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Originally Posted by Nesti
Aren't there 16 components of a [symmetric] Metric Tensor which define the potential of the field? (Figure attached).
such as in Metric Gij = g(1,1), g(1,2)...g(4,4). (16 in total)
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Although a tensor of rank 2 in 4 dimensional space has 4^2=16 components there are only 10 independent components since the metric tensor is symmetric.
For example if you express the tensor as a 4 X 4 matrix you will find the components above the leading diagonal are the same as below the diagonal. Hence the total number of independent or unique components is 16-6=10.
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Classifying a 'Symmetric Metric', or , 'Flat Metric':
1. All components other than (1,1 / 2,2 / 3,3 / 4,4) must equal 0 if the field is to be classed as symmetric.
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No as explained above the non diagonal components don't have to equal zero.
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2. If the field is 'flat', then all 16 components must equal 0.
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If the field is flat (and orthogonal) the leading diagonal are non zero constants (usually 1,-1,-1,-1 Lorentzian or -1,-1,-1,-1, Minkowski) . All other components are zero.
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So now I come back to my earlier comments, in that the speed at which the metric may change (Yes, defined by the speed at which the gravitational wave may propagate) must also be defined by the spacetime topology in which the wave must travel. That goes with light too. So if light normally travels at c, but is hindered by the topology of a particular region, how can the Metric (the wave) possibly be measured against light if both are affected by the same topology. Both light and wave are joined at the hips (so to speak). And since the variations you are looking for are so minute and the ability to survey the Metrics at all points within the travelled path are so limited, there's no hope of getting any meaningful verification of the speed of the gravitational wave.
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The worldline of a photon is a null geodesic irrespective of the topology.
The metric of the perturbed space doesn't change. That's because even though the wave is moving the "shape" of the wave remains the same.
Unless of course if you consider space-time in front of the wave being flat and the metric assuming the perturbed space as the wave passes through.
In the real world however it is impossible to measure a metric as it is a mathematical equation not a value.
Gravitational waves will hopefully be detected by the forces they exert.
Regards
Steven