In this course we will seek to “understand Einstein,” especially focusing on the special theory of relativity that Albert Einstein, as a twenty-six year old patent clerk, introduced in his “miracle year” of 1905. Our goal will be to go behind the myth-making and beyond the popularized presentations of relativity in order to gain a deeper understanding of both Einstein the person and the concepts, predictions, and strange paradoxes of his theory. Some of the questions we will address include: How did Einstein come up with his ideas? What was the nature of his genius? What is the meaning of relativity? What’s “special” about the special theory of relativity? Why did the theory initially seem to be dead on arrival? What does it mean to say that time is the “fourth dimension”? Can time actually run more slowly for one person than another, and the size of things change depending on their velocity? Is time travel possible, and if so, how? Why can’t things travel faster than the speed of light? Is it possible to travel to the center of the galaxy and return in one lifetime? Is there any evidence that definitively confirms the theory, or is it mainly speculation? Why didn’t Einstein win the Nobel Prize for the theory of relativity? About the instructor: Dr. Larry Lagerstrom is the Director of Academic Programs at Stanford University’s Center for Professional Development, which offers graduate certificates in subjects such as artificial intelligence, cyber security, data mining, nanotechnology, innovation, and management science. He holds degrees in physics, mathematics, and the history of science, has published a book and a TED Ed video on "Young Einstein: From the Doxerl Affair to the Miracle Year," and has had over 30,000 students worldwide enroll in his online course on the special theory of relativity (this course!).
Cause and effect: spacetime diagram
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From the course by Stanford University
Understanding Einstein: The Special Theory of Relativity
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From the lesson
Paradoxes to Ponder
Week 7: Paradoxes to Ponder
Meet the Instructors
Larry Randles LagerstromAcademic Director
Last week, we did an example involving cause and effect.
And what we showed is that if you can have a situation where something is traveling
faster than the speed of light.
A material object like a spaceship or it actually generalizes to any message,
any information type message, where you can get from one point to another point.
Then you run into problems with cause and effect.
There are contradictions with all.
So let's just review what that situation was, and
then we're also going to apply our space time diagram to the situation and
visualize it in another way to get a better idea, perhaps of what was going on.
So remember that the situation we set up was that we had the bad guys and
the good guys.
And they signed a treaty at the bad guy's planet and you're zero, at which point
the good guys took off in their ship at 0.6 times the speed of light.
And remember of course, they're traveling along the X axis, this is their
world line on the space-time diagram of the bad guy's frame of reference.
Frame of reference of the bad guy's planet, so it's XBG here, and
TBG for the bad guys.
In the intervening years, the four years between when the treaty was signed,
and the good guys took off.
The bad guys invented a faster than light spaceship.
And so, in year four, by which time they perfected it, they set off in their faster
than light spaceship, which would go a speed three times the speed of light and
caught up to the good guys at time equals five years.
At which time the good guys have traveled three light years away.
Remember, they were traveling at 0.6C.
So what that enables them to do within five years,
five times 0.6C gives you that the three light years.
Bad guys could travel three times the speed of light and therefore,
it only took them one year between four and five to get to that point.
So, they met actually, remember if you're visualizing this along the X axis,
this is not the actual intersection point on the X axis where it actually occur.
That would be right here.
In other words, the good guys are traveling along the X axis.
This is a space-time diagram, showing you their progress through time as well,
and the bad guys catch up to them and pull off their sneak attack at that point.
And we analyze it from the bad guy frame.
The launch occurred at four years and the attack was at five years.
And then we analyzed it from the good guys frame of reference
using Lorentz transformation, and discovered that the attack occurred
at four years in their frame of reference but the launch occurred at five years.
In other words, the attack occurred even before the super spaceship had
been launched and therefore, clearly a violation of cause and effect.
You can't have an attack if the spaceship hasn't maybe even been invented yet and
certainly hasn't been launched at that point.
And therefore,
the conclusion is that faster than light travel is impossible because
we've never seen in all our observations a violation of cause and effect.
Now, that doesn't mean perhaps in some case in the future,
we may discover something like this, there's a violation.
But everything we know,
all the experiments we do, says there's no violation like this of cause and effect.
Again, some of you may be thinking of things like in quantum physics
which get a little strange.
