all know (actually that has been standardized long ago).
So, 24 hours = 24 x 60 minutes,
As any person with a basic knowledge of geography would know, to estimate our location on earth, we would need the knowledge of two parameters, the latitude and the longitude. Latitude could easily be estimated when you were on the northern hemisphere, since the angle made by the north star (Polaris) with the horizontal on earth (connecting the place and the north pole) is the value of the latitude. In the southern hemisphere, things became a little bit more difficult, since there’s no star directly overhead the south pole, but using the polaris itself, that is possible with some extra calculations.
However, the real problem came with the longitudes. We know that the earth rotates 360 degrees to complete one term of rotation, and this causes the phenomena of day and night. Now, in this rotation takes place exactly in 24 hours, as we all know (actually that has been standardized long ago).
So, 24 hours = 24 x 60 minutes,
Or, 360 degrees of rotation => 24 x 60 minutes,
or, 1 degree of rotation => 4 minutes.
So, 1 degree of rotation (i.e 1 degree of longitude change) would contribute to a time difference of 4 minutes. The Greenwich Mean Time (GMT) is considered to be the standard time on earth, and so it was during that period. So, since the earth rotates in the direction west-east, any place east of the GMT will have a GMT + X value of time, where X stands for the time difference contributed by the longitudinal difference. Similarly, west of the GMT we would have GMT - X.
Now, the theorists of the time realized that 12:00 PM at any place on earth is the time when the sun is directly overhead, at the maximum apparent height (peak of the sky, commonly called). So, the procedure to estimate longitude correctly would be:
1. First, keep a clock synchronized with the GMT, so that it says the time of the GMT region at any point of the day.
2. Now, in your position, look out for when the sun is directly overhead. It is definitely 12:00 PM at the place then.
3. Find out what’s the time on the GMT-synced clock at this moment. Say it is Y. So, the longitude of the place is ( Y - 12:00 ) / 4, with ( Y - 12:00 PM ) expressed in minutes.
So, for example if you are at the centre of India, and you’ll find that when the clock strikes 12:00 PM here, it is 6:30 PM on the GMT-synced clock. So, the longitude of the centre of India would be:
or, 1 degree of rotation => 4 minutes.
So, 1 degree of rotation (i.e 1 degree of longitude change) would contribute to a time difference of 4 minutes. The Greenwich Mean Time (GMT) is considered to be the standard time on earth, and so it was during that period. So, since the earth rotates in the direction west-east, any place east of the GMT will have a GMT + X value of time, where X stands for the time difference contributed by the longitudinal difference. Similarly, west of the GMT we would have GMT - X.
Now, the theorists of the time realized that 12:00 PM at any place on earth is the time when the sun is directly overhead, at the maximum apparent height (peak of the sky, commonly called). So, the procedure to estimate longitude correctly would be:
1. First, keep a clock synchronized with the GMT, so that it says the time of the GMT region at any point of the day.
2. Now, in your position, look out for when the sun is directly overhead. It is definitely 12:00 PM at the place then.
3. Find out what’s the time on the GMT-synced clock at this moment. Say it is Y. So, the longitude of the place is ( Y - 12:00 ) / 4, with ( Y - 12:00 PM ) expressed in minutes.
So, for example if you are at the centre of India, and you’ll find that when the clock strikes 12:00 PM here, it is 6:30 PM on the GMT-synced clock. So, the longitude of the centre of India would be:
( 12:00 - 6:30 ) / 4
= 5:30 / 4
= 5.5 x 60 / 4
= 5.5 x 15
= +82.5 degrees.
The simple beauty of this process continues to dazzle me even today. Because in science, simplicity = elegance.
= 5:30 / 4
= 5.5 x 60 / 4
= 5.5 x 15
= +82.5 degrees.
The simple beauty of this process continues to dazzle me even today. Because in science, simplicity = elegance.
Devising an accurate clock
.jpg) |
| Ole Romer |
This was when Galileo and Romer entered. Galileo wanted to tackle the problem using what he liked most, physics. After discovering the moons of Jupiter, Galileo observed that in the Jovian system, the moon Io orbits Jupiter once in every 1.769 days. Now, those of you are familiar with the concept of Light Clock, the simplest clock in existence, should know that any periodic phenomenon can be turned into a clock. As such, the revolution of Io, the innermost moon of Jupiter, would serve (theoretically) as a clock for keeping the proper time. Synchronizing this data with the results of the GMT-synced clock at regularly, after your ship has departed from the London docks, would be a cakewalk.
