Minkowski on mathematician and physicist

十二月 19, 2012

Minkowski concluded this section with a remark that
clarifies his understanding of the basic motivations behind Einstein’s contribution to the
latest developments in electrodynamics: mathematicians—Minkowski said—accustomed
as they are to discuss many-dimensional manifolds and non-Euclidean geometries, will
have no serious difficulties in adapting their concept of time to the new one, implied by
the application of the Lorentz transformation; on the other hand, the task of making physical
sense out of the essence of these transformations had been addressed by Einstein in the
introduction to his 1905 relativity article.26


Time is Relative

十月 7, 2012


I recently had an argument about why time is relative. My friend found it hard to conceive how time could not be absolute. At that time, when I tried to explain it, I found that it was hard for me see WHY time really is relative. All I could do was state experimental results that indicate it is the case. This shows that thinking you can understand a deep concept at a very shallow, philosophical level doesn’t really mean much, but at least it’s better than no understanding at all.

Newton’s laws put an end to the idea of absolute motion. Two people playing ping pong on a moving train would measure the distance the ball bounces on the table (between consequecutive bounce) as lesser than a person standing along the track watching the ball bounce. Both measurements are equally valid since there is no absolute standard of rest.

Maxwell’s equations predicted that light waves travel at a fixed speed. This is something we have to believe, and it arises out of the fact that his equations predicted “there would be wavelike disturbances in the combined electromagnetic field, and that these would travel at a fixed speed, like ripples on a pond.”

But Newton’s theory got rid of the idea of absolute rest and if we believe that there’s no ambiguity about time, then if light is supposed to travel at a fixed speed, we need say what the fixed speed is measured relative to. It was suggested that there was a substance called “ether” that was present everywhere (even in “empty space”) and that light waves travelled through this ether at a fixed speed. However, different observers, moving relative to the ether, would see light coming at them at different speeds.

The way to verify this is to measure the speed of light as the earth moves through the ether on its orbit around the sun. So, light speed measured in the direction of earth’s orbit (when we are moving towards the source of light) should be higher than the speed of light measured at right angles from the direction of earth’s motion (when we are not moving towards the source of light). This experiment, called the Michaelson-Morley experiment, was performed in 1887, and to their great surprise, it was shown that the speed of light was exactly the same in both cases!

Einstein then pointed out that the whole idea of ether was unnecessary as long as one was willing to abandon the idea of absolute time. He postulated that the laws of science should be the same for all freely moving observers regardless of the speed. This was true for Newton’s laws of motion, but it also now applied to Maxwell’s theory and the speed of light: all observers should measure the same speed of light, no matter how fast they are moving.

One of the remarkable consequences of this theory is that how it has changed our thinking regarding space and time. In Newton’s theory, if a pulse of light is sent from one place to another, different observers would agree on the time the journey took (since time is absolute) but will not agree on how far the light travelled (since space is not absolute). Since the speed of light is measured as distance travelled/time taken, different observers would measure different speeds for the speed of light. But this clearly violates what Maxwell’s equations say and what has been shown by experiment. In relativity, all observers MUST agree on how fast light travels, but they don’t have to agree regarding the distance and consequently, they must disagree over the time it has taken! (The time taken is the distance (which they don’t agree on) divided by the speed of light (which they do agree on).) That is, observers must have their own measure of time, as recorded by a clock carried by them and even identical clocks carried by different observers would not necessarily agree.

From all this we can make the statement that time MUST be relative, because if it weren’t, it would contradict experimental results. But this doesn’t, at least to me, shed light on WHY this is the case. This might be a question more metaphysical than physical (i.e., time is relative just as Newton’s laws are the way they are). I shall attempt to address my question of “WHY” in greater detail now. (Though ultimately I don’t have a good answer I think—anyone?)

This theory (special relativity) and the more general form (which takes gravitional effects into account) of it makes several other important predictions which have withstood empirical observations. But one of the prediction of general relativity is time should appear to run slower near a massive body like the earth. This is because the energy of light is proportional its frequency (waves/second). As light travels upward in earth’s gravitional field, it loses energy, and therefore its frequency goes down. (The length of time between one wave crest and the next goes up.) To someone high up, it would appear that that everything down below was taking longer to happen. Apparently, this prediction can be tested by using a pair of very accurate clocks, one at the bottom nearer to the earth, and the other at the top. This was done in 1962, and the clock at the bottom was found to run slower, in exact agreement with general relativity! This is of practical importance, if you wish to trust navigational signals from satellites. If you ignore relativistic effects, you could be off by several miles!

