Mordred

May not denote it in terms of equations of state. Another format to describe Newton limit is [latex]G_{\mu\nu}=\eta_{\mu\nu}+H_{\mu\nu}[/latex] [latex]\eta_{\mu\nu}[/latex] denotes Minkowskii note there is no stress term. Either way the geodesic equation we have is time-like interval. Null geodesic for requires time-like intervals. Space-like intervals are tachyonic. So this doesn't include redshift. For sure there is always some time dilation unless your in the same reference frame. The term non relativistic simply means approximately Euclidean flat. For the [latex]G_{\mu\nu}[/latex] essentailly describes the geodesic line element. The stress tensor and the Ricci/Riehmann modify the metric tensor and vise versa

Actually no your right. I misread the conditions. Which is the matter must be approximately static. Lol wonder how I got off tangent there lol. Got into my head inside a uniform distribution. Probably because I'm more often modelling that in the FLRW metric. Lol gotta hate handwritten personal notes. Ah its specified better under weak field. Pressure contribution is neglibible Newton limit is [latex]p=0[/latex] [Latex]\Lambda=0[/latex] So pressureless matter no relativistic radiation. No cosmological constant. What does Ryder set the system as for Newton limit? Does he give the same equations of state? Edit: Newton limit is still non-relativistic meaning your infalling particle. The geodesic equation we derived is space-time component. Not the time-time needed for redshift. Time-time is a null geodesic.

Yes apply the shell theorem in the Newton limit. Two terms describe uniform. Homogeneous-essentailly no preferred location. Isotropic- no preferred direction. In the plasma case you model system as an ideal gas or fluid that homogeneous and isotropic in distribution. For gravitational redshift you want to map this in the Scwartzchild metric. Which has a preferred direction. Planets and BHs etc. In a uniform mass distribution there is no time dilation. Newton limit. In an increasing or decreasing mass distribution there is Schwartzchild metric.

Yes on the distribution. Yes on the geodesic path. The other thing to watch for in regards to the 1+ 1- is your sign convention. So if you add position Hoo to Goo keep track of your sign notation. The other trick is normalized units within the tensors. My advise is to study matrix math Then work into GR àtensors Because of centre of mass two parallel geodesics will deviate from a straight line. This is what's plotted in the metric tensor. The Ricci tensor reflects the curvature changes due time dilation. The Ryder reference you have may be in the (-+++) convention. The workup I did is in the (+---) convention. I don't know if your familiar with tensor calculus. So this link will help. https://www.google.ca/url?sa=t&source=web&rct=j&url=http://www.ita.uni-heidelberg.de/~dullemond/lectures/tensor/tensor.pdf&ved=0ahUKEwiI26G0_d_NAhVS-mMKHZPzBVYQFgg4MAc&usg=AFQjCNH28SMShwZP6np1MNQZ56WEtHGUHg&sig2=L0nCitet3oBLAmv9x-kGJg

Thats really one of those questions that we haven't been able to accurately answer yet. Theoretically gravity would symmetry break first. However in order for that symmetry break to occur a guage boson would need to drop out of thermal equilibrium. Theoretically this would be the graviton. Which would then mean that the graviton would be the heaviest boson in the total energy equivalent. If that turns out to be the case then gravity would be considered a force in every aspect. Without the graviton were stuck with gravity as a result spacetime geometry. More accurately the influences upon the stress-momentum tensor. This tensor can be influenced by any and all other particle to field relations. In this sense it the sum of all field interactions. Rather than its own unique field. (Bit of an oversimplification) but I know how much you don't see math relations. We won't know until we can quantize gravity. Or produce a graviton at an accelerator. Gravity exists before the Higgs symmetry breaks. In some books a fully unified field has been called super gravity. A system of nothing but photons can generate gravity. When all particles are in thermal equilibrium. There is no way to distinquish any particle from a photon. This condition exists prior to the Higgs field symmetry break.

