Definition of Minkowski spacetime

[ygor.geurts@gmail.com]

Hi,

I'm trying to get the correct definition of Minkowski spacetime. I
have basically five different (looking) definitions and I'm wondering
which one is correct or whether they are equivalent perhaps?

1. a d-dimensional vector space with a lorentz, ie (d-1,1) signature,
inner product.

2. an affine space (ie. principal homogeneous space) for a lorentz
inner product vector space.

3. a (simply-)connected flat lorentz manifold.

4. an abelian lie group whose lie algebra is a lorentz inner product
vector space.

5. the quotient of the poincare group by the lorentz group

Then in all of the above definitions, sometimes people dont just
demand a lorentz metric, but explicitely demand the explicit minkowski
metric diag(-1,1,...,1). Does that make sense? Doesnt that mean that
for example one is not allowed to change to for example polar
coordinates to describe the physical situation, since this changes the
metric components?

So, how are these definitions related? I would say the affine space
definition is more general than the vector space definition, so then I
wonder how the affine space definition is related to the manifold one?
Is the affine space also a manifold?

Also how is the poincare group definided in this context? I have seen
it defined as the isometry group of Minkowski space, but how can i see
that this then equals V x' O(d-1,1) (or does this only hold in the
case of the minkowsi metric) ?

I hope anyone could help me out,

Ygor

Now my question is,

[Chalky]

On Jun 18, 5:40 pm, ygor.geu...@gmail.com wrote:
> Hi,
>
> I'm trying to get the correct definition of Minkowski spacetime. I
> have basically five different (looking) definitions and I'm wondering
> which one is correct or whether they are equivalent perhaps?
>
> 1. a d-dimensional vector space with a lorentz, ie (d-1,1) signature,
> inner product.
>
> 2. an affine space (ie. principal homogeneous space) for a lorentz
> inner product vector space.
>
> 3. a (simply-)connected flat lorentz manifold.
>
> 4. an abelian lie group whose lie algebra is a lorentz inner product
> vector space.
>
> 5. the quotient of the poincare group by the lorentz group
>
> Then in all of the above definitions, sometimes people dont just
> demand a lorentz metric, but explicitely demand the explicit minkowski
> metric diag(-1,1,...,1). Does that make sense? Doesnt that mean that
> for example one is not allowed to change to for example polar
> coordinates to describe the physical situation, since this changes the
> metric components?
>
> So, how are these definitions related? I would say the affine space
> definition is more general than the vector space definition, so then I
> wonder how the affine space definition is related to the manifold one?
> Is the affine space also a manifold?
>
> Also how is the poincare group definided in this context? I have seen
> it defined as the isometry group of Minkowski space, but how can i see
> that this then equals V x' O(d-1,1) (or does this only hold in the
> case of the minkowsi metric) ?
>
> I hope anyone could help me out,
>
> Ygor
>
> Now my question is,

Wow, why are all these definitions so scary to the uninitiated?

When I was an undergrad, Minkowski spacetime was introduced as simply
Euclidean (i.e. flat) space (often represented in 1 dimension), + time
inicated as an extra dimension with units chosen such that a light ray
propagates at 45 degrees to both the horizontal and vertical axes.
Adding a second dimension of space then gives the familiar 'light
cone'

What is wrong with that definition?

[Igor Khavkine]

On Jun 18, 12:40 pm, ygor.geu...@gmail.com wrote:
> I'm trying to get the correct definition of Minkowski spacetime. I
> have basically five different (looking) definitions and I'm wondering
> which one is correct or whether they are equivalent perhaps?
>
> 1. a d-dimensional vector space with a lorentz, ie (d-1,1) signature,
> inner product.

This is the basic definition.

> 2. an affine space (ie. principal homogeneous space) for a lorentz
> inner product vector space.

This is a theorem that follows from 1. Any vector space is also an
affine space.

> 3. a (simply-)connected flat lorentz manifold.

