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Discussion and Debate
Discussion and Debate
Physical & Life Sciences
Evidence of gravitational waves, or evidence of confirmation bias?
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<blockquote data-quote="sjastro" data-source="post: 71627670" data-attributes="member: 352921"><p>Selfsim,</p><p></p><p>This thread needs to be developed with an injection of real science instead of the opinion based zero knowledge zero understanding that has prevailed.</p><p></p><p>A good starting point is to look at the mathematics behind gravitational waves and the resultant physical interpretations that have been confirmed by LIGO.</p><p>This can be the propagation of GWs (gravitational waves) through spacetime or their creation.</p><p>The propagation aspect is the “easy part” and will be described as the mathematics behind the creation of gravitational waves, apart from being very complicated, is outside the boundaries of conventional general relativity and the scope of this forum.</p><p></p><p>I’ll rehash some of the maths that I described in the “We almost live in a Static Universe” thread on metrics and how this is applied to the propagation of gravitational waves through spacetime.</p><p></p><p>The concept of a metric.</p><p>You have heard of the saying the shortest distance between two points is a straight line.</p><p>This is only true in flat space and can be mathematically expressed using Pythagoras theorem for a right angle triangle. C²=A²+B².</p><p>In this case C is the distance, A and B being the horizontal and vertical distances respectively.</p><p>In x-y coordinates with segments dx and dy the equation is ds²=dx²+dy².</p><p>ds²=dx²+dy² is known as a metric.</p><p>In 3D flat space the metric is ds²=dx²+dy²+dz².</p><p></p><p>On the surface of a sphere, the shortest distance between two points is not a straight line but an arc.</p><p>In this case the metric is ds²=r²(dθ²+sin²θdΦ) where r is the radius of the sphere, θ, Φ are the latitudinal and longitudinal angles respectively.</p><p></p><p>These metrics are spatial metrics, however since relativity uses spacetime there is an extra time term c²dt² where c is the speed of light.</p><p>The metrics for space time are the difference between the time term and the spatial terms.</p><p>Metrics simply describe a property of spacetime.</p><p></p><p>In 3D flat spacetime the metric is ds²=c²dt²-dx²-dy²-dz².</p><p></p><p>This metric is known as the Lorentz metric and describes the properties of spacetime in the absence of matter and gravity.</p><p>This is a static metric whose properties do not change with time.</p><p></p><p>Metrics are solutions to Einstein’s field equations</p><p></p><p>Rₐₑ - (1/2)Rₐₑ + Λgₐₑ = -(8πG/c^4)Tₐₑ</p><p></p><p>This equation tells us of the relationship between gravity and spacetime.</p><p>When gravity is absent (no mass) spacetime is flat otherwise gravity curves spacetime.</p><p>Λ is the cosmological constant for an accelerating Universe.</p><p>The right hand term indicates the presence of matter.</p><p></p><p>For the Lorentz metric we can simplify the field equations.</p><p>Since the Lorentz metric describes static flat space time in the absence of matter Λ=0 and the right hand term vanishes.</p><p>The field equations are reduced to the vacuum form.</p><p></p><p>Rₐₑ = 0.</p><p></p><p>We can now mathematically model the propagation of a GW through spacetime using the Lorentz metric as a starting point.</p><p>Before the onset of the GW spacetime is static and flat and is described by the Lorentz metric</p><p></p><p>ds²=c²dt²-dx²-dy²-dz².</p><p></p><p>When the GW passes through this static flat space time we use a principle known as perturbation theory to describe how the metric changes when a GW is present.</p><p>We assume the metric undergoes a small perturbation or variation which effects the c²dt², dx²,dy² and dz² coefficients of the metric.</p><p></p><p>This perturbed Lorentz metric has the form.</p><p></p><p>ds²=(1+F00)c²dt²-(1+F11)dx²-(1+F22)dy²-(1+F33)dz².</p><p></p><p>At this stage all we know about F00, F11, F22 and F33 is that they are time dependant functions.</p><p>Collectively they are described as the general term Fab.</p><p></p><p>If we assume that the Fab terms are very small, the perturbed metric is flat enough to be a solution to the field equations Rₐₑ = 0.</p><p>We can plug the metric into the field equations and after some very tedious calculations this leads to a more general equation.</p><p></p><p>□²Fab= δ²Fab/δx²+ δ²Fab/δy²+ δ²Fab/δz²-(1/c²) δ²Fab/δt² where □ is the D’Alembert operator.</p><p></p><p>This complicated equation known as the Helmholtz equation which is well known in electrodynamics theory gives us a wealth of information and allows us to make some physical interpretations of the Fab terms.</p><p>The Fab terms are gravitational potentials of a gravitational field and travel at the speed of light c.</p><p>The Helmoltz equation being a wave equation immediately tells us that the Fab terms have wave like properties.</p><p>The gravitational forces act transversely to the direction of motion of the wave and have the unusual property of compressing objects in spacetime in one direction while stretching them in the 90 degree direction.</p><p>Whereas gravity has a spherical symmetry, gravitational waves have a quadrupole symmetry which can be visualized as an ellipsoid or football shape.</p><p></p><p>These are not assumptions but what the mathematics tells us and allows predictions on which experiments are developed.</p><p></p><p>The stretching/compression of objects in the presence of gravitational waves in theory can be detectable by the use of masses attached to springs with specific damping and elastic characteristics, the assembly tuned to a particular resonance frequency.</p><p>This formed the basis of Weber’s experiment.</p><p>Unfortunately this method failed due to lack of sensitivity and the extreme difficulty of isolating the experiment from external vibrations.</p><p>A much more sensitive method is to exploit the quadrupole symmetry of the GWs by the use of laser interferometers as used by LIGO.</p><p>Due to quadrupole symmetry of a passing GW one arm of the interferometer is displaced differently to the other arm resulting in a laser interference pattern.</p><p></p><p>LIGO’s discovery of GWs is a triumph of quantum limited laser interferometry.</p></blockquote><p></p>