But even in quantum physics, the special theory of relativity has been shown
to hold in situations like this where you can't travel faster than light.
So here's a redrawn diagram, same diagram except what we've done here is drawn
our combined diagram so again, we have the axis for the bad guys frame of reference.
And then we put in the good guy axis.
The X axis and the time axis for the good guys.
And we've also drawn in the lines of simultaneity, and I've tried as much as
possible to make this to scale, didn't quite get it all right there.
But the key thing here is that, of course,
you've got your two axis as we've shown here.
The speed of light in these units, with again, it's light years and years.
Speed of light would be a 45 degree line there.
And so what do we see from this?
Well, remember the lines of simultaneity in any frame of reference
are going to be parallel to the X axis.
The X axis defines everything that occurs at a certain time, or
any given horizontal line in the bad guy diagram.
And you're given a horizontal line to find the line of simultaneity,
everything that's going on say, T equals four, is on this line.
When we have the skewed axis,
everything that's going on at a given time is a line parallel to the X axis.
And so I've drawn in a number of these lines here, so
they're just all parallel to the X axis.
And if we analyze this, okay, so just to be clear here,
it means the X axis is when T, the XGG axis is all events when TGG equals zero.
In the good guy's frame, that happens at T equals zero.
That's when they set off from the planet.
We defined that as our origin point.
So that's a zero.
Then here is a little while later, this line.
And then a little while after that is this line and this line and so on and so forth.
And that tracks their measurement of time as they move along.
So, what happens here then?
Well, we get up to where the sneak attack occurs.
And note that, if we're just counting lines here, it's not quite accurate, but
we've got one, two, three, four, about four-and-a-half lines of simultaneity.
Technically, if I draw in the lines exactly as they should be,
we'd be getting the answer four here.
It'd be at four years.
But these lines are not drawn exactly to one year per line.
Little bit less than one year per line there.
But clearly, it's one, two, three, four and a little bit more
is where the sneak attack occurs, but look where the launch occurs.
It's one, two, three, four, five and almost six there.
So clearly from the diagram without even doing
any calculations as long as we get our axis drawn correctly.
We can see that in the good guy's frame of reference,
the launch occurs after the attack.
And in fact, in a sense, the good guys would see the attack occurring here,
assuming they're still around.
Some may observing, maybe the good guys had two ships and one ship got attacked,
the other ship would see actually, the bad guy's ship go.
Instead of go from the launch point here to the attack position.
It would go from the attack point position there and travel backwards to
the launch point, and then presumably see it get disassembled and all that.
So in a sense, it'd be like time running backwards for
the good guy's frame of reference.
Again, we just don't see things like that happening in real life.
Time flows in one direction.
And so we can't have a situation where time flows in the positive direction in
one frame of reference.
And the negative direction in another frame of reference and
make it consistent with physical reality as we know it.
Okay. So that's a nice example of how we can use
space-time diagrams when we get both things on here to see what's going on
with the situation.
Now one other point here I want to make, I'm going to try to draw in
the 45-degree line here for a world line of a light beam, say.
So we'll, I'm just going to eyeball this, but actually we can do a little bit there,
we'll do four here and that's about right there.
Roughly speaking, I might be off a little bit there, but, something like this.
So.
Okay, so there is blue light there.
So this is the world line of a light beam traveling along.
It should bisect these two axis, it got off slightly,
but it bisects the two axis here, bisects any of the axis we draw.
And from this, we can note that if you have a line whose slope is
less than the speed of light, world line.
And which we've talked about before.
That means this is a velocity greater than the speed of light.
And that's where you can get this crossing of lines of simultaneity.
Note that at the speed of light, it does not cross
lines of simultaneity in the backwards direction like this one does.
As soon as you get something bending down, so
that it's greater than the speed of light, then you can get this crossing of
the lines of simultaneity in a reverse direction, in a backwards direction.
And this is just a graphical way of saying this thing we've said several times now,
is that you can't go faster than the speed of light.
And so that is an example then of this cause and
effect relationship that we need to keep consistent.
And we'll actually, this week as we look at several different paradoxes,
we'll come back to as certain cases using our space-time diagrams to
visualize what's going on.
As well as do the Lorenz transformation analysis that we've done before.
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