Thats really one of those questions that we haven't been able to accurately answer yet. Theoretically gravity would symmetry break first. However in order for that symmetry break to occur a guage boson would need to drop out of thermal equilibrium. Theoretically this would be the graviton. Which would then mean that the graviton would be the heaviest boson in the total energy equivalent. If that turns out to be the case then gravity would be considered a force in every aspect. Without the graviton were stuck with gravity as a result spacetime geometry. More accurately the influences upon the stress-momentum tensor. This tensor can be influenced by any and all other particle to field relations. In this sense it the sum of all field interactions. Rather than its own unique field. (Bit of an oversimplification) but I know how much you don't see math relations. We won't know until we can quantize gravity. Or produce a graviton at an accelerator. Gravity exists before the Higgs symmetry breaks. In some books a fully unified field has been called super gravity.

Well in Dungeons and Dragons. They have a magic item called a portable hole. Lol Sorry off topic but I just couldn't help myself on that comment😈

So from that above. Take a cloud of plasma with uniform distribution. The centre of mass of that cloud is where your gravitational potential vectors will be pointing to. There isn't enough anistropy mass density in that cloud to have time dilation. Newton limit describes this system beautifully. Let's ask a counter question. What is the in the function of the metric tensor and the Ricci tensor/ Ricci scalar? https://en.m.wikipedia.org/wiki/Ricci_curvature Hint on the metric tensor. Think back to the time like geodesics. Read that section here. https://en.m.wikipedia.org/wiki/Metric_tensor Remember geodesics also occur in Euclidean flat space. Looking at that section then think about this earlier statement Its a bit tricky but if you think about the quoted statement above you should be able to answer the question below If not let me know and I will post the parallel transport Kronecker delta relations I have ! Moderator Note Strong Hint. Uniform Density+ Centre of mass. If it helps plot this on a graph paper. Place a particle of equal mass at each x,y coordinate. Then place centre of mass at coordinate 0,0. What type of path will particles free fall. What is the deviation between two of those geodesic paths

Yes thats essentially correct. I don't know the model procedure Ryder used. The procedure I have was done via the geodesic equation by modelling the parallel transport of two photons null geodesics. Then calculating calculating the slight deviation of those two null geodesics. Which is why the procedure I have is lengthy and complex. Newton limit is modelled as a scalar field. The [latex]H_{\mu\nu}[/latex] is used to model permutations of that scalar field. Its one of those hidden tensors in the EFE. For example I can model a vector or scalar field. Then model a gravity wave of that field with the [latex]H_{\mu\nu}[/latex] tensor. Newton gravity didn't include time dilation. This one of the key aspects that seperate Newton gravity from SR. Think of the Newton limit as pre Minkowskii. To model the Newton limit you need to describe a system where time dilation is neglibible. Which is why a uniform matter distribution is often used. If all observers is in the same mass density there is no time dilation. ( this makes the Newton limit handy to model gravity without the added complexity of time dilation)

Cool there's one I hadn't read yet. I may just pick up a copy to go with my collection. 40 plus textbooks just isn't enough. Lol that and my collection drives my wife crazy hehe

I'm not sure I would be able to find that solution on the Internet. Most Textbooks and articles usually skip right to the Scwartzchild metric as an example. Instead of covering the non relativistic Newton potential. However I do have the solution in my notes. It will take me some time to post the latex. (Which I will do once I am free for sufficient time) lol hopefully my old notes are still detailed enough and legible. This isn't precisely the same solution but its close. The symbology used is slightly different for the G_00 element. http://www.google.ca/url?sa=t&source=web&cd=8&ved=0ahUKEwiU9OvO9N3NAhUQHGMKHUqWA1YQFgg1MAc&url=http%3A%2F%2Fwww.math.uchicago.edu%2F~may%2FVIGRE%2FVIGRE2010%2FREUPapers%2FTolish.pdf&usg=AFQjCNF2yRQkUsx2hfa2by5mSWoIOxymlA&sig2=Py9ERsx_AKyp79263uc9UA Its essentially the same as you are deriving the Newton potential limit from the Einstein field equations. So will be variations though roughly equivalent. In the notation I used the N subscript simply denotes Newton as a identifier. I'LL still post the solution I have but having another example solution never hurts. PS how I found this solution is google. "Deriving Newton potential from Einstein field equations" then add pdf to refine your search. Other note... the solution in the link above is far easier to follow than the solution I have.

