Hello, this is I'm not sure if this is Lecture 7 or Lecture 8,
but anyways, today's lecture
we're going on to talk at the beginning of Chapter
6, Foot kinematics. Now this is a bit of a a shift from what
we've been doing up to this point, not not conceptually, but
we're we're moving on to the next stage
as I mentioned the Lorentz transformation equations which
we talked about the last two lectures and we found some of
the paradoxes puzzles with them in the last one
are very important and it would pay. It would repay you to go to
be quite diligent about the exercises for
chapter 5 because they will settle down. A lot of your
understanding of the ideas here and there are also nicely
examinable that they're the sort of things that are really just
easy to see. Ohh yeah I've got something about that in the
exam. That's a hint. It's pretty obvious hint, since the only
objective for chapter five was was one.
Anyway, we're now moving on to what we've been talking up to
now about events, and I've been stressing again and again that
events are frame independent things. They have a position and
a time
and that the the story that that the point of the recent
transformation is to talk about the coordinates of that event in
this frame and in that frame which are moving with respect to
each other.
All the only thing that been moving up to this point have
been framed.
But this is a physics course as well as an astronomy course. So
we're interested in things moving. We're interested in in
the description of things moving.
And before we can describe, before we can explain how things
move dynamics, we have to describe how things move
kinematics.
And so this is where we talk about
velocity, we'll talk about acceleration.
And
that brings in talking about four dimensional vectors, 4D
vectors, 4 vectors.
So that's the goal I the plan is to to to to. There's much
relation 8 should come to think of it, because the plan is to do
the next few chapters in
three lectures. So we're ready to move on to GR. Talking about
GR, some some topics in GR in lecture 11, we seem to be on
schedule, well
schedule at this year. So any questions about structure, about
things like supervisions, about things like the product, which I
hadn't actually checked, but which I hope that is, we'll find
your questions there.
Anything else that folks are thinking about, worrying about,
puzzled about,
And it was Mr Tape. OK,
let's go.
So in a way, we're returning to physics here by talk, by talking
about velocity, acceleration and things like that.
So before we can go and talk about those, we're gonna have to
talk about to to, to expand the notion of what you understand
about vectors to to to to to to to discover what the vectors are
in that, in that context. And then we can move on.
So,
and I'll move this to here.
What are we
so really understanding? The concept of all vector is is the
key thing in this whole thing, and that's various things that
you one can do. So this is
and it seems as there were technique have used several
times in this course so far. I'm going to describe from the
you're familiar with in unfamiliar terms because if
those terms, they're going to adapt those terms to use in a
context that you are unfamiliar with. So
this is a vector a in the XY plane,
which are completely familiar.
The vector has components.
There's a basis vector, a unit vector particle on the X axis of
the unit vector pointing along the Y axis. And you can
construct the vector A
with
so much times the
a unit vector on the X axis plus so much with the unit vector
along the Y axis. And you can get the vector E back, and so
those ex and EY are the components of that vector.
OK, nothing exotic there. If you were to change your mind about
the frame and rotate the frame to the frame frame
and the vector doesn't change, the vector doesn't care what
frame you're talking about,
but the components of the vector
would change
and be different in the other frame. So if there's a basis
vector pony X prime axis, a basis vector Polygon that the
the Y prime axis and the same vector E is ex primed
times
the X plus AY primed
times. EY,
nothing exotic there.
The key thing is, the vector is the same in both cases,
right? It's a displacement vector. It's not a position
vector. It's not. How do you get the order into a? It's a a
length and A and a direction in the plane.
Good.
Any questions?
Right. So that that's really the the the terminology that I I'm
using here
and
what we could do then is ask well how do we get if we know
what axe and AY are, how do we work out what ex prime and AY
primes are? And again, that's not. I mean you might not write
this down off the top of your head, but you're not surprised
to see something like that. I trust
the if instead of of the the that's a that are we talked
about a displacement vector does XYZ
in three-dimensional space.
That seemed Victor. That seemed displacement of. How do you get
from here to here?
We have different components in different frame and the related.
By something like that I mean as I say you may not write that
down on top of your head but you're not surprised to see a
you cost sign - cost in that expression
and and and and the the point here is that
that displacement vector I in the previous thing I used a
general vector a but I'm going to talk about displacement
vector as the just the displacement from this point to
this point. So delta X out of Y, delta Z, I'm going to use that
displacement vector as our prototype vector. OK. And by
prototype I mean that anything which looks like that vector,
any which looks at that, we're going to call a vector as well,
right?
