Greg Egan
Nov4-06, 03:39 PM
In article <1160061191.526136.270010@c28g2000cwb.googlegroups. com>, "km"
<mailkaran@gmail.com> wrote:
> I am reading sean carrol and cannot understand the definition of
> tangent vectors on manifolds. Could somebody please eexplain?
>
> Thanks
> KM
I don't have the book you're referring to, but I can tell you what a
tangent vector on a manifold is.
First start with the idea of a parameterised path. Imagine a function
that takes some interval of the real numbers into a curve on your
manifold which passes through the point P, g:[a,c]->M, with g(b)=P for
some a<b<c.
Now suppose that you have a real-valued function f defined on some
neighbourhood N of P, f:N->R. You can compose the functions g and f to
get a function h that takes you from [a,c] to R:
h:[a,c]->R, h(t) = f(g(t))
Here's an example in the real world. Suppose you have a railway line
that passes through Grand Central Station in New York. As a particular
train travels along that railway line, its varying position on the
surface of the Earth gives you a map g from some interval of time [a,c]
(measured in seconds from midday yesterday, perhaps) to points on the
manifold S^2, the two-dimensional sphere.
Now, the height above sea level around Grand Central station is a
real-valued function f defined on a neigbourhood N of P (P being the
station). So we can find the height of the train above sea level at any
given moment by h(t) = f(g(t)).
Associated with the map g and the point P is a tangent vector v defined
as an operation on all real-valued functions f in the neighbourhood of P:
v(f) = df(g(t))/dt |_{t=b, where g(b)=P}
In our real-world example, v(f) for our choices of g and f gives us the
rate of change with time of the height above sea level of a certain train
that passes through P.
Now if we imagine all possible railways lines that could pass through
Grand Central Station, and all possible speeds at which trains could
travel along those lines, we will encompass all the possible ways in
which real-valued functions defined in the neighbourhood of the station
could change with time. But not every variation in railway line and
journey speed will give a different result at P: if two railway lines
are tangent to each other at P, and the train has the same speed as it
moves along them, then any function f will change with time in the same
way for those two trains.
The space of parameterised paths through P, where we treat two paths as
equivalent if they're the same in that sense, is one way to define what's
known as the tangent space at P. This is a vector space of differential
operators on a neighbourhood of P. You can add these vectors and
multiply them by scalars in the obvious way:
(alpha v + beta w)(f) = alpha v(f) + beta w(f)
If you have coordinate functions {x^i} defined on your manifold M, then
you can assign coordinates to a vector v:
v^i = v(x^i) = dx^i/dt (evaluated at the t that maps to P)
For example, if you have latitude and longitude as coordinates around
Grand Central Station, then the coordinates of the tangent vector for our
train will be the rate of change with time of the train's latitude and
longitude.
In general relativity, the simplest and most important tangent vector
you'll ever learn about is an object's 4-velocity. If you consider the
world line of an object through spacetime, and you parameterise that
curve with the proper time tau measured by a clock carried along with the
object, then the tangent vector at some event P is just the operation of
taking the rate of change with respect to tau of any function defined on
the spacetime around P. This tangent vector, the object's 4-velocity, is
often written as "@_tau" (where "@" is meant to be a partial derivative
curly d), because when you apply an object's 4-velocity to a function,
you're just taking the partial derivative of that function with respect
to tau.
Ditto for other curve parameters. If you have coordinates {x^i} on your
manifold, you can get a basis of the tangent vectors at P from the
coordinate vectors {@_x^i}, defined by:
@_x^i(f) = @f/@x^i
and obviously if you apply a coordinate vector to a coordinate function:
@_x^i(x^j) = {1 if i=j
{0 otherwise
You can also expand a given vector v in terms of this basis, and the
coordinates of v as we previously defined them:
v = sum over i of v(x^i) @_x^i
because
v(f) = sum over i of v(x^i) @f/@x^i
= sum over i of dx^i/dt @f/@x^i
where t is the parameter along a suitable curve.
