Apparently I just need enough coffee to get it right.
I had forgotten that the Hubble rate which is defined as ##H = a_{,0}/a## also has to be converted to a conformal time derivative. By using ##H = \dot{a}/a^2## where dot-derivative means ##\frac{\partial}{\partial \eta}## I arrived at the...
Homework Statement
Problem in question is problem 5.6 in Dodelson's Modern Cosmology (https://www.amazon.com/dp/0122191412/?tag=pfamazon01-20)
Take the Newtonian limit of Einstein's equations. Combine the time-time equation (5.27) with the time-space equations of exercise 5 to obtain the...
Wow, thanks for the link! It helps to know the transformation law for the Christoffel symbols.
So, by using the fact that ##\Gamma^a_{bc} = \frac{\partial^2 x'^i}{\partial x^b \partial x^c}\frac{\partial x^a}{\partial x'^i}## I was able to find expressions for the two fractions involved...
I now notice that my original metric is can be written as
## \bar{g}_{ab} = \eta_{ab} + h_{ab}## with
##h_{ab} = diag(2\phi,-2\phi,-2\phi,-2\phi)## and that the transformation matrix I am after will satisfy
##\eta_{ab} = A_a^c A_b^d \bar{g}_{cd}##
Thank you for clearing that up. :) I've spent the last 40 minutes or so trying to grasp how to "cleverly use the fact that ##\Gamma^{\mu'}_{\nu'\gamma'}(p) = 0##" but I have yet to see the light. :(
So far all I have is the formula for the Christoffel symbols:
##\Gamma^{a'}_{b'c'} =...
I hate it when I miss something obvious and somehow end up making stuff way more complicated than it needs to be - and I already find relativity quite complicated! :(
Maybe I've stared at the problem for so long that I miss something here, but I have some questions...
Using a Taylor expansion up to and including first order to keep stuff linear I get
##\phi(t,x,y,z) \approx \phi^0 + (t-t_0)\phi^0_{,t} + (x-x_0)\phi^0_{,x} +(y-y_0)\phi^0_{,y} +(z-z_0)\phi^0_{,z}##.
Putting this into the metric I get
##ds^2 \approx (1+2\phi^0)dt^2 -...
Hm, this leaves us with a global transformation transforming the entire of spacetime to Minkowskian which I take it is not the purpose of the problem.
We might have to assume (as was assumed in a previous problem in the set I'm working with but not stated explicitly in this problem) that ##\phi...
Thank you for your reply. :) Swift action in here!
I am not sure I understand your notation, so let me make a quick check to see if I got it right:
{\lambda^a}_\hat{a} : The matrix that transforms coordinates ##x^a## to what I denoted as primed coordinates ##x'^a##?
The middle part of...
I've been working on a problem that I can't seem to get started on. Here is how it is posted:
Metric of a space is:
ds^2 = (1+2\phi^2)dt^2 - (1-2\phi)(dx^2+dy^2+dz^2), where |\phi | << 1 everywhere. Given a point (t_0 , x_0 , y_0, z_0) find a coordinate transformation to a locally...
Homework Statement
Problem as stated: Consider a vector A^a. Is the four-component object \left( \frac{1}{A^0},\frac{1}{A^1},\frac{1}{A^2},\frac{1}{A^3}\right) a tensor?
Homework Equations
Roman indices run from 0 to 3. Einstein summation convention is used.
Tensors of rank 1 (vectors)...
Thank you for your reply.
I am, however, unsure of the interpretation of the vy part of the assignment. The exercise is written as in the OP. In my solution I have assumed vy was K'' with respect to K'.
Of course I could check whether or not ds² is invariant with respect to the double...
Homework Statement
Question as stated: In special relativity consider the following coordinate transformation between inertial frames: first make a velocity boost v_x in the x-direction, then make a velocity boost v_y in the y-direction. 1) Is this a Lorentz transformation? 2) Find the matrix...