And in the
in the lecture notes folder on the middle there's the numbered
chapters. There's also a document called Recipe and
Document called Maths. The document called Maths at the end
of it has a little review of linear algebra and it's 2
paragraphs which just lists basically just a number of words
that I'm going to presume you're familiar with. You've done in
previous years maths, so. So you might have to go back and look
at you, just remember what those words mean. But I'm going to
presume you're familiar with those words,
right?
So much for three vectors.
We saw the displacement vector in
feed in three dimensions dot XYZ.
We can now talk about the displacement vector in 4
dimensions
and and at the moment right now I'm only talking these numbers
haven't haven't got to the vector but yet
delta T, delta Y, delta X and delta Z. And we know from
chapter 5 transformation that they are related to the
displacements. That doesn't frame the delta XYZ frame by an
expression like that. If you multiply that out, multiply that
matrix equation, you'll get the Lorentz transformation equations
from halfway through chapter
3, and that creates the notes, let's say
6.2.
And if we
that, that's just a matrix, so the inverse of that matrix is
just the the matrix inverse of that.
Now if we talk about
notice you park that thought
and and think about what a vector in Minkowski space would
look like.
Here, rather than the X&Y axis, we've got X and the T axis
and the X prime and the T prime axis and as usual, rather than
being rotated like that, they've gone like that When you go from
when you when you project from one frame into the other
and you have a vector there, and think of that displacement
vector for now, delta X, delta T
white dotted as well,
that links to points in
space-time.
It links to events.
There's an event to the beginning of the at the end, and
this is the difference between them. There's just exist what
and T in there and that vector we can decide, OK, we're going
to call that vector as well.
Has component A1
when you project it onto the X axis
component. East Naughty.
You project it onto the T axis. Some rather old fashioned
textbooks refer to that as as the Force dimension. We refer to
the zeroth dimension because just a bit tidier,
much more common nowadays. And similarly, if you instead not
project it vertically downwards, but project the vector parallel,
you know, projected parallel to the T frame axis and ending up
on the X frame axis, that's A1A primed one.
So the component of the vector E
and the basis vector pointing along the X prime axis and
similarly project it from a onto the the T frame axis, moving it
parallel to X prime.
You end up with an E prime knot,
and the idea
is these are the components of that vector
4 vector in the two different frames, exactly analogously to
the way we saw the components of the three vector in the two
rotated frames.
OK.
And if that vector E is this delta X, delta Y, delta Z, delta
T vector, then we discover
that the relationship between those components is just that
thing that we are, that it's fairly obvious that there's no
fairly obvious from what we learned about the Lorentz
transformation equation. So this is the point that this is the
bit where we're going from Chapter 5 to chapter six. OK,
the stuff we learned. Chapter 5 we're now applying
but really grasping hold of the of the idea of geometry in
chapter 6.
And again we're going to take the displacement vector in 4
dimensions. There's delta T dot XY&Z are the prototype
displacement vector and so anything that looks like that
and that behaves like that is a vector in 4 dimensions, A4
vector.
OK,
um,
so just as before, an arbitrary vector has components in
different frames. The vector is the same in all the frames, it's
just the numbers that change.
OK, so that's and that that's a key thing that that that that
sounds like a fairly neutral thing to say, but it's actually
a very important thing to see. The victor is the if like the
physical thing we're going to talk about velocities in a bit,
the velocity vector and if something is moving that's a
physical thing that frame independent thing. It's the the
numbers that the components in particular frame
that that you choose that are frame dependent.
Uh, so
we've got that and justice as with the displacement vector.
Therefore for an arbitrary vector A
the components are going to transform like this, so just the
same as the displacement vector but with for an operative vector
components with components A.
So the risk of just banging on with this at too many times.
These are described in the same vector a
but with the numbers a naughty to a three and a not primed to a
three. Prime is being different simply because you're choosing
different frames,
right? So I want to just clear that too many times just so you
that's it doesn't sound as important when I see it first
time as it is.
So
just just really for completeness,
the same thing is true if you draw this in the in the whole
thing, this whole thing in the other way around as it were,
with the
the framed frame being sort of the basic one and the other one
being the one distorted away.
And what we are then going to
how do we introduce them, we come to that.
So we'll we'll, I think we'll do a bit more linear algebra here
and
and uh
how do I?