Umm, I'm not 100% sure this is true. But if you got it from a reliable
source, then it should also be a theorem. It probably uses the fact that
there are topological obstructions to putting a flat metric on a
simply-connected space that is not R^d.

> 4. an abelian lie group whose lie algebra is a lorentz inner product
> vector space.

I think this is necessary, but not sufficient. The d-dimensional flat
torus fits the same description.

> 5. the quotient of the poincare group by the lorentz group

Another theorem.

> Then in all of the above definitions, sometimes people dont just
> demand a lorentz metric, but explicitely demand the explicit minkowski
> metric diag(-1,1,...,1). Does that make sense? Doesnt that mean that
> for example one is not allowed to change to for example polar
> coordinates to describe the physical situation, since this changes the
> metric components?

Another theorem is that a flat metric admits a set of coordinates
(called inertial coordinates) in which the metric has the component
form diag(-1,1,...,1). You are allowed to use any other coordinate
system you like, but these coordinate systems are preferred in a
certain sense.

> Also how is the poincare group definided in this context? I have seen
> it defined as the isometry group of Minkowski space, but how can i see
> that this then equals V x' O(d-1,1) (or does this only hold in the
> case of the minkowsi metric) ?

That's another theorem. The Poincare group is defined precisely as you
stated (the group of isometries of Minkowski space). Proving that any
Minkowski isometry is an affine transformation should not be too
difficult. Afterward, the semidirect product decomposition
V x' O(d-1,1) falls out quite naturally.

Hope this helps.

Igor

[Oh No]

Thus spake ygor.geurts@gmail.com
>Hi,
>
>I'm trying to get the correct definition of Minkowski spacetime. I
>have basically five different (looking) definitions and I'm wondering
>which one is correct or whether they are equivalent perhaps?
>
>1. a d-dimensional vector space with a lorentz, ie (d-1,1) signature,
>inner product.

It is sensible to use a basic definition, then treat the others as
theorems, as Igor said on s.p.r.

Regards

--
Charles Francis
moderator sci.physics.foundations.
charles (dot) e (dot) h (dot) francis (at) googlemail.com (remove spaces and
braces)

http://www.teleconnection.info/rqg/MainIndex

[illywhacker]

On Jun 19, 4:51=A0am, Igor Khavkine <igor...@gmail.com> wrote:
> On Jun 18, 12:40 pm, ygor.geu...@gmail.com wrote:
>
> > I'm trying to get the correct definition of Minkowski spacetime. I
> > have basically five different (looking) definitions and I'm wondering
> > which one is correct or whether they are equivalent perhaps?
>
> > 1. a d-dimensional vector space with a lorentz, ie (d-1,1) signature,
> > inner product.
>
> This is the basic definition.

Doesn't a vector space have a distinguished point, i.e. 0? It seems
that 2 is more like the basic definition, which is basically the same
as 5.

illywhacker;

[Pmb]

"Chalky" <chalkyspam@bleachboys.co.uk> wrote in message
news:deb34b68-0005-4abb-984c-8c630363be58@26g2000hsk.googlegroups.com...
> On Jun 18, 5:40 pm, ygor.geu...@gmail.com wrote:
>> Hi,
>>
>> I'm trying to get the correct definition of Minkowski spacetime. I
>> have basically five different (looking) definitions and I'm wondering
>> which one is correct or whether they are equivalent perhaps?
>>
>> 1. a d-dimensional vector space with a lorentz, ie (d-1,1) signature,
>> inner product.
>>
>> 2. an affine space (ie. principal homogeneous space) for a lorentz
>> inner product vector space.
>>
>> 3. a (simply-)connected flat lorentz manifold.
>>
>> 4. an abelian lie group whose lie algebra is a lorentz inner product
>> vector space.
>>
>> 5. the quotient of the poincare group by the lorentz group
>>
>> Then in all of the above definitions, sometimes people dont just
>> demand a lorentz metric, but explicitely demand the explicit minkowski
>> metric diag(-1,1,...,1). Does that make sense? Doesnt that mean that
>> for example one is not allowed to change to for example polar
>> coordinates to describe the physical situation, since this changes the
>> metric components?
>>
>> So, how are these definitions related? I would say the affine space
>> definition is more general than the vector space definition, so then I
>> wonder how the affine space definition is related to the manifold one?
>> Is the affine space also a manifold?
>>
>> Also how is the poincare group definided in this context? I have seen
>> it defined as the isometry group of Minkowski space, but how can i see
>> that this then equals V x' O(d-1,1) (or does this only hold in the
>> case of the minkowsi metric) ?
>>
>> I hope anyone could help me out,
>>
>> Ygor
>>
>> Now my question is,
>
> Wow, why are all these definitions so scary to the uninitiated?
>
> When I was an undergrad, Minkowski spacetime was introduced as simply
> Euclidean (i.e. flat) space (often represented in 1 dimension), + time
> inicated as an extra dimension with units chosen such that a light ray
> propagates at 45 degrees to both the horizontal and vertical axes.
> Adding a second dimension of space then gives the familiar 'light
> cone'
>
> What is wrong with that definition?