Mike I'm sure you've seen a topographical map. One that shows elevation change. On that topographical map distance scale changes as the elevation gets higher. (Picture the lines indicating elevation gradients). Now lets make a map showing vectors. A vector has two quantities. Scalar (magnitude) and direction. (Direction of influence) So we wish to map gravitational potential Now instead of mapping elevation we are mapping spacetime. Where time is a coordinate of a vector. So our vector coordinates are 3 of space, one of time. Now instead of mapping distance and elevation were mapping units of force. Which is in metres/s^2. Now just like the topographical map where you have a greater elevation change shown as lines being closer together. In spacetime the elevation is greater units of force (gravitational field potential). Though this is commonly mapped not as an elevation change. But the exact opposite. (A hole). In 2d visualize a rubber sheet with s ball in the centre. The greater the mass of the ball. The greater the depression. The path of light follows this curvature. Just in a different path than particles with mass.

I think have misread what you posted here

Visualize a galaxy 1 billion light years away. Draw a line from the centre of that galaxy to observer on Earth. Now draw a line from the outer edge of that galaxy the same observation point. How much angle between those two lines will you have. (Assume the galaxy has the same radius as the Milky way) Now your statement about two way signals. Were certainly not going to sendva signal from Earth to that Galaxy and wait for its return to take s measurement. (Assuming no expansion) that would take us 2 billion years to make a single measurement. When we measure a distant stellar object were making a measurement as it was in the past. Through Cosmological redshift we can approximate its distance. With calculation we can calculate its proper distance now and its proper distance when the light source originated. However for that we must work initally in commoving coordinates prior to Proper coordinates. Here is two articles I wrote a few years back. One is on redshift and expansion. The other is universe geometry and how it affects light paths. http://cosmology101.wikidot.com/redshift-and-expansion http://cosmology101.wikidot.com/universe-geometry Thankfully your now familiar with line element for the metrics for page two of the last article. http://cosmology101.wikidot.com/geometry-flrw-metric/ Other good study guides are. http://arxiv.org/pdf/hep-ph/0004188v1.pdf :"ASTROPHYSICS AND COSMOLOGY"- A compilation of cosmology by Juan Garcıa-Bellido http://arxiv.org/abs/astro-ph/0409426 An overview of Cosmology Julien Lesgourgues PS the lightcone calculator in my signature does these calculations Probably one of the better articles is Lineweaver and Davies. I'll have to find a new link to it. Hrmm the full article has moved Ah found it. http://www.google.ca/url?sa=t&source=web&cd=9&ved=0ahUKEwiWw7OLwd3NAhUM_mMKHdc0BqwQFgg1MAg&url=http%3A%2F%2Fwww.dark-cosmology.dk%2F~tamarad%2Fpapers%2Fthesis_complete.pdf&usg=AFQjCNHLzxKUp15sqgaDF2B8NU6i4xnBdg&sig2=GwUOC4EW-TeSeL2l6a2mnA

Why would measuring the rotation curve of the Andromeda Galaxy require a round trip for light? Secondly why would the speed of light make any difference to measurement points of a distant galaxy, when the galaxy is far enough away the light paths from each edge of the galaxy essentially follows the same path as it approaches Earth.? Your last post makes little sense