[QUOTE="sjastro, post: 71627670, member: 352921"] Selfsim, This thread needs to be developed with an injection of real science instead of the opinion based zero knowledge zero understanding that has prevailed. A good starting point is to look at the mathematics behind gravitational waves and the resultant physical interpretations that have been confirmed by LIGO. This can be the propagation of GWs (gravitational waves) through spacetime or their creation. The propagation aspect is the “easy part” and will be described as the mathematics behind the creation of gravitational waves, apart from being very complicated, is outside the boundaries of conventional general relativity and the scope of this forum. I’ll rehash some of the maths that I described in the “We almost live in a Static Universe” thread on metrics and how this is applied to the propagation of gravitational waves through spacetime. The concept of a metric. You have heard of the saying the shortest distance between two points is a straight line. This is only true in flat space and can be mathematically expressed using Pythagoras theorem for a right angle triangle. C²=A²+B². In this case C is the distance, A and B being the horizontal and vertical distances respectively. In x-y coordinates with segments dx and dy the equation is ds²=dx²+dy². ds²=dx²+dy² is known as a metric. In 3D flat space the metric is ds²=dx²+dy²+dz². On the surface of a sphere, the shortest distance between two points is not a straight line but an arc. In this case the metric is ds²=r²(dθ²+sin²θdΦ) where r is the radius of the sphere, θ, Φ are the latitudinal and longitudinal angles respectively. These metrics are spatial metrics, however since relativity uses spacetime there is an extra time term c²dt² where c is the speed of light. The metrics for space time are the difference between the time term and the spatial terms. Metrics simply describe a property of spacetime. In 3D flat spacetime the metric is ds²=c²dt²-dx²-dy²-dz². This metric is known as the Lorentz metric and describes the properties of spacetime in the absence of matter and gravity. This is a static metric whose properties do not change with time. Metrics are solutions to Einstein’s field equations Rₐₑ - (1/2)Rₐₑ + Λgₐₑ = -(8πG/c^4)Tₐₑ This equation tells us of the relationship between gravity and spacetime. When gravity is absent (no mass) spacetime is flat otherwise gravity curves spacetime. Λ is the cosmological constant for an accelerating Universe. The right hand term indicates the presence of matter. For the Lorentz metric we can simplify the field equations. Since the Lorentz metric describes static flat space time in the absence of matter Λ=0 and the right hand term vanishes. The field equations are reduced to the vacuum form. Rₐₑ = 0. We can now mathematically model the propagation of a GW through spacetime using the Lorentz metric as a starting point. Before the onset of the GW spacetime is static and flat and is described by the Lorentz metric ds²=c²dt²-dx²-dy²-dz². When the GW passes through this static flat space time we use a principle known as perturbation theory to describe how the metric changes when a GW is present. We assume the metric undergoes a small perturbation or variation which effects the c²dt², dx²,dy² and dz² coefficients of the metric. This perturbed Lorentz metric has the form. ds²=(1+F00)c²dt²-(1+F11)dx²-(1+F22)dy²-(1+F33)dz². At this stage all we know about F00, F11, F22 and F33 is that they are time dependant functions. Collectively they are described as the general term Fab. If we assume that the Fab terms are very small, the perturbed metric is flat enough to be a solution to the field equations Rₐₑ = 0. We can plug the metric into the field equations and after some very tedious calculations this leads to a more general equation. □²Fab= δ²Fab/δx²+ δ²Fab/δy²+ δ²Fab/δz²-(1/c²) δ²Fab/δt² where □ is the D’Alembert operator. This complicated equation known as the Helmholtz equation which is well known in electrodynamics theory gives us a wealth of information and allows us to make some physical interpretations of the Fab terms. The Fab terms are gravitational potentials of a gravitational field and travel at the speed of light c. The Helmoltz equation being a wave equation immediately tells us that the Fab terms have wave like properties. The gravitational forces act transversely to the direction of motion of the wave and have the unusual property of compressing objects in spacetime in one direction while stretching them in the 90 degree direction. Whereas gravity has a spherical symmetry, gravitational waves have a quadrupole symmetry which can be visualized as an ellipsoid or football shape. These are not assumptions but what the mathematics tells us and allows predictions on which experiments are developed. The stretching/compression of objects in the presence of gravitational waves in theory can be detectable by the use of masses attached to springs with specific damping and elastic characteristics, the assembly tuned to a particular resonance frequency. This formed the basis of Weber’s experiment. Unfortunately this method failed due to lack of sensitivity and the extreme difficulty of isolating the experiment from external vibrations. A much more sensitive method is to exploit the quadrupole symmetry of the GWs by the use of laser interferometers as used by LIGO. Due to quadrupole symmetry of a passing GW one arm of the interferometer is displaced differently to the other arm resulting in a laser interference pattern. LIGO’s discovery of GWs is a triumph of quantum limited laser interferometry. [/QUOTE]
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