Episode 42: The Lorentz Transformation - The Mechanical Universe
Summary
TLDRThis script explores the historical and theoretical foundations of the theory of relativity, highlighting the contributions of physicists like Lorentz, Fitzgerald, and Einstein. It discusses the Michelson-Morley experiment, which failed to detect the luminiferous aether, leading to the development of the Lorentz transformation equations. The script delves into the concept of the constancy of the speed of light and how it challenges traditional notions of time and space. It also touches on the personal influences and the significance of Einstein's approach to relativity, which redefined our understanding of the universe.
Takeaways
- 🌌 The conventional history of relativity starts with the Ether theory, which was disproven by the Michelson-Morley experiment, leading to Einstein's theory of relativity.
- 🔍 The real story of relativity involves the Lorentz transformation, which includes the constant speed of light derived from Maxwell's equations.
- 🚂 At the end of the 19th century, the speed of trains was the fastest human experience, with the Earth's rotation around the Sun being faster but unnoticed by inhabitants.
- 🌟 The Michelson-Morley experiment aimed to detect motion through the Ether by measuring the speed of light, but found that the speed of light is constant regardless of the observer's motion.
- 🤔 GF Fitzgerald and Hendrik Lorentz proposed that motion through the Ether causes a contraction in the direction of motion, which was a key insight for the theory of relativity.
- 🕰 Lorentz's work suggested that time, as well as distance, is affected by motion, leading to the idea that time dilation occurs in moving frames of reference.
- 📏 The Lorentz transformation equations express how time slows and distances contract in a moving frame, and they join time and space into a single continuum.
- 🌐 Einstein's special theory of relativity was based on two postulates: the laws of physics are the same in all inertial frames, and the speed of light is constant for all observers.
- 🚀 Einstein's theory provided a deeper understanding of the Lorentz transformation, viewing the constant speed of light as a fundamental principle rather than an appearance.
- 📚 There are two independent histories of the theory of relativity: one through the Ether theory and Lorentz's work, and the other through Einstein's independent development of the same theory with a different perspective.
Q & A
What was the conventional history of relativity mentioned in the script?
-The conventional history of relativity mentioned in the script includes the theory of the Ether, the Michelson-Morley experiment which showed the Ether theory was wrong, and then Albert Einstein's formulation of the theory of relativity.
What is the Lorentz transformation and why is it significant?
-The Lorentz transformation is a set of equations that describe how space and time coordinates transform from one inertial frame of reference to another moving relative to the first. It is significant because it incorporates the constant speed of light and introduces the concepts of length contraction and time dilation, which are fundamental to the theory of special relativity.
What was the Michelson-Morley experiment aiming to prove?
-The Michelson-Morley experiment aimed to detect the motion of the Earth through the luminiferous Ether by measuring the effects of the Ether on the speed of light.
What was the unexpected result of the Michelson-Morley experiment?
-The unexpected result of the Michelson-Morley experiment was that the speed of light was found to be the same regardless of the motion of the observer, which contradicted the expectation that the speed of light would vary depending on the observer's motion through the Ether.
Who were Albert A. Michelson and Edward Morley, and what was their contribution to the development of the theory of relativity?
-Albert A. Michelson and Edward Morley were American physicists who conducted the Michelson-Morley experiment. Their experiment, which failed to detect the luminiferous Ether, contributed to the development of the theory of relativity by challenging the classical understanding of space and time.
What was Hendrik Lorentz's contribution to the understanding of the Michelson-Morley experiment results?
-Hendrik Lorentz contributed to the understanding of the Michelson-Morley experiment results by proposing the concept of length contraction, suggesting that objects contract in the direction of motion relative to the Ether, which could explain why the speed of light appeared constant to all observers.
What is the significance of the speed of light in the theory of relativity?
-In the theory of relativity, the significance of the speed of light is that it is constant for all observers, regardless of their relative motion. This constancy leads to the concepts of time dilation and length contraction, which are fundamental to understanding the behavior of objects at speeds close to the speed of light.