Now, my question is, what is the basis for this phenomenon? Why should the clocks have different times? Why should it be related to light and its frequency? This is saying that time slows down when it is exposed to intense gravity. Why is this the case?

Similarly consider the twins paradox where one twin flies off into space at the speed of light and comes back and hasn’t aged much more than the twin who is on earth. It’s not a paradox if you have relative time, but what is the physiological basis (do the cells actually undergo cell division slower?) for this phenomenon?

We must accept that time isn’t independent of space and it forms a 4-dimensional construct called space-time. General relativity shows that gravity is really not a force, but rather a consequence of the fact that space-time is curved due to the distribution of mass and energy. Bodies like the earth don’t follow a curved orbit due to a force called gravity, but they follow the nearest thing to a straight path in curved space, called a geodesic. On the surface of the earth, a geodesic is a circle (the equator, for example). In general relativity, bodies always follow straight lines in four-dimensional space-time, but they appear to us to move along curved paths in three-dimensional space. (Hawking says this is like watching an airplane over an hilly ground. Although it follows a straight line in 3D space, its shadow follows a curved path on the ground.)

I imagine this as a huge rubber sheet with bodies like the sun, the earth, etc. are laying on the sheet causing a depression (a “gravity well”). An object travelling through this sheet might just go past the end of a well, causing it follow a curved path as thought it suffered a gravitional pull. An object might also just go inside the well and follow an elliptical path around the walls of the well. An object may finally, due to friction, decay and eventually fall in the greater object at the bottom of the well.

A postulate of general relativity, which follows from logical deductions, is that if two events are close together, there is an interval between them which can be calculated by some function of their coordinates. We know, according to the mathematics of the theory, that if we choose a region of space-time where the gravitation is the same throughout the region, that we can obtain very nearly a Euclidean space. We have a second postulate which states that a body travels in a geodesic in space-time unless non-gravitional forces act on it. The third postulate says that light travels on a geodesic such that the interval between any parts of it is zero.

In the general theory, It is only neighbouring events that have a definite interval and this is INDEPENDENT of the route pursued. The interval between distant events depends on the route pursued, and it can be calculated by dividing up the route into small enough parts (each part has constant gravitional effects and thus we can calculate the interval between the two neighbouring events), and adding up the intervals for all parts. If the interval is spacelike, a body cannot travel from one event to the other. Therefore, the interval has to be timelike. The interval between neighbouring events when it is timelike is the time between them for observers who travel from one event to the other. And so the TOTAL interval between two events will be judged by people who travel from one of the other by what their clocks show to to be the time they have taken on the journey. The slower they travel, the longer they will think they have been on the journey. This is not platitude—if you travel from DC to NY and leave at 6a and arrive at 10a, the more slowly you travel, the longer you will take according to your watch. So if you travel at the speed of light (third postulate), going all across the solar system before reaching your destination, your watch would say that you had taken no time at all! If you had gone by any circuitous route, which enabled you arrive in time by travelling fast, the longer your route the less time you will take. The diminision of time is continual as you approach the speed of light.

A body when left to itself travels so that the time, measured by its clocks, is the longest. If it had travelled by any other route from one event to another, the time would be shorter. This is saying that bodies left to themselves make their journeys as slowly as they can. Russell refers to this as a law of cosmic laziness. Mathematically, they travel in geodesics, in which the total interval between any two events on the journey is GREATER than by any other alternative route. (The fact that it is greater and not less is because the sort of interval we are considering is more analogous to time than to distance.) So, if someone flies off into space and comes back to earth after a while, the time between departure and return would be less by their clocks than by the clocks on earth, since the earth, during its journey around the sun, chooses the route so that any bit of it is measured longer than any other alternative route.

“Space and time are now dynamic quantities: when a body moves, or a force acts, it affects the curvature of space and time—and in turn the curvature of space-time affects the way in which bodies move and forces act. Space and time not only affect, but are also affected by everything that that happens in the universe.” Hofstadter would call this a Strange Loop.


  • A Brief History of Time by Stephen Hawking
  • Einstein’s Law of Gravition by Bertrand Russell
  • The Two Masses by Isaac Asimov

Pseudointellectual ramblings || Ram Samudrala || me@ram.org

Inertialsystem that is accelerating

九月 9, 2012

Inertialsystem that is accelerating

Einstein and Tummler, his sailing yacht

七月 29, 2012

Einstein and Tummler, his sailing yacht

In general relativity, there is no gravitational force. Gravity is geometry

七月 28, 2012

In general relativity, there is no gravitational force. Gravity is geometry.