Its wrong because Minkowski space is not Euclidean. For a space to be
Euclidean it has to have the same metric that Euclidean space does, i.e., in
Cartesian coordinates in 4-space g_00 = g_11 = g_22 = g_33 = 1, while all
other components are zero. However the metric for Minkowski space is g_00
= -g_11 = -g_22 = -g_33 = -1.

More generally a maifold is neither Euclidean or non-Euclidean etc until
more structure is added, i.e. when a metric is defined.

Pete

[Rock Brentwood]

ygor.geu...@gmail.com wrote:
> Hi,
>
> I'm trying to get the correct definition of Minkowski spacetime.
>
> 2. an affine space (ie. principal homogeneous space) for a lorentz
> inner product vector space.
>
> how the affine space definition is related to the manifold one?
> Is the affine space also a manifold?

An affine geometry can actually be represented as an algebra with the
following ternary operation:
[a, r, b] = (1-r)a + rb
and a suitable set of axioms; e.g.,
[a, 0, b] = a; [a, 1, b] = b
[a, rt(1-t), [b, s, c]] = [[a, rt(1-s), b], t, [a, rs(1-t), c]].
Alternatively, one can mimick the standard definition of a vector
space by bringing in "affine addition"
a-b+c,
but this can be defined in terms of the ternary operation by
a-b+c = [[b, 1/(1-r), a], r, [b, 1/r, c]]
for any r other than 0 or 1. The axioms can then be used to prove this
is independent of r.

Alternatively, one can start with addition itself and define the
abelian group structure of addition by the axioms
a - b + c = c - b + a (Abelian)
a - b + b = a = b - b + a (Zero & Identity)
a - b + (c - d + e) = (a - b + c) - d + e.

To answer your question, directly, this construction allows one to
define a vector space "fixed at a point O", defining vector space
operations by
a + b = a - O + b
ra = [O, r, a].
So, if A is the affine space, then the above construction gives you a
vector space A_O.

One can also define a vector space structure intrinsically by the
formal differences of affine points, dA = (A x A)/R, where the formal
difference "a-b" is the equivalence class (a,b) with respect to the
equivalence relation R; and where R is defined by the equivalences
generated by the identity (a-b+c)-d = c-(b-a+d).

The additive group operations are then just (a-b) + (c-d) = (a-b+c)-d
= c-(b-a+d), with 0 vector a-a = b-b (equality can be proven from the
identity above and the axioms); and inverse -a = (a-a)-a.

The resulting vector space delta(A) is identical in structure to each
A_O, with the correspondence given by a-b --> a-b+O, and inverse a -->
a-O.

Hence, each space A_O is a TANGENT SPACE of A, when A is regarded as a
manifold; and each tangent space A_O is identical in structure to the
vector space delta(A).

Thus, not only is A a manifold, but it's a "homogeneous space".

The inner product operation also needs to be "affinized" into a
ternary operation ... I won't discuss this in too much detail here but
leave it as an exercise.