Has no rest mass. It does have inertial mass. Massless particles follow whats called a null geodesic. Massive particles will follow a space-time geodesic. The math behind geodesics are complex. I just recently did a post on it in Speculations to demonstrate how the geodesic equation is derived. Here is the pertinant section. In the presence of matter or when matter is not too distant physical distances between two points change. For example an approximately static distribution of matter in region D. Can be replaced by tve equivalent mass [latex]M=\int_Dd^3x\rho(\overrightarrow{x})[/latex] concentrated at a point [latex]\overrightarrow{x}_0=M^{-1}\int_Dd^3x\overrightarrow{x}\rho(\overrightarrow{x})[/latex] Which we can choose to be at the origin [latex]\overrightarrow{x}=\overrightarrow{0}[/latex] Sources outside region D the following Newton potential at [latex]\overrightarrow{x}[/latex] [latex]\phi_N(\overrightarrow{x})=-G_N\frac{M}{r}[/latex] Where [latex] G_n=6.673*10^{-11}m^3/KG s^2[/latex] and [latex]r\equiv||\overrightarrow{x}||[/latex] According to Einsteins theory the physical distance of objects in the gravitational field of this mass distribution is described by the line element. [latex]ds^2=c^2(1+\frac{2\phi_N}{c^2})-\frac{dr^2}{1+2\phi_N/c^2}-r^2d\Omega^2[/latex] Where [latex]d\Omega^2=d\theta^2+sin^2(\theta)d\varphi^2[/latex] denotes the volume element of a 2d sphere [latex]\theta\in(0,\pi)[/latex] and [latex]\varphi\in(0,\pi)[/latex] are the two angles fully covering the sphere. The general relativistic form is. [latex]ds^2=g_{\mu\nu}(x)dx^\mu x^\nu[/latex] By comparing the last two equations we can find the static mass distribution in spherical coordinates. [latex](r,\theta\varphi)[/latex] [latex]G_{\mu\nu}=\begin{pmatrix}1+2\phi_N/c^2&0&0&0\\0&-(1+2\phi_N/c^2)^{-1}&0&0\\0&0&-r^2&0\\0&0&0&-r^2sin^2(\theta)\end{pmatrix}[/latex] Now that we have defined our static multi particle field. Our next step is to define the geodesic to include the principle of equivalence. Followed by General Covariance. Ok so now the Principle of Equivalence. You can google that term for more detail but in the same format as above [latex]m_i=m_g...m_i\frac{d^2\overrightarrow{x}}{dt^2}=m_g\overrightarrow{g}[/latex] [latex]\overrightarrow{g}-\bigtriangledown\phi_N[/latex] Denotes the gravitational field above. Now General Covariance. Which use the ds^2 line elements above and the Einstein tensor it follows that the line element above is invariant under general coordinate transformation(diffeomorphism) [latex]x\mu\rightarrow\tilde{x}^\mu(x)[/latex] Provided ds^2 is invariant [latex]ds^2=d\tilde{s}^2[/latex] an infinitesimal coordinate transformation [latex]d\tilde{x}^\mu=\frac{\partial\tilde{x}^\mu}{\partial x^\alpha}dx^\alpha[/latex] With the line element invariance [latex]\tilde{g}_{\mu\nu}(\tilde{x})=\frac{\partial\tilde{x}^\mu \partial\tilde{x}^\nu}{\partial x^\alpha\partial x^\beta} g_{\alpha\beta}x[/latex] The inverse of the metric tensor transforms as [latex]\tilde{g}^{\mu\nu}(\tilde{x})=\frac{\partial\tilde{x}^\mu \partial\tilde{x}^\nu}{\partial x^\alpha\partial x^\beta} g^{\alpha\beta}x[/latex] In GR one introduces the notion of covariant vectors [latex]A_\mu[/latex] and contravariant [latex]A^\mu[/latex] which is related as [latex]A_\mu=G_{\mu\nu} A^\nu[/latex] conversely the inverse is [latex]A^\mu=G^{\mu\nu} A_\nu[/latex] the metric tensor can be defined as [latex]g^{\mu\rho}g_{\rho\nu}=\delta^\mu_\mu[/latex] where [latex]\delta^\mu_nu[/latex]=diag(1,1,1,1) which denotes the Kronecker delta. Finally we can start to look at geodesics. Let us consider a free falling observer. O who erects a special coordinate system such that particles move along trajectories [latex]\xi^\mu=\xi^\mu (t)=(\xi^0,x^i)[/latex] Specified by a non accelerated motion. Described as [latex]\frac{d^2\xi^\mu}{ds^2}[/latex] Where the line element ds=cdt such that [latex]ds^2=c^2dt^2=\eta_{\mu\nu}d\xi^\mu d\xi^\nu[/latex] Now assunme that the motion of O changes in such a way that it can be described by a coordinate transformation. [latex]d\xi^\mu=\frac{\partial\xi^\mu}{\partial x^\alpha}dx^\alpha, x^\mu=(ct,x^0)[/latex] This and the previous non accelerated equation imply that the observer O, will percieve an accelerated motion of particles governed by the Geodesic equation. [latex]\frac{d^2x^\mu}{ds^2}+\Gamma^\mu_{\alpha\beta}(x)\frac{dx^\alpha}{ds}\frac{dx^\beta}{ds}=0[/latex] Where the new line element is given by [latex]ds^2=g_{\mu\nu}(x)dx^\mu dx^\nu[/latex] and [latex] g_{\mu\nu}=\frac{\partial\xi^\alpha}{\partial\xi x^\mu}\frac{\partial\xi^\beta}{\partial x^\nu}\eta_{\alpha\beta}[/latex] and [latex]\Gamma^\mu_{\alpha\beta}=\frac{\partial x^\mu}{\partial\eta^\nu}\frac{\partial^2\xi^\nu}{\partial x^\alpha\partial x^\beta}[/latex] Denote the metric tensor and the affine Levi-Civita connection respectively.

