On Friday, January 12, 2018 at 10:33:29 PM UTC-6, agrays...@gmail.com wrote:
>
>
>
> On Friday, January 12, 2018 at 5:40:27 PM UTC-7, Lawrence Crowell wrote:
>>
>> On Friday, January 12, 2018 at 12:33:51 PM UTC-6, John Clark wrote:
>>>
>>>
>>>
>>> On Fri, Jan 12, 2018 at 11:06 AM, <agrays...@gmail.com> wrote:
>>>
>>> For me, the problem is space vs spacetime. In LIGO, the recombined waves
>>>> of light show offsets due to different path lengths. So this seems to be a
>>>> differential distortion of *space *as the wave passes. So what has
>>>> *time* got to do with the phenomenon? AG
>>>>
>>>
>>>
>>> The gravity wave changes the *time* it takes for light to go down those
>>> different paths, that's how we know the length i
>>> n
>>> *space* must have changed because the one thing that nothing can do,
>>> not even a gravity wave, is change the speed of light in a vacuum.
>>> So its best not to think of space and time as 2 seperate things, there
>>> is just spacetime.
>>>
>>>
>>> John K Clark
>>>
>>
>> The metric components that vary are the spatial parts of the metric. In
>> the weak field limit the metric may be written as g_{ab} = η_{ab} + h_{ab},
>> where η_{ab} is the flat spacetime metric the h_{ab} are the perturbation
>> terms on the flat space metric. The elements h_{11} = h_{22} and h_{12} =
>> h_{21} are the + and x polarization directions of the helicity = 2
>> field-wave. Then for technical reasons one takes the traceless part of this
>> metric and runs it through the Einstein field equations. Since the field is
>> weak these field equations are linear and the wave equation is a standard
>> EM-like wave equation.
>>
>> LC
>>
>
> *I was going to post that since the metric field is a function of space
> and time, and we can speak of space-time, the same can be said of any field
> dependent on space and time, such as the EM field. AG *
>
The difference is that with an ordinary field φ = φ(r,t) a derivativea are
∂φ/∂t, ∂φ/∂r. If φ is a vector field this still holds. However, with
spacetime physics you have the frame a local field operator is on also
being dynamical. Then for a vector field V^μ a derivative ∂V^μ/∂x^ν, for
the Greek indices running over (t, r, y, z) is modified to
∂V^μ/∂x^ν → DV^μ/∂x^ν = ∂V^μ/∂x^ν - Γ^μ_{να}V^α.
This is the covariant derivative that contains the connection term
Γ^μ_{να}involving derivatives of the metric tensor. Since this is from a
covariant derivative of the metric tensor we have
Dg_{μν}/∂x^ρ = ∂g_{μν}/∂x^ρ + Γ^α{μρ}g_{αν} + Γ^α_{νρ}g_{μα }.
The covariant derivative of the metric is zero, this is a consequence of
the equivalence principle, and so one can then derive what the connection
terms are according to derivatives of the metric.
The equivalence principle tells us that no observer in a local frame can
determine whether they are falling or in a free region of flat spacetime.
This means their motion is not dependent on any direction so the covariant
directional derivative of a vector must be zero or ∇_uU = 0, and because
the metric defines the interval ds^2 = g_{μν}dx^μdx^ ν, and U^μ = dx^μ/ds, we
then also have ∇_ug = 0 or Dg_{μν}/∂x^ρ = 0. Just write
1 = g_{μν}(dx^μ/ds)(dx^ ν/ds)
and use the fact the derivative of 1 is zero.
LC
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