However, this wasn’t a very practical solution since it would be difficult for humans to observe the eclipse and reemergence of Io from the deck of a moving ship. However, it opened the doors to a profound new set of ideas.
In 1671, Ole Romer and Giovanni Cassini observed and recorded a number of Io’s eclipses and reemergences. However, Romer was thrilled to see that the Jovian system is far from being perfectly accurate. Instead, during some times in the year, the eclipse of Io would be upto 22 minutes before/after the predicted time of the eclipse. This definitely, was initially regarded as a blow to the entire method. But Romer came up with a better explanation, that would later enable Christian Huygens, another great scientist, to finally give us an estimate of the value of the speed of light. [Ref. 1]
Romer and Huygens
Romer’s theory was simple, and it rested on a very basic premise: light takes time to reach earth from Jupiter. Now, the distance between Jupiter and Earth (more precisely, Io and earth) is not a fixed value. As Jupiter and the earth, both revolve around the sun, at time the earth comes directly in the middle of Jupiter and the Sun, forming a straight line. This is when the distance betwee Jupiter and the earth is shortest. Likewise, there are times when the Sun comes in the middle, forming a straight light between Jupiter and the earth. This is when the distance is the longest.
Now, let us say that the distance of the earth from the sun is SE, and that in the case of Jupiter is SJ.
So, the shortest distance would be JE (min) = SJ - SE,
And the longest distance would be JE (max) - SJ + SE,
Now, let us assume that light has a finite (but large) speed, v. So, the time taken by light (and the news of Io’s eclipse) to reach the earth would be t (max) = ( SJ + SE ) / v seconds when earth and jupiter are farthest apart.
Likewise, when earth and jupiter are closest, this time taken would be
t (min) = ( SJ - SE ) / v seconds.
Since Romer’s observations and trials-and-errors showed us that the maximum time-error would be 22 minutes, we can say:
t (max) - t (min) = 22 minutes = 22 x 60 s = 1320,
or, ( SJ + SE ) / v - ( SJ - SE ) / v = 1320 s,
or, ( SJ + SE - SJ + SE ) / v = 1320,
or, 2SE / v = 1320 s,
or, v = 2SE / 1320 km/s
Now, 2SE = 2 x radius of the earth’s orbit = diameter of the earth’s orbit. Since we have the data, 14,96,00,000 km is the radius of the earth’s orbit, we can easily find out the speed of light using Romer’s findings, which Christian Huygens did.
So, v = 2 x 149600000 / 1320
= 226666 km/s (appx)
Obviously, this is not very far from the modern value of the speed of light, which is 299792.458 km/s.
Suddenly, it changed everything. Light now was known to have a limited and finite speed.
For centuries, the Newtonian mechanics had ruled the heartland of the scientific realm. To quote from one of the History Channel’s documentaries, “His laws explained everything, and bound the earth to the heavens”. Using his sophisticated realizations that he extracted out of his mathematical findings, he endowed the scientific world with what can be called gems of science.
Now, let us say that the distance of the earth from the sun is SE, and that in the case of Jupiter is SJ.
So, the shortest distance would be JE (min) = SJ - SE,
And the longest distance would be JE (max) - SJ + SE,
Now, let us assume that light has a finite (but large) speed, v. So, the time taken by light (and the news of Io’s eclipse) to reach the earth would be t (max) = ( SJ + SE ) / v seconds when earth and jupiter are farthest apart.
Likewise, when earth and jupiter are closest, this time taken would be
t (min) = ( SJ - SE ) / v seconds.
 |
| In Romer and Huygen's predictions, the earth-sun distance alone is required to estimate the speed of light |
t (max) - t (min) = 22 minutes = 22 x 60 s = 1320,
or, ( SJ + SE ) / v - ( SJ - SE ) / v = 1320 s,
or, ( SJ + SE - SJ + SE ) / v = 1320,
or, 2SE / v = 1320 s,
or, v = 2SE / 1320 km/s
Now, 2SE = 2 x radius of the earth’s orbit = diameter of the earth’s orbit. Since we have the data, 14,96,00,000 km is the radius of the earth’s orbit, we can easily find out the speed of light using Romer’s findings, which Christian Huygens did.