😉

Yes thats correct it is the non relativistic Newton weak field limit. The number of steps to derive that, is lengthier than the post itself.

Thanks I plan on posting that post into the Relativity forum. I might pin it into the What is space thread.

Man that took hours to latex lol. Ajb if you would for errors when you get a chance I'd appreciate it. Now what did we learn. First I described a static field of uniform density. This means that the stress tensor is also static. (You cannot change one without the other). Then I described in mathematical detail the equivalence principle. Follows by the principle of covariance. From there I detailed the Kronecker delta. Once I had these established I then detailed how the Geodesic equation is derived. (Which details particle freefall) within that field. As an added bonus I described the Levi-Civita connection.

Here this link will explain it better and save me a ton of latex. https://en.m.wikipedia.org/wiki/Line_element Keep in mind I still haven't shown a particles movement. All the steps above just detail the static geometry. GR is all about relating events. So you don't want your coordinates that describe those events to move. (Though in some cases you have no choice ie expansion) Right now I need to take a break and check through my old notes. Make sure I got the covariant and contravariant terms correct. My old handwriting is a bit messy lol I'm placing these into a single post as I may use this again. Though I messed up some of the original posts lol In the presence of matter or when matter is not too distant physical distances between two points change. For example an approximately static distribution of matter in region D. Can be replaced by tve equivalent mass [latex]M=\int_Dd^3x\rho(\overrightarrow{x})[/latex] concentrated at a point [latex]\overrightarrow{x}_0=M^{-1}\int_Dd^3x\overrightarrow{x}\rho(\overrightarrow{x})[/latex] Which we can choose to be at the origin [latex]\overrightarrow{x}=\overrightarrow{0}[/latex] Sources outside region D the following Newton potential at [latex]\overrightarrow{x}[/latex] [latex]\phi_N(\overrightarrow{x})=-G_N\frac{M}{r}[/latex] Where [latex] G_n=6.673*10^{-11}m^3/KG s^2[/latex] and [latex]r\equiv||\overrightarrow{x}||[/latex] According to Einsteins theory the physical distance of objects in the gravitational field of this mass distribution is described by the line element. [latex]ds^2=c^2(1+\frac{2\phi_N}{c^2})-\frac{dr^2}{1+2\phi_N/c^2}-r^2d\Omega^2[/latex] Where [latex]d\Omega^2=d\theta^2+sin^2(\theta)d\varphi^2[/latex] denotes the volume element of a 2d sphere [latex]\theta\in(0,\pi)[/latex] and [latex]\varphi\in(0,\pi)[/latex] are the two angles fully covering the sphere. The general relativistic form is. [latex]ds^2=g_{\mu\nu}(x)dx^\mu x^\nu[/latex] By comparing the last two equations we can find the static mass distribution in spherical coordinates. [latex](r,\theta\varphi)[/latex] [latex]G_{\mu\nu}=\begin{pmatrix}1+2\phi_N/c^2&0&0&0\\0&-(1+2\phi_N/c^2)^{-1}&0&0\\0&0&-r^2&0\\0&0&0&-r^2sin^2(\theta)\end{pmatrix}[/latex] Now