So now at this point it's slightly I'll just switch both
of the visualizer. There's one or other of these is the one
that gets onto the E360. I don't know which one it is. I'll just
switch both over and
OK, I think, I think, I think this is actually 8.
So
what do we want to write down here?
So I think the standard things if A
and B our four vectors
then
A+B
is also 4 vector and the components of that are C
See naughty will be a naughty plus
be naughty. C1 equals A1 plus B1
and so on, so so so the add in that in in in the expected way.
The way that you're familiar with adding vectors in three
dimensions.
Now the scalar product. If you think back to to to the vector
stuff with vectors you learned. One of the things you learn
about the scalar product is the which measures how how separate
the projection of one right onto onto another. In this context,
the
scalar product
is odd
and the
the scalar product of A&B
is going to be a Naughty. B naughty what? What? What if we
have a dot B? They're both free vectors, you remember? That's a
naughty
being. Naughty plus A1B1 plus A2B2. That's familiar, I trust.
Yeah. Here is minus A1
B 1 -, a two, B 2 -, a
three,
B three,
and that can be written
as
E
Peter B,
where these are to pick up
slipping into a matrix version of this where the the matrix ETA
is
a diagonal matrix 1 -, 1, -, 1, -, 1.
Now I've done several things in those couple of lines.
The first thing is, I've just pulled this definition of scale
part out of the hat. Really, you know, minus A1B1 and so on. And
you think those might seem completely arbitrary. But if you
think back to the definition of the scale of the invariant
interval squared,
that's fundamental for those maintains come from and we'll
see the link again in a moment. So, So that's a bit of a I've
stopped that out of the hat. We shouldn't be too much of a
surprise.
The next line is just that that that same thing written in.
In metric terms
we have introduced this 4 by 4 diagonal matrix ETA
where the central the diagonal 1 -, 1 mitral -1 and everything
else zero. And you can and if you start it a bit you'll see
that if you form that matrix product with that and the and
the components of the matrix E then you'll get that term back
again. Why do I do this other than to make it look
complicated? I don't do it to make it look complicated. I do
it because this matrix eater
is terrifically important.
If you remember in the in the
discussion of the scale of product in in in 3D space, the
scale of production 2 vectors is.
It tells you it includes information about how long the
vectors are and what the angle is between them and so on. It's
AB Cos Theta as you recall.
If you take the the the the scale of product of a vector
with itself,
that gives you the length of the vector.
If you recall, because of course cost zero is 1 and a a a Cos
Theta is going to be a ^2. So the the the the dot product. The
scalar product of a vector with itself gives the length of a
vector.
And here if I ask what is the
a scalar product of of the vector E with itself,
that's going to be a not a naught minus A1, A 1 -, a two A
2 -, a three A3.
And if we talk about the
the displacement vector R
which is equal to Delta T, delta Y&X, delta Y
doctor Z. So that's just the displacement from this event to
this event
and ask what is
our dot
dot R?
We see that it's just a t ^2 minus Delta X ^2 minus Delta y
^2 minus
Doctor Seth Green.
So we've recover the invariant interval
from this definition of the scalar product.
That's. So it's it's not really pulled out of a hat.
Was not pulled out of the hat. But it's entirely reasonable
because we end up with the right answer
that's telling us that if that displacement vector doesn't re
goes from this event of that event,
the length of that vector is the invariant interval between those
the the the delta squared that we found in two chapters ago. So
that's why this
matrix here meta is important because it recovers this
and that's why that matrix there
is encoding the definition of length in this space.
A hand up.
What does that say?
After
or diag.
Lots of tea,
yeah.
So what do you mean
to the
what you transpose? Etab
yeah yeah sorry. He transpose yes because you've both got a
column vectors. So so if you think of the the matrix product
of the row vector matrix, column vector.
So this is the in the same way that the
scalar product of
a free vector a with itself gives you a square plus a
naughty squared.
So that should be so that that that that would be a dot B
equals A1B1
plus A2B2 plus A
BB3 as well.
So there the scalar product of a vector with itself would be a 1
^2 plus A 2 ^2 + a ^2 or X ^2 + y ^2 + Z ^2, which gives the
length of a vector code Pythagoras theorem
when you
the scale.
Or why the minus signs? Why the minus signs here? Yeah, that's
that's the bit I pulled out of a hat. But it makes sense because
we end up if we do the same thing with the displacement
vector dot R which is this separation between two events.
We get this thing here
which you recognise
at the invariant interval.
