The Schwarzschild metric describes the gravity of a spherically symmetric mass $M$ in spherical coordinates: $$ds^2 =-\left(1-\frac{2GM}{c^2r}\right)c^2 \, dt^2+\left(1-\frac{2GM}{c^2r}\right)^{-1}dr^2+r^2 \,d\Omega^2 \tag{1}$$ Naively, I would expect the classical Newtonian limit to be $\frac{2GM}{c^2r}\ll1$ (), which yields $$ds^2 =-\left(1-\frac{2GM}{c^2r}\right)c^2 \, dt^2+\left(1+\frac{2GM}{c^2r}\right)dr^2+r^2 \,d\Omega^2 \tag{2}$$ However, the correct &quot;Newtonian limit&quot; as can be found for example in , eq.(6.29), is $$ds^2 =-\left(1-\frac{2GM}{c^2r}\right)c^2 \, dt^2+\left(1+\frac{2GM}{c^2r}\right)\left(dr^2+r^2 \,d\Omega^2\right) \tag{3}$$ Question: Why is the first procedure of obtaining the Newtonian limit from the Schwarzschild solution incorrect?
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For Minkowski space with a weak gravitational field the metric takes the form $$ ds^2 = (1+2\phi/c^2)c^2dt^2 -(1-2\phi/c^2)(dx^2+dy^2+dz^2), $$ where $\phi$ is the Newtonian gravitational potential. You can get the $(1+2\phi/c^2)$ factor in front of the $dt^2$ by starting with the geodesic equation and going to the &quot;Newtonian limit&quot; of slow speeds and a slowly changing field. But is there a way to get the $(1-2\phi/c^2)$ factor for the spatial part of the metric by a similar procedure? What about some clever thought experiment?
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From Newton's law of gravitation we know that  $$F=G\frac{m_1m_2}{r^2}$$  where $G$ is gravitational constant. We can also see that it has dimensions  $$[G]=\frac{[L]^3}{[M][T]^2}$$  and we have a good numerical estimate of its value ($G\simeq 6.67\times 10^{-11} N(m/kg)^2$).  I've read (for instance in the paper I'm currently mostly interested in, Kerr/CFT correspondence) that in higher dimensions, there are 4 dimensional gravitational constant, and in the search for explanation of it, I've also found that in string theory we have gravitational constants of higher dimensions.  But I haven't seen any formula that would give me mean to calculate it in higher dimensions.  Is there such a thing, or is the definition of it theory dependent?
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The Runge-Lenz vector is an "extra" conserved quantity for Keplerian $\frac{1}{r}$ potentials, which is in addition to the usual energy and angular momentum conservation present in all central force potentials.    I suppose it is not a coincidence that $\frac{1}{r}$ potentials have this extra conserved quantity and are also one of the two central force potentials that have closed orbits.  Indeed, one may think of the RL vector as an expression for the orientation and eccentricity of the elliptical orbit, which is conserved iff the orbit closes on itself.    This then leads me to think that there must be an analog to the RL vector in an $r^2$ potential, which is the other central force potnetial that has closed orbits.  Surely, we can define a vector pointing along the direction of the orbit's major axis, that has magnitude proprtional to the eccentricity, and this will be conserved.  Is there any way to write down such a vector in terms of dynamical variables, and thus obtain an $r^2$ analog for the RL vector?
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The height of a string in a gravitational field in 2-dimensions is bounded by $h(x_0)=h(x_l)=0$ (nails in the wall)  and also $\int_0^l ds= l$. ($h(0)=h(l)=0$, if you take $h$ as a function of arc length) .  What shape does it take?  My try so far: minimise potential energy of the whole string,  $$J(x,h, \dot{h})=\int_0^l gh(x) \rho \frac{ds}{l}=\frac{g \rho }{l}\int_0^l h(x) \sqrt{1+\dot{h}^2} dx$$  With the constraint  $$\int_0^l \sqrt{1+\dot{h}^2} dx- l=0$$ If it helps, it's evident that $\dot{h}(\frac{l}{2})=0$.  Generally, this kind of equation is a case of a constrained variational problem, meaning that the integrand in  $$\int_0^l \frac{g \rho }{l}h(x) \sqrt{1+\dot{h}^2} +\lambda(\int_0^l \sqrt{1+\dot{h}^2} dx- l)dx$$  Must satisfy the Euler Lagrange equation. The constraint must also be satisfied.  But, in truth, by this point I am clueless. $\lambda$ is worked through $\nabla J  = \lambda \nabla(\int_0^l \sqrt{1+\dot{h}^2} dx- l)$. I have tried this , but get nonsensical answers.  Is this method the best? If so, in what ways am I going about it wrongly thusfar?