So, v = 2 x 149600000 / 1320
= 226666 km/s (appx)
Obviously, this is not very far from the modern value of the speed of light, which is 299792.458 km/s.
Suddenly, it changed everything. Light now was known to have a limited and finite speed.
Newton vs Einstein: The advent of relativistic mechanics
For more than 2 centuries, Newtonian mechanics had reigned supreme. However, during the middle of the 19th century, some problems had arisen. The first was that, the calculations according to Newtonian mechanics could not accurately account for the findings in the case of Mercury. Also the variations were tiny, it intrigued the astrophysicists. Secondly, Maxwell’s elegant, single framework of Electromagnetic theory was incompatible with Newtonian mechanics. [Ref. 2]
Maxwell’s theory indicated that no one could ever catch a beam of light, no matter how fast he ran. However, according to Newtonian mechanics, since light has a finite speed, one should be able to catch a beam of light if he runs equally fast. Hypothetically speaking, everything would appear still then. This is not the case in Maxwell’s theory. No matter how fast you run, that’s not going to happen. This is where Einstein entered, with a theory that would change the course of physics forever.
Einstein, using his professor Hermann Minkowski’s 4-D coordinate system (with axes x , y , z and t) devised a unified framework of spacetime (space and time brought together). In Einstein’s wonderful theory, time itself is relative, unlike the Newtonian model where time is universal. In fact, if we apply Newtonian mechanics celestially, we’d see that in the case of Mercury and all other planets:
mv^2 / r = GmM / r^2,
or, v = sqrt ( GM / r ),
where v = Mercury’s velocity,
G = universal constant of gravitation,
m = Mass of Mercury,
M = Mass of the sun,
r = distance between the Sun and Mercury.
Naturally, it happens to give us an estimate of Mercury’s speed. Putting this information together with our knowledge of Mercury’s distance from the sun (r), we can predict the time taken by Mercury to complete one complete revolution, which is t = 2 x pi x r / v (appx, because more precisely speaking, planets travel in elliptical paths).
However, no matter how precise these calculations were performed, the errors prevailed. This was because, as later explained by Einstein, Newton considered the effect of the sun’s gravity (and any gravitational influence, in general) to be instantaneous. Thus, the effect of any influence would proceed at an infinite speed. This was not the case in Einstein’s theory, since it predicted that any influence can proceed at the maximum speed of light, and never beat it.
The second point, as later explained by Einstein’s General Theory of Relativity, was that, the mass of a body isn’t constant when it’s moving. The rest mass of the body chances, relative to its velocity. This is given by Einstein’s famous equation:
m(v) = m(0) / sqrt ( 1 - v^2 / c^2 ),
where m(v) => mass of the body at velocity v,
m(0) = rest mass of the body,
v = velocity of the body,
c = speed of light (constant).
So Mercury, moving at an extreme speed around the sun (and with a small rest-mass), would experience a different mass when moving around the sun at that speed. This was one of the main causes of the problem.
Einstein, by establishing his glorious theory of gravity, had replaced the Newtonian mechanics, and limited to latter only to the earthly domain. Even today, we are using the Newtonian mechanics for terrestrial calculations, because it is actually an approximate version of the larger Relativistic Mechanics that is applied celestially. Even in the spaceship flights today, for reaching the moon and beyond, Newtonian mechanics can be safely and securely used.
However, Einstein’s theory established some fundamental facts about our universe. One such fact is that, nothing ever can beat the speed of light. In fact, light is the fastest moving wave in the entire universe.
Using the same equation as above, we see that if v = c, i.e if a body is to be moving at the speed of light, then:
m(v) = m(0) / sqrt ( 1 - v^2 / c^2 )
= m(0) /sqrt ( 1 - 1 )
= m(0) / 0
= Infinite!
And lets not talk about anything that traverses beyond the speed of light, because the mass of such as a body would have to be expressed using i, the imaginary or complex number. [Ref. 3]
As per further calculations, anything can only travel at the speed of light, it and only if it was a rest-mass of 0, i.e m(0)=0. Photons, the bundles of energy that serve as the corpuscles of light, have 0 rest-mass, and so they travel at the speed of light. For anything else, we’d need to provide an infinite amount of energy to accelerate it to the speed of light.