It’s similar for the clocks in the spaceship. The signal that the group at the front receives is “Doppler-shifted” to stretched waves because these scientists are moving away. Their detector counts fewer of these stretched waves per second, and the scientists conclude that the clock in the back of the ship runs more slowly than the clock at the front.
To confirm, the group of scientists at the back of the ship detects the light from the front clock. Because this group is accelerating toward the place where the signal was emitted, they detect a compressed wave, and their detector counts more of these waves per second. For them, the clock at the front runs faster than theirs. The two groups are in agreement about the clocks

Lorentz-Fitzgerald Contraction

七月 14, 2012

In 1889, the Irish physicist George Francis Fitzgerald proposed
a radical explanation of why the Michelson-Morley
experiment failed to detect the luminiferous ether. His
explanation came at a time when he, like most scientists,
firmly believed in the ether. Movement through the ether,
Fitzgerald said, shortened the arm of Michelson’s interferometer
just enough to cancel the decrease in the speed
of light caused by the ether wind. This length contraction
took place along the line of motion and was almost impossible
to detect because any meter stick used to measure it
would contract, too.
Two years after Fitzgerald published his proposal, Hendrik
Lorentz, a prominent Dutch physicist who was also a
staunch believer in the ether, developed the idea further.
The shortening of objects in motion relative to an observer
became known as the Lorentz-Fitzgerald contraction.
Lorentz also came up with a general method for transforming
the space and time coordinates of events from one
inertial frame of reference to another. The equations he
derived to do this are called Lorentz transformations, and
they proved useful to Einstein as he developed the special
Lorentz’s formulas for calculating time dilation and
length contraction are identical to those Einstein developed
for special relativity. Why, then, are Lorentz and Fitzgerald
not considered to be the authors of the theory of special
relativity? The answer lies in the two men’s wrong interpretation
of the Michelson-Morley experiment. According
to Lorentz and Fitzgerald, the ether existed and the speed
of light was constant relative to it. Einstein’s bold leap forward
was to ignore the ether and accept what Maxwell’s
equations were telling him: The speed of light is the same
for every observer. It is this key conclusion that led Einstein
to relativity—and kept Lorentz and Fitzgerald from discovering
it themselves.

Nevertheless, Einstein knew he owed much to the two
men’s groundbreaking ideas and was quick to recognize
them. In an after-dinner speech he delivered in California,
Einstein credited “the ideas of Lorentz and Fitzgerald, out
of which the Special Theory of Relativity developed.”

Lorentz took the distortions that he discovered in fast-moving material objects to be laws of nature

七月 12, 2012

Time dilation. There is, however, another distortion that material objects undergo as a function of their absolute motion. That is a slowing down of clocks (and physical processes generally) at the same rate as the length contractions, or the so-called “time dilation,” which took somewhat longer for Lorentz to discover.

The Galilean transformation for time in Newtonian physics is simply t = t’ , because Newtonian physics assumes that time is the same everywhere. But by using transformation equations to describe the distortions in material objects, Lorentz found that he had to introduce a special equation for transforming time: t’ = t – vx/c2  (Goldberg, p. 94). The new factor in the transformation equation, vx/c2, implied that time on the moving frame varies with location in that frame. Lorentz called it “local time,” but he did not attribute any physical significance to it. “Local time” is not compatible with the belief in absolute space and time, and Lorentz described it as “no more than an auxiliary mathematical quantity” (Torretti, p. 45, 85), insisting that his transformation equations were merely “an aid to calculation” (Goldberg, p. 96).

The slowing down of physical processes is called “time dilation.” Lorentz discovered this distortion by tinkering with various ways of calculating the coordinates used on inertial reference frames in relative motion. Thus, it is natural to describe time dilation as the slowing down of clocks on the moving reference frame. It was included in the final version of Lorentz’s explanation, now called the “Lorentz transformation equations.” (Lorentz 1904) Those equations contained not only the length contraction and transformation for “local time”, but also the implication that clocks on moving frames are slowed down at the same rate as lengths are contracted (that is, ). The final Lorentz equation for time transformation included both the variation in local time and time dilation: .

Though Lorentz took the distortions that he discovered in fast-moving material objects to be laws of nature, he did not think that they were basic. He thought they were effects of motion on the interactions between electrons and the ether which could be explained by his electronic theory of matter, and he saw explaining this effect as the the main challenge to Newtonian physics. The transformation equations themselves never seemed puzzling to Lorentz, because he never took them to more than just a mathematical aid to calculation.