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Question: As we know, (1) the macroscopic spatial dimension of our universe is 3 dimension, and (2) gravity attracts massive objects together and the gravitational force is isotropic without directional preferences. Why do we have the spiral 2D plane-like Galaxy(galaxies), instead of spherical or elliptic-like galaxies?   Input: Gravity is (at least, seems to be) isotropic from its force law (Newtonian gravity). It should show no directional preferences from the form of force vector $\vec{F}=\frac{GM(r_1)m(r_2)}{(\vec{r_1}-\vec{r_2})^2} \hat{r_{12}}$. The Einstein gravity also does not show directional dependence at least microscopically.  If the gravity attracts massive objects together isotropically, and the macroscopic space dimension is 3-dimensional, it seems to be natural to have a spherical shape of massive objects gather together. Such as the globular clusters, or GC, are roughly spherical groupings , as shown in the Wiki picture:     However, my impression is that, even if we have observed some more spherical or more-ball-like , it is more common to find more-planar  such as our ? (Is this statement correct? Let me know if I am wrong.)  Also, such have a look at this more-planar Spiral "galaxy" as this NGC 4414 galaxy:    Is there some physics or math theory explains why the  turns out to be planar-like (or spiral-like) instead of spherical-like?  See also a   p.s. Other than the classical stability of a 2D plane perpendicular to a classical angular momentum, is there an interpretation in terms of the quantum theory of vortices in a macroscopic manner (just my personal speculation)?  Thank you for your comments/answers!
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On this site, the Phys.SE question  has been asked. See also  Phys.SE question. The 'answer' provided on this forum has been that the curvature of spacetime explains both. The answer is still cryptic for me as I am more a concrete thinker.  Newton said $F=ma$. I can use this formula to measure inertial mass. Experimentally I can measure the motion of an object while applying a constant force to it. Newton also said $F=\frac{GMm}{r^2}$. In this case, what simple experiment will allow the measurement of gravitational mass?
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If we call the velocity of the mass $u$, then we can say: $u \cos θ = v$ and hence $u= \frac v{\cos θ}$. But why can't we write  $v \cos θ= u$ and add the contribution from both of the strings: $u= \frac{2v}{\cos θ}$ ? Why is it only $u= \frac v{\cos θ}$? Please note that it is not a homework based  question. I just want someone to solve the doubt arising in my mind.
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For some holomorphic function $f$ show that    $$\lim_{t\to0} \frac{cr(f(ta),f(tb),f(tc),f(td))-cr(a,b,c,d)}{t^2cr(a,b,c,d)}=\frac{(a-b)(c-d)}{6}S(f)(0)$$   where $cr(a,b,c,d)$ is the cross-ratio of four points in $\Bbb C$ and $S(f)$ is the Schwarzian derivative of $f$.   I am unsure on how to really make any progress after inserting the definition of the cross-ratio and turning it all into one fraction (rather than fractions in a fraction). I tried working from the other side but made even less progress, since I don't even know how I am supposed to get the factor $(a-b)(c-d)$ in there, since this is nowhere in the fraction.   Any help would be greatly appreciated.
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Can someone suggest a textbook that treats general relativity from a rigorous mathematical perspective? Ideally, such a book would   Prove all theorems used. Use modern "mathematical notation" as opposed to "physics notation", especially with respect to linear algebra and differential geometry. Have examples that illustrate both computational and theoretical aspects. Have a range of exercises with varying degrees of difficulty, with answers.   An ideal text would read a lot more like a math book than a physics book and would demand few prerequisites in physics. Bottom line is that I would like a book that provides an axiomatic development of general relativity clearly and with mathematical precision works out the details of the theory.  Addendum (1): I did not intend to start a war over notation. As I said in one of the comments below, I think indicial notation together with the summation convention is very useful. The coordinate-free approach has its uses as well and I see no reason why the two can't peacefully coexist. What I meant by "mathematics notation" vs. "physics notation" is the following: Consider, as an example, one of the leading texts on smooth manifolds, John Lee's Introduction to Smooth Manifolds. I am very accustomed to this notation and it very similar to the notation used by Tu's Introduction to Manifolds, for instance, and other popular texts on differential geometry. On the other hand, take Frankel's Geometry of Physics. Now, this is a nice book but it is very difficult for me to follow it because 1) Lack of proofs and 2)the notation does not agree with other math texts that I'm accustomed to. Of course, there are commonalities but enough is different that I find it really annoying to try to translate between the two...  Addendum (2):  For the benefit of future readers, In addition to suggestions below, I have found another text that also closely-aligns with the criteria I stated above. It is, Spacetime: Foundations of General Relativity and Differential Geometry by Marcus Kriele. The author begins by discussing affine geometry, analysis on manifolds, multilinear algebra and other underpinnings and leads into general relativity at roughly the midpoint of the text. The notation is also fairly consistent with the books on differential geometry I mentioned above.