Hope this helps.
<mailkaran@gmail.com> wrote:
> I am reading sean carrol and cannot understand the definition of
> tangent vectors on manifolds. Could somebody please eexplain?
>
> Thanks
> KM
I don't have the book you're referring to, but I can tell you what a
tangent vector on a manifold is.
First start with the idea of a parameterised path. Imagine a function
that takes some interval of the real numbers into a curve on your
manifold which passes through the point P, g:[a,c]->M, with g(b)=P for
some a<b<c.
Now suppose that you have a real-valued function f defined on some
neighbourhood N of P, f:N->R. You can compose the functions g and f to
get a function h that takes you from [a,c] to R:
h:[a,c]->R, h(t) = f(g(t))
Here's an example in the real world. Suppose you have a railway line
that passes through Grand Central Station in New York. As a particular
train travels along that railway line, its varying position on the
surface of the Earth gives you a map g from some interval of time [a,c]
(measured in seconds from midday yesterday, perhaps) to points on the
manifold S^2, the two-dimensional sphere.
Now, the height above sea level around Grand Central station is a
real-valued function f defined on a neigbourhood N of P (P being the
station). So we can find the height of the train above sea level at any
given moment by h(t) = f(g(t)).
Associated with the map g and the point P is a tangent vector v defined
as an operation on all real-valued functions f in the neighbourhood of P:
v(f) = df(g(t))/dt |_{t=b, where g(b)=P}
In our real-world example, v(f) for our choices of g and f gives us the
rate of change with time of the height above sea level of a certain train
that passes through P.
Now if we imagine all possible railways lines that could pass through
Grand Central Station, and all possible speeds at which trains could
travel along those lines, we will encompass all the possible ways in
which real-valued functions defined in the neighbourhood of the station
could change with time. But not every variation in railway line and
journey speed will give a different result at P: if two railway lines
are tangent to each other at P, and the train has the same speed as it
moves along them, then any function f will change with time in the same
way for those two trains.
The space of parameterised paths through P, where we treat two paths as
equivalent if they're the same in that sense, is one way to define what's
known as the tangent space at P. This is a vector space of differential
operators on a neighbourhood of P. You can add these vectors and
multiply them by scalars in the obvious way:
(alpha v + beta w)(f) = alpha v(f) + beta w(f)
If you have coordinate functions {x^i} defined on your manifold M, then
you can assign coordinates to a vector v:
v^i = v(x^i) = dx^i/dt (evaluated at the t that maps to P)
For example, if you have latitude and longitude as coordinates around
Grand Central Station, then the coordinates of the tangent vector for our
train will be the rate of change with time of the train's latitude and
longitude.
In general relativity, the simplest and most important tangent vector
you'll ever learn about is an object's 4-velocity. If you consider the
world line of an object through spacetime, and you parameterise that
curve with the proper time tau measured by a clock carried along with the
object, then the tangent vector at some event P is just the operation of
taking the rate of change with respect to tau of any function defined on
the spacetime around P. This tangent vector, the object's 4-velocity, is
often written as "@_tau" (where "@" is meant to be a partial derivative
curly d), because when you apply an object's 4-velocity to a function,
you're just taking the partial derivative of that function with respect
to tau.
Ditto for other curve parameters. If you have coordinates {x^i} on your
manifold, you can get a basis of the tangent vectors at P from the
coordinate vectors {@_x^i}, defined by:
@_x^i(f) = @f/@x^i
and obviously if you apply a coordinate vector to a coordinate function:
@_x^i(x^j) = {1 if i=j
{0 otherwise
You can also expand a given vector v in terms of this basis, and the
coordinates of v as we previously defined them:
v = sum over i of v(x^i) @_x^i
because
v(f) = sum over i of v(x^i) @f/@x^i
= sum over i of dx^i/dt @f/@x^i
where t is the parameter along a suitable curve.
Hope this helps.