What is the principle of relativity as mentioned in the script?
-The principle of relativity, as mentioned in the script, states that the laws of physical phenomena should be the same for an observer at rest and for an observer in uniform motion. This principle is a cornerstone of the theory of special relativity.
How did Albert Einstein's approach to the theory of relativity differ from Hendrik Lorentz's?
-Albert Einstein's approach to the theory of relativity differed from Hendrik Lorentz's in that Einstein based his theory on two postulates: the principle of relativity and the constancy of the speed of light for all observers. Einstein's approach was more fundamental and universal, while Lorentz was focused on explaining specific experimental results like those from the Michelson-Morley experiment.
What is the Lorentz factor (gamma) and how does it relate to time dilation and length contraction?
-The Lorentz factor, denoted by gamma (γ), is a mathematical factor that relates the time and space measurements of an object in motion to those in a stationary frame of reference. It is given by the equation γ = 1 / √(1 - v²/c²), where v is the relative velocity and c is the speed of light. The Lorentz factor is used to calculate time dilation (moving clocks run slower) and length contraction (objects in motion contract in the direction of motion).
Outlines
🌌 Introduction to the History of Relativity
The paragraph introduces the historical context of the theory of relativity, starting with the Ether theory and the Michelson-Morley experiment that disproved its existence. It discusses how Albert Einstein's theory of relativity emerged as a result. The narrator corrects the common misconception about the historical development of the theory, hinting at a more complex and nuanced story that will be explored later. The paragraph also introduces the Lorentz transformation, emphasizing the importance of the speed of light as a constant in modern physics, derived from Maxwell's equations and measured by Michelson with remarkable precision.
🚂 The Impact of Technology on Perceptions of Speed
This paragraph delves into the concept of speed at the end of the 19th century, where trains represented the epitome of speed. It contrasts the human experience of speed through trains with the speed of the Earth orbiting the Sun, highlighting the relative nature of speed. The paragraph discusses the Ether theory and how it was tested by Michelson and Morley, aiming to detect the Earth's motion through the Ether by measuring the speed of light. The unexpected result of the experiment, showing that the speed of light is constant regardless of the observer's motion, is explained, leading to the initial skepticism and eventual acceptance of this phenomenon through the work of Fitzgerald and Lorentz.
🔬 The Lorentz Transformation and the Contraction Hypothesis
The paragraph explores the Lorentz transformation and the concept of length contraction. It explains how Lorentz and Fitzgerald proposed that an object moving through the Ether would contract in the direction of motion, a phenomenon later attributed to the properties of electrons. The discussion continues with the implications of this contraction for understanding the Michelson-Morley experiment and the constancy of the speed of light. The narrative also touches on the idea that time, like distance, is affected by motion, setting the stage for a deeper understanding of relativity.
⏱️ The Relativity of Time and the Principle of Relativity
This paragraph introduces the concept of time dilation and the principle of relativity as proposed by Galileo and later formalized by Henri Poincaré. It discusses how the laws of physics should be the same for all observers in uniform motion, challenging the traditional notions of time and space. The narrative explains how Lorentz developed equations to describe the effects of motion on rulers and clocks, leading to the understanding that a moving clock appears to run slower to a stationary observer. The paragraph also highlights the significance of the Lorentz transformation in unifying time and space.
🌐 The Development and Implications of the Lorentz Transformation
The paragraph discusses the Lorentz transformation in detail, explaining how it accounts for the observed phenomena of time dilation and length contraction. It describes the historical context in which Lorentz published his electron theory, which included the essential equations of the theory of relativity. The narrative then shifts to Albert Einstein's perspective, who was influenced by Lorentz's work and developed his own theory of relativity based on two fundamental postulates. The paragraph concludes by emphasizing the importance of the Lorentz transformation in understanding the special theory of relativity and its impact on the concepts of space and time.