So those minus signs are there so that we get the right answer.
Right answer here.
In other words, that matrix ETA,
now four matrix heater push a diagonal is telling us about the
geometry of Minkowski space. It's all encoded in that matrix.
That matrix is called the metric
metric, as an metro and as in metric system. But it means
measurement. The metric. It's what defines distance in this
space and in special activity. It's always that
in general relativity,
where generally comes in
or is that we're we're not talking about special activity
anymore. We're talking about things which are moving and are
accelerating, or which are moving on the influence of
gravity. And the effect of that, it turns out,
is the metric. In a curved space-time is curved because the
metric is different from that diagonal.
So this is the first time you've you you'll you'll see this that
that that that that this method but that stays with the whole
study of relativity all the way up to the
the key part of general activity. The Einstein's
equations for general activity are basically what is the metric
given that there are matters here and here, that that, that,
that, that, that, that's that's what solving Einstein
ingredients means.
But
in special activity it all with that, because the geometry of
Minkowski space of
of speaking, moving, moving at constant speed is nice and
simple.
I said it's always that sometimes it will be different
from that because some textbooks choose
this to be minus, plus, plus, plus,
and that and both, in which case the metric is minus, diagonal,
minus, plus, plus, plus, and both are reasonable.
I think that there's one that's more common in the ecology, one
that's more common in particle physics. I don't know which ones
which, but there's just our conventions in different areas.
So if you're looking at another textbook on relativity,
be careful and justice check what the signs in this
expression are or in this expression here are, because
they might be exactly the other way around
and that caused a couple of things elsewhere to change. But
it's not fundamentally. It's not fundamentally important
now to keep a good place to pause briefly have any questions
about
or happy with that or at least
unhappy about it. To happy extent
content question. Ohh, excellent. Well look, other
questions could change the metric to be -1. Yeah. Does that
mean that
like the way
derive the invariant interval forward that will change the
settings?
It would be because there will be another - somewhere else in
that so. So at one point in that definition.
I can't remember exactly how it would be different, but there
will be another minor saying somewhere in in in chapter 4
and I like this way of doing it for this class because this way
proper time is a positive.
So so if if two things are
two events which at the same place and and and and the simply
in different times. In other words, with is a watch observer
at both events that invent interval between those is the
proper time property means and it's positive
it. With this definition, which is. I think it's nice, yeah, but
but but but yes, so there's an arbitrariness in here. We all
have to be consistent, but it is basically arbitrary.
OK,
umm
and.
I think I have
alright. Yes,
today.
OK.
Great question.
And
the history of vectors
what's the put hands up here but have I think I'll give you I'll
come to think what's E dot B and what's the magnitude of and the
magnitude of just in your in your heads what's the
have opposed and talk to your neighbour and work out what it
would be is alright
right
yeah
quick quick
you know we.
OK,
having discussed that until over that a bit,
any books
and and any what I want, I want to show what a dot B would be
zero. Yeah, one 1 * 2 - 2 * 1
a. What's the length of a?
What are you there?
No, no, no Route 5.
It would be 1 ^2 -, 2 ^2.
The remaining three would be the but the length squared of that
and similarly
the other one would be 2 ^2 -, 1 ^2.
So what that shows is that those two vectors A&B 12002100 are
orthogonal to each other.
Now look orthogonal to each other as you as you visualise
them on on the on the plane. But to the extent that orthogonal
means that the scale of product is 0, those two vectors are
orthogonal to each other
and and and So in the geometry of of of Minkowski space, those
two vectors are are at right angles. We don't know.
Orthogonal right angles is the wrong word to use. They're
orthogonal.
The length of V, the length squared of east is negative. So
picking up the terminology from the end of chapter four, we call
that a space like vector because its length is length squared is
negative
be B is positive. It's something that is pointing along roughly
along the time axis. We call that a time like vector.
OK, so that's about terminology, which is just picking up the
terminology you used a couple of chapters ago.
The other thing I will mention, because oddly enough I don't
have it the notes is that
and this matrix
Gamma, Gamma V
Gamma V Gamma
that we used to transform from one stream to another. We
universally live with that Lambda
and and I I'll I'll use that term happily from no one
negative orientation in space.
