Re: unit 4-vectors

["RAUBER, JOEL" <JOEL_RAUBER@SDSTATE.EDU>]

This is simply a nomenclature quibble post.

When I studied SR, we happily referred to the "unitoid" in the time-like
direction as a unit vector or often as "the unit_time_like vector".  I think
its easier to simply generalize the meaning of unit vector to mean any
member of the basis set whose inner product with itself is +- one, rather
than to invent a new name.  Essentially, in most, if not all, respects one
treats e_t (see below) like one treats "ordinary" unit vectors, taking into
proper account your metric signature.

Which is to say that it is nice to have the same name referring to any of
the elements in the basis set
{e_t,e_1,e_2,e_3} in the linear expansion,

V = V_t*e_t + V_1*e_1 + V_2*e_2 + V_3*e_3

Why not "unit vector" or "unit basis vector"?

I'm in good company, e.g. MTW (Misner, Thorne and Wheeler) refer to the
above 4 vectors as "basis vectors of unit length"; (fig. 2.3 page 52)

{Note: Obviously they've generalized the idea of length to include negative
real numbers for the case of indefinite signature metrics}

So while the math is the same, I'd answer the original question as follows:

> Is there such a thing as a unit 4-vector?

Yes

Are there time-like unit vectors?

Yes.

(N.b. , in SR the inner product of a vector with itself may or may not be
positive; hence unit vectors do not necessarily have a norm that is +1, but
may have a norm of -1.  This is in contra-distinction with the situation in
ordinary euclidean 3-dim space.)

Joel R.

I'll let John discuss the oddity that this generalizes to with "unit" null
vectors.  I'll bet he prefers to call these "unitoid" as well.

-----Original Message-----
From: John S. Denker [mailto:jsd@AV8N.COM]
Sent: Friday, November 28, 2003 10:25 PM
To: PHYS-L@lists.nau.edu
Subject: Re: unit 4-vectors


On 11/28/2003 09:47 PM, Joe Heafner wrote:
> I'm thinking particularly
> with the context of special relativity.
...
> Is there such a thing as a unit 4-vector?

Yes.

There are not, however, any _timelike_ unit vectors,
which is probably what the question was driving at.

But not to worry.  In the timelike direction we
can define "unitoid" vectors, which play the
role corresponding to the bog-standard spacelike
unit vectors.

Statement 1:
The entire machinery of vector space -- including
vectors, dot products, projection operators,
basis vectors, and components -- goes over almost
unchanged from D=3 to D=1+3.

I say "almost" because in SR the fourth dimension
(time) picks up a minus sign in the definition of
the dot product.  That's what makes a four-vector
in SR (Minkowski space) different from what you
would have guessed if you blithely generalized
from D=3 Euclidean space to D=4 Euclidean space.

Forming a unit vector in any spatial direction
is completely routine: take any spacelike vector
v and divide by its norm |v| and you've got a unit
vector.

For any timelike vector w, you're stuck.  By
definition of timelike, any timelike vector
corresponds to a negative interval w dot w,
and no unit vector can have that property.

I hereby define a _unitoid_ vector to be one
with the property
        w dot w = +- 1  (plus or minus)

[I know the word is an execrable mongrelization
of Latin and Greek roots.]

Unit vectors are a subset of the unitoid vectors.

If/when one chooses a basis in SR, the sensible
thing is to choose three orthogonal spacelike
unit vectors and one timelike unitoid vector:
   e_t  e_x  e_y  e_z

Any four-vector can be expressed as a linear
combination of these four basis vectors.

So SR doesn't have a timelike unit vector, and
it doesn't need one.  It has something else
instead.

=====================

There is a converse to statement 1 that is even
stronger than statement 1 itself:

Statement 2:
Starting from SR, there is a subspace consisting
of the three spatial directions.  This is strictly
and exactly a subspace, no "almost" about it,
which means that anything that happens in this
subspace can be described in exactly the same
terms using the SR machinery and/or the D=3
Euclidean machinery.