🚀 The Space-Time Diagram and the Different Perspectives on Relativity
This paragraph introduces the concept of space-time diagrams as a tool for visualizing the effects of relativity, such as the relativity of simultaneity and the contraction of rulers in motion. It contrasts the perspectives of Lorentz and Einstein on the theory of relativity, highlighting how Lorentz viewed the constant speed of light as an appearance, while Einstein considered it a fundamental principle. The narrative concludes by acknowledging the independent yet converging paths of Lorentz and Einstein in the development of the theory of relativity, and it sets the stage for further discussions on the topic.
Mindmap
Keywords
💡Theory of Relativity
💡Lorentz Transformation
💡Speed of Light
💡Ether
💡Michelson-Morley Experiment
💡Length Contraction
💡Time Dilation
💡Galileo's Relativity Principle
💡Space-Time
💡Hendrik Lorentz
Highlights
The conventional history of relativity is logically correct but not historically accurate.
Lorentz transformation is a key part of the theory of relativity, focusing on the speed of light.
The speed of light, derived from Maxwell's equations, is essential for understanding modern physics.
The Michelson-Morley experiment showed that the speed of light is the same for all observers, contradicting the Ether theory.
GF Fitzgerald and Hendrik Lorentz proposed length contraction to explain the Michelson-Morley results.
Lorentz's explanation for the contraction phenomenon was linked to the properties of electrons.
Lorentz believed that the electron was responsible for the constancy of the speed of light for all observers.
The concept of length contraction was used to explain why the Michelson-Morley experiment failed to detect motion through the Ether.
Lorentz's work led to the development of the Lorentz transformation equations.
The Lorentz transformation equations express the mathematical essence of the special theory of relativity.
The theory of relativity implies that time and distance are affected by motion.
Henri Poincaré objected to the limited nature of Lorentz's explanation, proposing the principle of relativity.
Albert Einstein's work on relativity was influenced by Lorentz's equations and the principle of relativity.
Einstein's theory of relativity was based on two postulates: the relativity principle and the constancy of the speed of light.
Einstein's theory deduced the same equations as Lorentz but with a deeper understanding of their implications.
The Lorentz transformation equations join time and space together, changing the traditional meanings of these concepts.
The theory of relativity has two independent historical narratives, one through Lorentz and the other through Einstein.
Einstein's contribution to the theory of relativity was not just minor, but represented a profound shift in understanding.
Transcripts
foreign media
[Music]
[Music]
thank you
thank you
last time I told you the conventional
history of relativity
there was the theory of The Ether
and then the Michaelson Molly experiment
showed that it was wrong
and then Albert Einstein produced the
theory of relativity
as I told you then that story is
logically correct but not historically
correct
well what is the real story
I'll get back to that in a little while
first of all I would like to tell you
about a part of the theory which is
still called the lorentz transformation
[Music]
in the lorentz transformation there is
above all
the speed of light
the speed that sprang from Maxwell's
equations
the three hundred thousand kilometers
per second that Michelson measured with
Landmark accuracy
the speed which is essential in modern
physics
and vital to understanding the universe
as it really is
and at the end of the 19th century when
Lawrence derived his transformation
equations
when it came to speed the most moving
Human Experience
was provided by trains
the automobile
which would eventually outpace the train
wasn't quite in gear
and air travel was barely off the ground
big or small
sophisticated or Child's Play
trains were the very image of speed
and in fact the only thing on Earth that
moved people faster
was the Earth itself
orbiting the Sun at 108 000 kilometers
per hour
the inhabitants of the planet may not be
aware of the speed but speed it is
and to physicists at the turn of the
century it was speed relative to
The Ether
since the time of Aristotle The Ether
was believed to be the very stuff of
which the heavens themselves were made
that concept remained mere speculation
until two Americans set out to prove
once and for all
that what was then called The