Ohh, right. And orientation to the negative sign given
orientation of space
right. There's a mathematical question. I think this is a an
orientable space in the in the sense that if you move around
it, it's not a movie type of space. Is what you meant
and
it it it's all right. I think you both what you meant. Does he
give an orientation to space? I think sort of in the sense that
if you remember the
the that diagram with the light cone that indicates there's a
direction in. It's based in the sense that there's a direction
into the future
if you like. The things along the time axis stay along the
time axis of any frame that you move into. So they are always in
that direction. So that's a sort of directionality to space.
Think about orientation is a of a space is something that is
done in topology. I don't know, which I think is probably not
the question you're asking, but I think this is an oriental
space. But not. I'll come back in a moment, Is that does that
sort of make sense?
Yeah. See, you're already asked to write up and. Would you write
it as that matrix or would you write it as like the
gamma minus gamma? Or does it just depend on the situation
that you've got? Alright, and I think it depends on the
situation because there this is is going from a metric space to
another one in standard configuration. I think that you
know those movements positive V
so the inverse matrix to that.
If you if you were to to look at that and and and invert it then
you would get a a - Here and here we would make sense because
from the point of view of the prime frame the unprimed frame
is moving in the negative direction. So you can you can
get that inverse either by thinking of that way, that way
or minus V or you can take the matrix inverse of that and
discover that ministry.
So that transformation matrix Lambda
is uh.
Well, I I will use that notation to refer to it. And it has the
property that if we,
uh,
yeah,
this, you know, motivated. But if we were to
you, you'll have done matrix transformations at some point.
You know how to get from this vector to that vector using
transformation matrix?
Possibly, possibly not
Have the property that if you
to the matrix product of
the this diagonal matrix ETA and transform that matrix matrix ETA
into
the other frame, what you get is
eta
through the this metric transformed into the other frame
transformed into itself. In other words, the metrics, the
metric is frame independent. In other words, the would be an
interval is framed dependent. In other words, the separation
between two events at either end of a displacement vector is
Freeman variant that's not dependent on the even though the
components of those vectors will change. That's also very
important point. My question there,
well the the the gamma, the gamma is are the gamma that's
one and one. The gammas are just the Gammas that we let of
Yeah, that's gonna be, that's gonna be as well, yeah. Sorry,
00.
So it's it's block diagonal
and you know this bit diagonal. So the wine, the this is just
that me scribbling out the matrix that's in green 63-B.
For some reason I didn't seem to have pulled out that that, that.
That's the point. We were label that as Lambda.
Good. We're stepping through this,
so
there's a lot going on in this section.
I've seemed to just
see vectors exist
and this may each exists and that it's important, but that's
quite a lot. Yeah, there's a lot going on in this section because
one could talk about the maths in the section quite a lot
because it's talking about symmetry, it's talking about
invariance, very important things in maths, and some of the
maths underlie generativity. So we could go on about this for
quite a long time. So this is a bit of mathematical technology
and sort of dumping on you, which you'll get familiar with
in the electorate to come. But it is a weirdly crucial in the
sense that if you, if you study more of relativity, you stay
with those things again and again and again, indefinitely.
More or less.
Good,
right? Sorry,
I think that there is not much more to say about that. Good.
OK, so we have therefore the displacement vector.
Delta R is delta T, Delta X, Delta Y.
What is it?
And we also have thing to have to play with.
Well, I think we have to play with. We have the proper time
and remember the proper time is the time between two events in
A-frame in which those two events are the same place. In
other words, in other words, where the the spatial separation
between those two events is 0,
So what we can. So if
that displacement vector got R is like that,
we can talk about
Dr which is an infinitesimal version of that
the
Deez
and we'll rate that just by that we as DX
mute
but rotation here. So
in the same way that you you, you might see a a vector with
components 123 whatever. This impressive vector has components
and so,
and I'll write that as
DX0DX1DX2DX3.
And the convention is that when I use a Greek index here,
I'm I'm I'm using it to refer to index 0123. When I use a Latin
index through GK, wherever I'm using to refer to indexes 123.
So it's just the spatial indexes. That's a standard
convention. So we have this,
if it's impossible displacement, we can divide it
yeah
by the
the proper time detour
and get
DX0 by D Tau,
the X1 by
tall the X tube ID tall DX, D by D Tau,
right?
And we define that to be the
good,
the
the, the, the four velocity.
You
so remember that if R is a vector, then so will DRB. It has
the same properties. Tower is just a number, so D tall. We're
just a number. So those four things will behave as a vector
still. So therefore this thing behaves as the prototype dispute
vector does. Therefore it's a vector.
OK.