that we have defined our static multi particle field. Our next step is to define the geodesic to include the principle of equivalence. Followed by General Covariance. Ok so now the Principle of Equivalence. You can google that term for more detail but in the same format as above [latex]m_i=m_g...m_i\frac{d^2\overrightarrow{x}}{dt^2}=m_g\overrightarrow{g}[/latex] [latex]\overrightarrow{g}-\bigtriangledown\phi_N[/latex] Denotes the gravitational field above. Now General Covariance. Which use the ds^2 line elements above and the Einstein tensor it follows that the line element above is invariant under general coordinate transformation(diffeomorphism) [latex]x\mu\rightarrow\tilde{x}^\mu(x)[/latex] Provided ds^2 is invariant [latex]ds^2=d\tilde{s}^2[/latex] an infinitesimal coordinate transformation [latex]d\tilde{x}^\mu=\frac{\partial\tilde{x}^\mu}{\partial x^\alpha}dx^\alpha[/latex] With the line element invariance [latex]\tilde{g}_{\mu\nu}(\tilde{x})=\frac{\partial\tilde{x}^\mu \partial\tilde{x}^\nu}{\partial x^\alpha\partial x^\beta} g_{\alpha\beta}x[/latex] The inverse of the metric tensor transforms as [latex]\tilde{g}^{\mu\nu}(\tilde{x})=\frac{\partial\tilde{x}^\mu \partial\tilde{x}^\nu}{\partial x^\alpha\partial x^\beta} g^{\alpha\beta}x[/latex] In GR one introduces the notion of covariant vectors [latex]A_\mu[/latex] and contravariant [latex]A^\mu[/latex] which is related as [latex]A_\mu=G_{\mu\nu} A^\nu[/latex] conversely the inverse is [latex]A^\mu=G^{\mu\nu} A_\nu[/latex] the metric tensor can be defined as [latex]g^{\mu\rho}g_{\rho\nu}=\delta^\mu_\mu[/latex] where [latex]\delta^\mu_nu[/latex]=diag(1,1,1,1) which denotes the Kronecker delta. Finally we can start to look at geodesics. Let us consider a free falling observer. O who erects a special coordinate system such that particles move along trajectories [latex]\xi^\mu=\xi^\mu (t)=(\xi^0,x^i)[/latex] Specified by a non accelerated motion. Described as [latex]\frac{d^2\xi^\mu}{ds^2}[/latex] Where the line element ds=cdt such that [latex]ds^2=c^2dt^2=\eta_{\mu\nu}d\xi^\mu d\xi^\nu[/latex] Now assunme that the motion of O changes in such a way that it can be described by a coordinate transformation. [latex]d\xi^\mu=\frac{\partial\xi^\mu}{\partial x^\alpha}dx^\alpha, x^\mu=(ct,x^0)[/latex] This and the previous non accelerated equation imply that the observer O, will percieve an accelerated motion of particles governed by the Geodesic equation. [latex]\frac{d^2x^\mu}{ds^2}+\Gamma^\mu_{\alpha\beta}(x)\frac{dx^\alpha}{ds}\frac{dx^\beta}{ds}=0[/latex] Where the new line element is given by [latex]ds^2=g_{\mu\nu}(x)dx^\mu dx^\nu[/latex] and [latex] g_{\mu\nu}=\frac{\partial\xi^\alpha}{\partial\xi x^\mu}\frac{\partial\xi^\beta}{\partial x^\nu}\eta_{\alpha\beta}[/latex] and [latex]\Gamma^\mu_{\alpha\beta}=\frac{\partial x^\mu}{\partial\eta^\nu}\frac{\partial^2\xi^\nu}{\partial x^\alpha\partial x^\beta}[/latex] Denote the metric tensor and the affine Levi-Civita connection respectively. There done.