luminiferous Ether
did in fact exist they were Albert a
Michelson
and Edward Morley
and their goal was to detect motion
Through The Ether by measuring its
effects on the speed of light
[Music]
in one attempt after another
with the most sensitive experimental
device ever constructed
they found exactly what they weren't
looking for
the interferometer showed that
regardless of the motion of the Observer
the speed of light is the same
for Michelson and Morley this was bad
news
and it traveled fast
as fast as a boat could carry it to
Ireland
there physicist GF Fitzgerald an early
champion of Maxwell's electromagnetic
theory of light examine michelson's
otherwise disappointing results
Fitzgerald's explanation was that in its
motion Through The Ether one arm of the
interferometer was contracted by a
fraction of its length which was just
the right amount to allow the two light
beams to arrive simultaneously
immediately most scientists scoffed at
the idea
however when Henrik Lorenz arrived at
the same idea no one was laughing
Not only was he the world's expert on
Maxwell's electromagnetic Theory
but lorentz gave a tangible explanation
for the contraction phenomenon
he thought it had to be a property of
the electron
whose existence had just been confirmed
by J.J Thompson and his colleagues in
England
it seemed that in electrons they'd found
the inner parts of atoms
the ultimate constituents of matter
but they hadn't discovered all the
things that electrons actually do
one possibility Lawrence believed is
that an electron will contract in the
direction of motion
if that were true and since everything
is composed of electrons then an
interferometer would contract in the
direction of motion
so would a ruler
and so with a locomotive
while length contraction took care of
the Michelson Morley experiment
lorentz knew more was needed to explain
all the experiments that had failed to
detect motion Through The Ether
since he still believed in The Ether and
the fact that the Earth moved through it
he felt that somehow the electron must
be responsible for this amazing fact
about life
regardless of the speed with which they
themselves are moving all observers
measure the same speed of light
certainly other speeds say the speed of
a moving train
aren't the same to all Observers
to an observer on a platform for example
this Train's passing had a good clip
but to someone on the train
the train speed seems to be zero and the
ground outside seems to be moving
instead
therefore as Galileo was well aware in
the Renaissance
and as the Stella knows today
the speed of an object depends on the
observer's speed
but Lorenz said that perceptions of
light waves would be radically different
he suggested that even someone traveling
at nearly the speed of light
would still observe light moving at
three hundred thousand kilometers per
second
how could that be
consider two observers in relative
motion
in this case
Albert
and Henry
just for the sake of argument
at the exact place and time they pass
each other
they observe a flash of light
a sphere of light expands outward from
that point
since each measure the speed of light
relative to himself
each believes correctly that he is
always at the center of that expanding
sphere
even though they themselves move farther
and farther apart
how can two people
in different places
both be at the center of the same sphere
to confirm his perception
each sets up light detectors and equal
distance apart
however while Albert's detectors
register the light arriving
simultaneously he believes the light
strikes Henry's detectors at two
different times
[Music]
meanwhile Henry sees the same thing in
reverse
ah
[Music]
they agree on the speed of light
but they disagree on whether events
happen simultaneously or at different
times
this is not semantics nor a petty debate
it means that time as well as distance
has to be affected by motion
however as profound as this was
the French mathematician Ali parque
objected to the limited nature of
Lorenzo's explanation what was needed he
said was a new fundamental law of
physics
the principle of Relativity According to
which the laws of physical phenomena
should be the same
whether for an observer fixed or for an
observer carried along in a uniform
movement of translation
in other words as Galileo had suggested
one state of uniform motion is as good
as any other
after all this idea was the basis of
Galileo's reasoning and the law of
inertia almost 300 years earlier
[Music]
was suggesting that the idea of Galilean
relativity should be generalized to
include all physical phenomena including
