- #1
RyanH42
- 398
- 16
I want to calculate two things (This is not a homework question so I am posting here or actually I don't have homework like this)
First question is finding universe volume using spacetime metric approach.The second thing is find a smallest volume of a spacetime metric (related to plank distances I guess ).
I looked wikipedia and there said the volume of a matric equation is ##vol_g=√(detg)dx^0∧dx^1∧dx^2∧dx^3## (In local coordinates ##x^μ## of a manifold, the volume can be written like this)
##g=diag(-c^2,1,1,1)## the metric will be ##ds^2=-c^2dt^2+dx^2+dy^2+dz^2##.
The determinant of the flat spacetime metric is , ##detg=c^2## and then ##√(detg)=c##
The volume will be ##vol_g=cdx^0∧dx^1∧dx^2∧dx^3##
Let's start with my first question.How can we calculate universe volume using spacetime metric ?
##dx^0∧dx^1∧dx^2∧dx^3## this part is complicated for me.I want to make a assumption.In my first question (Universe volume)
##dx^0=(13.9,0,0,0)##
##dx^1=(0,46,0,0)##
##dx^2=(0,0,46,0)##
##dx^3=(0,0,0,46)##
Now the exterior product is complicated.If my assumption is true then How I can calculate the other exterior product ? If its not true what are the true numbers. Here the units are light year for distance and year for time.
And Also I want to make a assumption about my second question(The smallest volume of spacetime metric)
Here again ##vol_g=cdx^0∧dx^1∧dx^2∧dx^3##
##dx^0=(5.39 10E(-44),0,0,0)##
##dx^1=(0,1.61 10E(-35),0,0)##
##dx^2=(0,0,1.61 10E(-35),0)##
##dx^3=(0,0,0,1.61 10E(-35))##
here 5.39 10E(-44)=planck time (second) and 1.61 10E(-35)=planck length (meter)
First question is finding universe volume using spacetime metric approach.The second thing is find a smallest volume of a spacetime metric (related to plank distances I guess ).
I looked wikipedia and there said the volume of a matric equation is ##vol_g=√(detg)dx^0∧dx^1∧dx^2∧dx^3## (In local coordinates ##x^μ## of a manifold, the volume can be written like this)
##g=diag(-c^2,1,1,1)## the metric will be ##ds^2=-c^2dt^2+dx^2+dy^2+dz^2##.
The determinant of the flat spacetime metric is , ##detg=c^2## and then ##√(detg)=c##
The volume will be ##vol_g=cdx^0∧dx^1∧dx^2∧dx^3##
Let's start with my first question.How can we calculate universe volume using spacetime metric ?
##dx^0∧dx^1∧dx^2∧dx^3## this part is complicated for me.I want to make a assumption.In my first question (Universe volume)
##dx^0=(13.9,0,0,0)##
##dx^1=(0,46,0,0)##
##dx^2=(0,0,46,0)##
##dx^3=(0,0,0,46)##
Now the exterior product is complicated.If my assumption is true then How I can calculate the other exterior product ? If its not true what are the true numbers. Here the units are light year for distance and year for time.
And Also I want to make a assumption about my second question(The smallest volume of spacetime metric)
Here again ##vol_g=cdx^0∧dx^1∧dx^2∧dx^3##
##dx^0=(5.39 10E(-44),0,0,0)##
##dx^1=(0,1.61 10E(-35),0,0)##
##dx^2=(0,0,1.61 10E(-35),0)##
##dx^3=(0,0,0,1.61 10E(-35))##
here 5.39 10E(-44)=planck time (second) and 1.61 10E(-35)=planck length (meter)