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[Phys-l] Minkowski in the classroom (splinter from "how to prove relativity")



Hi John (Denker),

I have seen these ideas of yours before (probably on this forum) and I
always feel like they really do help with visualizing relativity.

I am curious if you (or others) know of textbooks for Modern Physics that do
SR using spacetime rather than what seems to be the more standard "time
dilation & length contraction" version.

Perhaps this is really about John Clements comments... that even students
who have gotten past a year of calc-based physics may not have the vector
skills to tackle relativity in this way.

I certainly gain a lot of insight from it, and as I contemplate teaching SR
to a new batch of physics sophomores in about a month I am left wondering
why it is that few textbooks have taken that path.

Jeff

--
Jeff Loats
Metropolitan State College of Denver

On Sat, Jun 5, 2010 at 1:39 AM, John Denker <jsd@av8n.com> wrote:

On 06/04/2010 10:07 PM, Stefan Jeglinski wrote:

What's missing, for me personally, is to add a discussion of
relativity of simultaneity.

That's another thing that's easy in the spacetime
approach (and not otherwise).

As the saying goes, learning proceeds from the
known to the unknown ... so let's start by
reviewing some things we already know:

Fact 1: In the XY plane, suppose you start out
with a vector purely in the X direction relative
to your frame of reference. Then you rotate your
frame CW a little bit. Relative to the new frame
the vector has picked up a small projection in the
Y direction. No problem. No surprise.

Fact 2: In the XY plane, suppose you start out
with a vector purely in the Y direction relative
to your frame of reference. Then you rotate your
frame CW a little bit. Relative to the new frame
the vector has picked up a small projection in the
-X direction. No problem. No surprise.

Fact 3: In the XT plane, a stationary object has
a 4-velocity that is purely in the T direction.
That is to say, it just sits there and gets later
at the rate of 60 minutes per hour. This is
something new, but not too much of a brain-strain.

Suppose we start with such a vector, initially purely
in the T direction relative to your frame of reference.
Then we rotate your frame a little bit in the XT
plane. The vector picks up a small projection in
the X direction. That means that as it moves along
its world line, mostly it moves in the T direction
but it also moves a little bit in the X direction.
The ratio, the amount of X per unit T, is called
the /velocity/.

So a boost is just a rotation in the XT plane.
A rotation in the XT plane mixes X and T in much
the same way as a rotation in the XY plane mixes
X and Y. This is new way of looking at things,
but it's perfectly plausible.

Note that we chose to rotate the frame of reference
rather than rotating the objects, because we wanted
to make it clear that *all* objects would be affected
equally. If we rotated some objects relative to others,
the ideas would be more open to misinterpretation.

Fact 4: There is one more case we need to consider.
Here is where the paydirt will be.

In the XT plane, suppose we start out with a vector
that is purely in the X direction relative to your
reference frame. This is obviously not a velocity
vector, but it could describe the displacement of
event A relative to event B.

Note that an /event/ is a point in four dimensional
spacetime. It can be specified by four coordinates
t,x,y,z. An event occurs at a specific place
_and time_. You can visualize the detonation of
a small firecracker as an example of an event,
i.e. something that happens at a specific place
at a specific time.

Anyway, we have two events A and B, and a vector
with the correct magnitude and direction to
specify the displacement from A to B. Initially
the vector is purely in the X direction. It
has no component in the T direction, which means
the two events (A and B) are _simultaneous_ in
your reference frame.

Now we rotate the reference frame a little bit in
the XT plane. As discussed a moment ago, this is
the same as saying the new reference frame is moving
with some velocity relative to the old reference
frame. But there is one more thing we need to
look at.

When we do the rotation, the vector that was purely
in the X direction picks up a small component in
the T direction. In the same way that the rotation
gave the T vector a small X component, it must give
the X vector a small T component.

That means that relative to the new reference frame,
events A and B are no longer simultaneous. The size
of the T-discrepancy is proportional to the length
of the original vector, since all we did is change
the angle in the XT plane, and basic notions of
trigonometry (or even more basic notions of leverage)
tell us things should be proportional. In particular,
if there is no separation between A and B, there is
no T discrepancy ... and conversely if A and B are
far apart, the T discrepancy can be rather large.
Therefore this phenomenon is officially called
_the breakdown of simultaneity at a distance_.

This is a remarkable feature of relativity. If you
think about it in terms of spactime, it makes sense.
It is item 4 on the following list:
If we rotate in the XY plane:
1) X turns into X plus a little bit of Y
2) Y turns into Y plus a little bit of -X
If we rotate in the XT plane:
3) T turns into T plus a little bit of X
4) X turns into X plus a little bit of T

Relativity is not weird. If we observed three of the
things on that list but not the fourth, now THAT would
be weird.

The fact is, events that are simultaneous in one frame
are generally not simultaneous in another frame. Again:
we call this the breakdown of simultaneity at a distance.
If you try to understand this in terms of time by itself
and three-dimensional space by itself, you are going to
have a very hard time. OTOH if you thing about it in
terms of spacetime, i.e. in terms of rotations in the
XT plane, it makes perfect sense.

The idea of simultaneity suffers a further beating when
we consider the effects of a gravitational field. That's
another thing that can be understood in terms of spacetime
and not otherwise.
http://www.av8n.com/physics/geodesics.htm#fig-darts

The history of relativity is amusing. Many of the things
we recognize as first-order manifestations of relativity
were discovered long before we had a unified view of
relativity, and were given non-unified names:
-- the first order correction to just sitting in one
place is called /velocity/
-- the first order correction to the electric field
is called the /magnetic field/
-- the first order correction to flat spacetime is
called Newtonian /gravitation/
-- the lowest-order correction to the rest energy mc^2
is called the /kinetic energy/ .5 mv^2

Those are old ideas with old names. In contrast, nobody
had any clue about (let alone a name for) the breakdown
of simultaneity until Lorentz (1892), Poincaré (1900),
and of course Einstein (1905). The idea of spacetime
was invented in 1908.

Von Stund′an sollen Raum für sich und Zeit für sich
völlig zu Schatten herabsinken
und nur noch eine Art Union der beiden
soll Selbständigkeit bewahren.

Hermann Minkowski (1908)

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