Well, I believe this concludes our lesson for today. In the next article of the series, I’d try to paint a picture of the nature of light, its properties.
Maxwell’s theory indicated that no one could ever catch a beam of light, no matter how fast he ran. However, according to Newtonian mechanics, since light has a finite speed, one should be able to catch a beam of light if he runs equally fast. Hypothetically speaking, everything would appear still then. This is not the case in Maxwell’s theory. No matter how fast you run, that’s not going to happen. This is where Einstein entered, with a theory that would change the course of physics forever.
Einstein, using his professor Hermann Minkowski’s 4-D coordinate system (with axes x , y , z and t) devised a unified framework of spacetime (space and time brought together). In Einstein’s wonderful theory, time itself is relative, unlike the Newtonian model where time is universal. In fact, if we apply Newtonian mechanics celestially, we’d see that in the case of Mercury and all other planets:
mv^2 / r = GmM / r^2,
or, v = sqrt ( GM / r ),
where v = Mercury’s velocity,
G = universal constant of gravitation,
m = Mass of Mercury,
M = Mass of the sun,
r = distance between the Sun and Mercury.
Naturally, it happens to give us an estimate of Mercury’s speed. Putting this information together with our knowledge of Mercury’s distance from the sun (r), we can predict the time taken by Mercury to complete one complete revolution, which is t = 2 x pi x r / v (appx, because more precisely speaking, planets travel in elliptical paths).
However, no matter how precise these calculations were performed, the errors prevailed. This was because, as later explained by Einstein, Newton considered the effect of the sun’s gravity (and any gravitational influence, in general) to be instantaneous. Thus, the effect of any influence would proceed at an infinite speed. This was not the case in Einstein’s theory, since it predicted that any influence can proceed at the maximum speed of light, and never beat it.
The second point, as later explained by Einstein’s General Theory of Relativity, was that, the mass of a body isn’t constant when it’s moving. The rest mass of the body chances, relative to its velocity. This is given by Einstein’s famous equation:
m(v) = m(0) / sqrt ( 1 - v^2 / c^2 ),
where m(v) => mass of the body at velocity v,
m(0) = rest mass of the body,
v = velocity of the body,
c = speed of light (constant).
So Mercury, moving at an extreme speed around the sun (and with a small rest-mass), would experience a different mass when moving around the sun at that speed. This was one of the main causes of the problem.
Einstein’s predictions on light: the profound speed-limit
Einstein, by establishing his glorious theory of gravity, had replaced the Newtonian mechanics, and limited to latter only to the earthly domain. Even today, we are using the Newtonian mechanics for terrestrial calculations, because it is actually an approximate version of the larger Relativistic Mechanics that is applied celestially. Even in the spaceship flights today, for reaching the moon and beyond, Newtonian mechanics can be safely and securely used.
However, Einstein’s theory established some fundamental facts about our universe. One such fact is that, nothing ever can beat the speed of light. In fact, light is the fastest moving wave in the entire universe.
Using the same equation as above, we see that if v = c, i.e if a body is to be moving at the speed of light, then:
m(v) = m(0) / sqrt ( 1 - v^2 / c^2 )
= m(0) /sqrt ( 1 - 1 )
= m(0) / 0
= Infinite!
And lets not talk about anything that traverses beyond the speed of light, because the mass of such as a body would have to be expressed using i, the imaginary or complex number. [Ref. 3]
As per further calculations, anything can only travel at the speed of light, it and only if it was a rest-mass of 0, i.e m(0)=0. Photons, the bundles of energy that serve as the corpuscles of light, have 0 rest-mass, and so they travel at the speed of light. For anything else, we’d need to provide an infinite amount of energy to accelerate it to the speed of light.
Well, I believe this concludes our lesson for today. In the next article of the series, I’d try to paint a picture of the nature of light, its properties.
References
[1]: Wonders of the Universe, Brian Cox
[2]: http://physics.ucr.edu/~wudka/Physics7/Notes_www/node98.html
[3]: Further reading: http://www.phys.unsw.edu.au/einsteinlight/jw/module4_time_dilation.htm
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