The s^2 indicates a light like seperation of two events on a 2d sphere.

Now General Covariance. Which use the ds^2 line elements above and the Einstein tensor it follows that the line element above is invariant under general coordinate transformation(diffeomorphism) [latex]x\mu\rightarrow\tilde{x}^\mu(x)[/latex] Provided ds^2 is invariant [latex]ds^2=d\tilde{s}^2[/latex] an infinitesimal coordinate transformation [latex]d\tilde{x}^\mu=\frac{\partial\tilde{x}^\mu}{\partial x^\alpha}dx^\alpha[/latex] With the line element invariance [latex]\tilde{g}_{\mu\nu}(\tilde{x})=\frac{\partial\tilde{x}^\mu \partial\tilde{x}^\nu}{\partial x^\alpha\partial x^\beta} g_{\alpha\beta}x[/latex] The inverse of the metric tensor transforms as [latex]\tilde{g}^{\mu\nu}(\tilde{x})=\frac{\partial\tilde{x}^\mu \partial\tilde{x}^\nu}{\partial x^\alpha\partial x^\beta} g^{\alpha\beta}x[/latex]

Thanks I forgot a formula. [latex]\phi_N(\overrightarrow{x})=-G_N\frac{M}{r}[/latex] The above is defining a uniform static field. [latex]\Omega[/latex] is a dimensionless value for density. As were just defining a static field we don't need the time parameter. We haven't detailed any particle motion yet. Ok so now the Principle of Equovalence. You can google that term for more detail but in the same format as above [latex]m_i=m_g...m_i\frac{d^2\overrightarrow{x}}{dt^2}=m_g\overrightarrow{g}[/latex] [latex]\overrightarrow{g}-\bigtriangledown\phi_N[/latex] Denotes the gravitational field above.

I'm placing these into a single post as I may use this again. In the presence of matter or when matter is not too distant physical distsnces between two points change. For example an approximately static distribution of matter in region D. Can be replaced by tve equivalent mass [latex]M=\int_Dd^3x\rho(\overrightarrow{x})[/latex] concentrated at a point [latex]\overrightarrow{x}_0=M^{-1}\int_Dd^3x\overrightarrow{x}\rho(\overrightarrow{x})[/latex] Which we can choose to be at the origin [latex]\overrightarrow{x}=\overrightarrow{0}[/latex] Sources outside region D the following Newton potential at [latex]\overrightarrow{x}[/latex] [latex]\phi_N(\overrightarrow{x})=-G_N\frac{M}{r}[/latex] Where [latex] G_n=6.673*10^{-11}m^3/KG s^2[/latex] and [latex]r\equiv||\overrightarrow{x}||[/latex] According to Einsteins theory the physical distance of objects in the gravitational field of this mass distribution is described by the line element. [latex]ds^2=c^2(1+\frac{2\phi_N}{c^2})-\frac{dr^2}{1+2\phi_N/c^2}-r^2d\Omega^2[/latex] Where [latex]d\Omega^2=d\theta^2+sin^2(\theta)d\varphi^2[/latex] denotes the volume element of a 2d sphere [latex]\theta\in(0,\pi)[/latex] and [latex]\varphi\in(0,\pi)[/latex] are the two angles fully covering the sphere. The general relativistic form is. [latex]ds^2=g_{\mu\nu}(x)dx^\mu x^\nu[/latex] By comparing the last two equations we can find the static mass distribution in spherical coordinates. [latex](r,\theta\varphi)[/latex] [latex]G_{\mu\nu}=\begin{pmatrix}1+2\phi_N/c^2&0&0&0\\0&-(1+2\phi_N/c^2)^{-1}&0&0\\0&0&-r^2&0\\0&0&0&-r^2sin^2(\theta)\end{pmatrix}[/latex] Now that we have defined our static multi particle field. Our next step is to define the geodesic to include the principle of equivalence. Followed by General Covariance.