light
for example an observer could not
determine whether he was in motion by
measuring the speed of light
since that speed is the same for all
Observers
and that meant age-old Notions about
time and space had to change
himself shied away from examining the
consequences
Lawrence developed the equations needed
to show precisely how much
rulers would have to contract
and clocks would have to slow down
when they were in motion
the essence of his reasoning can be seen
with the aid of the simplest possible
clock
two mirrors a fixed distance apart
with a light beam bouncing back and
forth between them
each bounce of the beam is a tick or a
talk of the timepiece
to Henry
his clock is stationary
and altogether ordinary but for Albert
that clock is moving
and between tick and talk
he sees the light beam trace a diagonal
path
which means it's traveling a longer
distance
but the speed of light is the same for
all Observers
so the light must take a longer time to
travel the longer distance therefore
Albert believes the moving clock runs
slow
but how slow
[Music]
the relativity of time is derived from
the right triangle formed by the
distances traveled
the Pythagorean theorem shows that the
path of the moving light is longer than
the distance between mirrors
by the factor one over the square root
of 1 minus V squared over c squared
this Factor occurs so often in
relativity that it has given its own
symbol
the Greek letter gamma
[Music]
so to an observer at rest
a moving like clock seems to be running
too slowly
by the factor gamma
a ruler or anything else in motion
also seems contracted by that same
factor
and it's called the Fitzgerald
contraction
for speeds much less than the speed of
light
gamma isn't very large
for example the Earth
in its headlong Dash around the Sun is
shortened by no more than the length of
one blade of grass
as for Speed on the Earth while Lawrence
was busy developing his theory
the steam locomotive barrier of 100
miles per hour was broken
at that speed the entire train is
shortened by less than the width of one
atom of the paint on its engine
and Lawrence himself didn't move slowly
he was the international physicist young
physicists from around the world
attended his lectures at the University
of Leiden in the Netherlands where he'd
been a professor for a quarter of a
century
among those who came to Leiden was
Albert Einstein
over the years the rents would have an
enormous professional and personal
influence on Einstein
[Music]
just before his death Einstein said
I have admired and loved Lawrence
perhaps more than anyone else I have
ever known
but it was Lawrence's work as a
theoretical physicist that went beyond
his privileged Circle of Friends
and traveled freely through time and
space
[Music]
knowing he was on the right track
the rents pursued the consequences as
far as he could
if moving things appear shorter
and moving clocks run slower
then how can two people moving in
relation to each other
agree on a consistent description of
where and when some event happened
to answer that what's needed is a set of
equations to replace the old Galilean
transformations
these equations weren't hard to find but
some of their implications
were hard to believe
in Galilean relativity the position of a
point x Prime in a moving frame is
related to its coordinate in a fixed
frame by x minus VT
lorentz found that for the new
relativity this must be multiplied by
the factor gamma
that's the equation along the direction
of motion
in directions perpendicular to the
motion distances are the same in both
frames
and as for time
although clocks can be synchronized in
any one frame
their readings in another frame may
depend on where they are
the time in the moving frame is gamma
times the quantity T minus v x over c
squared
[Music]
together these equations are the lorentz
transformation
they express the mathematical essence of
the special theory of relativity
the lorentz transformation slows time
and contracts distances in a moving
frame
no matter which frame is taken to be
moving
the observer in the moving frame thinks
he's at rest and that the other frame is
really moving
but these equations do more than that
they actually join time and space
together
when an event occurred has no meaning
without saying where it occurred
[Music]
1904 the Netherlands
lorentz publishes the definitive version
of his electron Theory
it contains the essential equations of
the theory of relativity
but Albert Einstein has not yet been
heard from which has caused some to say
that history has given him more credit
than he deserved 1905.