And we can do the same thing with
the for mentum
for acceleration rather
D2
ohh by D
tall squared
which is equal to
D2X naughty by D2 squared,
et cetera.
And we will write things like you MU is equal to DX MU
but he told me.
But that's just I can walk communication for the thing at
the top.
OK,
so let's look at the components in a bit more detail.
Yeah, this is what I'm going to follow my notes quite closely,
taking the art with the infinitesimal displacement of a
particle. So just moving through space and time
in A-frame in which the particle is not moving. So in the
particles rest frame,
the
just it it's in. In that frame, it's physical displacement.
Spatial displacement will be 0, so DX will be 0
and it's so the length of that discrete vector will be
Dr Dot Dr equals.
That are tough squared because that's just the the length of
the displacement in time in our frame in which the particle has
no displacement in space S that's do you DT squared
minus
the X ^2 where the equity is 0 and that the definition of the
property.
So that means that the thing of this particle that has a
displacement in physical displacement,
that displacement in I think will be the TD t ^2 minus DX
squared. And I'm going to just now confine this to movement in
the X direction and to press Y&Z that DT squared minus DX
squared
will be equal to D Tau squared.
No. If we
and
divide both sides of that by three t ^2, we get
You talk squared by DT. Squared is equal to 1 -, X ^2 by D
t ^2
and and notation is getting a bit confusing here
and we just clarify it with that,
so that that. But that's equal to 1 minus DX by DT squared, and
any pure mathematicians can look away just now which is
1 -, v ^2,
which is one over
gamma squared.
Or in other words, DT
by D Tau
is equal to gamma.
Or in other words, the rate of change at which the clock
I wish the the time coordinate in the frame of which the
particle is moving
with respect to the
time coordinate on the particles own clock
is equal to the time dilation factor. So that's another
expression of time dilation there.
Quite abstract one, but that's that's just time relation
happening there again.
So the difference between the coordinate time
in what I think was particles moving and the coordinate time
in A-frame of which the particle is not moving
is that
um
And. And what that tells us, turning the handle a bit, is
that you're not,
which is good. X naughty by D tall, which is DT by D Tau is
equal to gamma
and uh
you I
which is DX I. By
the tour
it ends up being gamma
VI and you see I've switched I'm I'm consistent with this
notation I've I've used MU as the index for something which
goes from zero to to 30123, and I've used I Latin index for the
spatial bits. With index goes 123 and so on.
So those are the components of the velocity vector
expect activity for a particle moving at speed V.
There's the components in our frame of of of the philosophy
vector of a particle which is moving with respect to us at
speed V
Umm.
And we could also write that out
as
gamma gamma VX,
Gamma BY gamma
VZ,
which is gamma 1V XVY,
he said.
And it looks like slangy. We will also write that as gamma
one
an underlying.
So this line here is just rewriting this line
make it look like a vector.
This line is just taking the gamma out. The common factor
gamma just number out, and this is just a slightly slangy
rewriting of that to indicate that this is still a four vector
where the three special vectors of written as a vector.
OK, that's good. No, we're making
you were going to OK, I'm going to get drink. One more thing
done
and what we stop our pick up next time in this in this is a a
good bit. In the frame in which
the particle is not moving,
the velocity is 0,
obviously.
So in that frame
you will be equal to gamma will be one, because if V is zero, we
won
zero
and in that frame
the length of the velocity vector,
he says. It's simple. It's one, It's one thing one.
So you don't. You
is equal to 1, so the length of the velocity vector in the frame
of which particle is moving is 1.
But remember that the
length of a vector,
the and the metric are framing variant. And what that means is
that the length of the of the velocity vector
it's invariant
and with the same in all frames. So in any other frame it was
part of moving, the length of the velocity vector in that
frame with those components in that frame will also be one,
so the velocity vector. So you're used to that in 3D space.
Used to the idea that something was moving faster. Has a bigger
velocity vector. It's the velocity vector is longer, V is
longer.
In this context we see that's not the case. The velocity
vector is always the same length,
but in the frame in which the particle is not moving, the
velocity vector is pointing directly along the T axis.
In another frame,
the velocity vector is pointing in a different direction. It's
the same length
and and in these terms, but it's pointing a different direction.
So when you see something moving at roughly 6 speed,
what you're seeing is it's
velocity vector, foreshortened if you like, and it ends up
looking a different shape,
but it's still the same length
and that's a natural place to stop. It will pick up that field