Why the time component of the FLWR metric doesn't have a scale factor of its own, analogous to a(t) for space components? In other words, why is only the space component of the space-time expanding and not the whole space-time? Why isn't also the time component expanding?
well if the only thing that changes here is α as a function of t, it doesn't really matter if t is changing, also changing in terms of what? some more absolute time? then maybe you could start talking about how it relates to other fields but so far there doesn't seem to be any effects to that, also would that really affect the stress energy tensor? anyways tl;dr a=a(t) so changing t to something else doesn't change much here, just complicates things
In the previous lecture the author talked about the assumptions of the reference frame used. We picked a frame such that all matters in the universe on average is at rest spatially and only move along the time direction ("Cosmic Rest Frame"), so dx_i = 0, g_tt = -1.
I don't understand if Lamda or Dark energy contributing negatively in that Friedman equation then how it causes the universe's expansion. It is helping gravity in that equation so it should be the cause for the contraction of the universe. please answer!
Vishal, that's a good question...It's true that the matter energy density rho_M and the dark energy density rho_D have the same sign, they're both positive. So one might think that the dark energy would "help" the matter density pull the universe back together. But positive energy (or mass) is not necessarily attractive. In Newtonian gravity, we think of positive mass as being attractive because it produces a gravitational potential with a positive gradient. (The Newtonian potential is V = -M/r so the gradient is positive if M>0. Correspondingly, the force - dV/dr is negative when M>0, which corresponds to an attractive force.) For the Friedmann equation, the potential is proportional to (rho_M + rho_D)*a^2, but rho_M is proportional to 1/a^3 and rho_D is independent of a. So, even though rho_M and rho_D are both positive, the gradient of rho_M*a^2 and the gradient of rho_D*a^2 have opposite signs. The ordinary matter is attractive, but the dark energy is repulsive.
The sign convention "-+++" is used throughout this video series, under which the norm of the 4-velocity is always -1. If the sign convention "+---" is used, then it'll be 1.
Why the time component of the FLWR metric doesn't have a scale factor of its own, analogous to a(t) for space components? In other words, why is only the space component of the space-time expanding and not the whole space-time? Why isn't also the time component expanding?
well if the only thing that changes here is α as a function of t, it doesn't really matter if t is changing, also changing in terms of what? some more absolute time? then maybe you could start talking about how it relates to other fields but so far there doesn't seem to be any effects to that, also would that really affect the stress energy tensor? anyways tl;dr a=a(t) so changing t to something else doesn't change much here, just complicates things
In the previous lecture the author talked about the assumptions of the reference frame used. We picked a frame such that all matters in the universe on average is at rest spatially and only move along the time direction ("Cosmic Rest Frame"), so dx_i = 0, g_tt = -1.
I don't understand if Lamda or Dark energy contributing negatively in that Friedman equation then how it causes the universe's expansion. It is helping gravity in that equation so it should be the cause for the contraction of the universe. please answer!
Vishal, that's a good question...It's true that the matter energy density rho_M and the dark energy density rho_D have the same sign, they're both positive. So one might think that the dark energy would "help" the matter density pull the universe back together. But positive energy (or mass) is not necessarily attractive. In Newtonian gravity, we think of positive mass as being attractive because it produces a gravitational potential with a positive gradient. (The Newtonian potential is V = -M/r so the gradient is positive if M>0. Correspondingly, the force - dV/dr is negative when M>0, which corresponds to an attractive force.) For the Friedmann equation, the potential is proportional to (rho_M + rho_D)*a^2, but rho_M is proportional to 1/a^3 and rho_D is independent of a. So, even though rho_M and rho_D are both positive, the gradient of rho_M*a^2 and the gradient of rho_D*a^2 have opposite signs. The ordinary matter is attractive, but the dark energy is repulsive.
Good Day Sir, in your Video 11:14 stated U_u*g^uv*U_v = - 1......why not 1 could you please explained it?.....thanks for these great Video... 👍👍
The sign convention "-+++" is used throughout this video series, under which the norm of the 4-velocity is always -1. If the sign convention "+---" is used, then it'll be 1.