Baron Switzerland
Einstein a young physics student
supporting himself as a patent Clerk
finds himself disturbed by seeming
inconsistencies at the very core of
physics Ken inertia and the laws of
mechanics be made consistent
[Music]
Maxwell's theory of Optics and
electromagnetism
[Music]
Einstein decides that they must
even if that means giving up not only
The Ether but the traditional meanings
of time and space
he sets forth two fundamental postulates
the first is
relativity principle the laws of physics
are the same for all inertial frames
his second postulate states that the
speed of light is the same for all
Observers
he simply assumes the phenomenon that
lorentz has been struggling to explain
from these two postulates alone Einstein
deduces exactly the same equations
lorentz discovered earlier
but now they have a very different
meaning
the fundamental concepts of space and
time
have become intertwined
the essence of the idea can be
understood by visualizing time as if it
were another dimension
Albert standing still in space flows
through time
so that a vertical line represents a
fixed point x equals zero in his
reference frame at different times
[Music]
while a horizontal cross-section
represents simultaneous times in
different places
on the other hand someone in motion
Galileo for example
traces an oblique path
foreign
so while what Albert thinks of as a
fixed Point makes a vertical line
Galileo's idea of nothing happening
appears as a tilted line at X Prime
equals zero or anywhere else in his
frame
but of course if Galileo had drawn the
picture his line for standing still
would be vertical
and Alberts would be tilted backward
the same idea can be used to show the
relativity of time
when Henry and Albert observed the same
expanding light sphere
it reaches their detectors at definite
points in time and space
[Music]
these are called events
[Music]
meanwhile the light itself traces out a
cone
to Albert events on the horizontal
cross-section are simultaneous
for him one of Henry's detectors flashes
first
then both of his own flash
simultaneously
and finally Henry's other detector
flashes
[Music]
so he thinks these two events are
simultaneous
but Henry thinks these two events are
simultaneous
so not only are Henry's lines of
constant position tilted
but so are his lines of simultaneous
time
for Henry simultaneous events take place
everywhere on a tilted cross-section
so he thinks one of Albert's signals
goes first
then both of his then Albert's other
signal
of course if Henry were drawing the
picture
he would draw his lines of constant
place and constant time perpendicular to
each other
amazingly that wouldn't change the light
cone at all
this way of looking at things is called
a space-time diagram
and many of the strange effects of
Relativity can be visualized this way
for example
Albert thinks that Henry's ruler isn't
quite a meter long
while Henry seeing Albert speed by
thinks Albert's ruler is shorter
[Music]
on the space-time diagram
Albert measures lengths on his space
axis where Henry's ruler is shorter
[Music]
but on Henry's axes the situation is
reversed
and Albert's ruler is shorter
and what about the mystery of the clocks
how can each think the other's clock is
slow
on the space-time diagram
just follow the bouncing light beams
foreign
[Music]
on Albert's time access Henry's ticks
are farther apart than his own
[Music]
but on Henry's time axis Albert's ticks
are farther apart no matter how he looks
at it
actually there's more than one way to
look at the lorentz transformation
itself
while it was first derived by lorentz
Einstein arrived at the same equations
but from a completely different
direction
lorentz used the equations to explain
the Michelson Morley experiment while
Einstein's goal was to establish
relativity as a fundamental and
Universal principle for all of physics
for Lawrence the constant speed of light
for all observers was a mere appearance
for Einstein this constant speed was a
principle from which all else should be
derived
Lawrence was perhaps the last great
classical physicist but the equations
that bear his name
are at the heart of relativity
and the future it created
in any case there are two completely
independent histories of the theory of
relativity
one of them goes this way there was The
Ether Theory and in the Michelson Morley
experiment and other experiments fail to
detect its existence and then cuancare
and Lorenz with great difficulty produce
the formulas that were necessary to
explain the results of those experiments
and then
completely separately and independently
young Albert Einstein was worried about
a deep problem having to do with the
nature of light and electricity and he
decided that this was the way the world
had to work
and he came up with exactly the same
Theory
but with a far far deeper understanding
of what it meant
so
it's possible to say that Einstein made
only minor contributions to the theory
of relativity just as it's possible to
say that Copernicus did nothing but a
trivial mathematical transformation of
coordinates
but to say that ignores in the most
profound possible way
the real history of both of those
subjects
we'll go on without discussion of the
theory of relativity next time
thank you
[Applause]
[Music]
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