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so I had my first approximation to the gravity equation  \begin{equation} F=GmM/r^2 \end{equation} and some questions arose that my teacher couldnt respond: if r approximates to 0 with mM being constant, then the force should tend to infinity, right? so that being true, then if we would travel to the center of the earth, we would be crushed by gravity? if so, how can there be anything right in the center of the earth? wouldnt anything there be compressed and destroyed by the infinite force?  having that in mind, my next question is as gravity force is measured from the center of an object, any part of the same object which is not at the center would be affected by the same object's gravity? if so, how can we differentiate between two objects that are not in a vacuum? my question originated thinking in us standing on the earth, in contact with it, are we, in a gravitational sense, one object? this lead to the last question, if gravity is a property of matter, then how applying the same logic as before, how does it work between two neutrons?(chose neutrons because i assumed it would be more difficult obtaining gravitational force if there was electrostatic forces from charges inbetween) do two newtrons in contact count as 1 gravitational object? and how thats its own gravity affect the newtron from center and out?
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My book (Concepts of Physics by H.C. Verma) writes:  It has been reported () that the force between two masses may be better represented by $$F = \frac{G_{\infty} m_{1} m_{2}}{r^2} \left[1 + \left(1 + \frac{r}{\lambda} \right) \alpha e^{-\frac{r}{\lambda}}\right]$$ where $\alpha \approx - 0.007$ and $\lambda \approx 200~\mathrm{m}$.  What is this? Such a horrendous formula! So, what about Newton's? And what's the difference between $G$ &amp; $G_{\infty}$?
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I am reading through this article   about the special case of the positive mass theorem in general relativity. I do not understand the section below:  In particular what does it mean to isometrically embed an $n$-dimensional Riemannian manifold $(M,g)$ in a higher dimensional space, and where does the embedding given for the  $n=3$ Schwarzschild metric come from? I assume we are referring to the standard definition of isometric embeddings using pullbacks of covariant tensor fields. I assume the author refers to the map between Riemannian manifolds $f: \mathbb{R}^3-B_{2m}\to\mathbb{R}^4$ given by $f(x,y,z)=(x,y,z,\sqrt{8m(r-2m)})$. I have tried to check if the Schwarzchild metric on $\mathbb{R}^3$ is the pullback of the flat metric on $\mathbb{R}^{4}$, however this doesn't appear to be true...
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In fact, when Sinnott published the   haversine formula, computational   precision was limited. Nowadays,   JavaScript (and most modern computers   &amp; languages) use IEEE 754 64-bit   floating-point numbers, which provide   15 significant figures of precision.   With this precision, the simple   spherical law of cosines formula (cos   c = cos a cos b + sin a sin b cos C)   gives well-conditioned results down to   distances as small as around 1 metre.   In view of this it is probably worth,   in most situations, using either the   simpler law of cosines or the more   accurate ellipsoidal Vincenty formula   in preference to haversine! (bearing   in mind notes below on the limitations   in accuracy of the spherical model).   Source:    What is the reason why law of cosines is more preferable?  Note: The quoted text has been updated by  as mentioned .
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Does the Schwarzschild radius exist at the exact tipping point where neutron degeneracy pressure fails?  For example, consider a hypothetical neutron star that is being held up just below the limit of neutron degeneracy pressure. If the neutron star gains enough mass (assume a constant radius), it will collapse into a black hole. Does neutron degeneracy pressure fail at the exact point where the neutron star gains enough mass appropriate for the resultant black hole's Schwarzschild radius (where the escape velocity = c)? If so, why do these seemingly unrelated limits match?  Or... does the existing neutron star's radius exceed the resultant black hole's radius? In other words, does neutron degeneracy pressure fail, and as the the neutron star's radius decreases when collapsing, does it eventually reach a smaller (than in the previous scenario) Schwarzschild radius? If this is the case, what is the "in-between" state of matter as the neutron star collapses into a black hole?  Thank you!
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Time dilation  using Schwarzschild metric for a non rotating spherical body is: $$t_0=t_f\sqrt{1-\frac{2GM}{rc^2}}$$  For such a non rotating spherical body, what would the time dilation of a clock in vacuum free-falling from infinity be?  (If the answer is non-trivial; a high level outline of the calculation would suffice / be appreciated)  Edit: I am currently working on an iOS app that is trying to model the mechanism underpinning relativity.  So, far the mechanism that I have created is shockingly simple and shockingly good at conforming to Relativity.  However, I am trying to break it.  I am trying to find any possible areas where the two may diverge.  I have noted that using my model a clock in freefall will experience no time dilation, i.e. $t_0=t_f$ and I want to make sure Relativity agrees.  I have noted the gravitational component of time dilation above.  Since my clock is moving one might also expect a kinematic time dilation.  I can calculate the velocity of my clock: $$E_k=\frac{1}{2}mv^2$$ $$E_p=\frac{-GMm}{r}$$ $$v=\sqrt{\frac{2GM}{r}}$$ Plugging this velocity into the kinematic time dilation equation: $$\Delta t'=\frac{\Delta t}{\sqrt{1-\frac{v^2}{c^2}}}$$ $$\Delta t'=\frac{\Delta t}{\sqrt{1-\frac{2GM}{rc^2}}}$$  At this point one might make the observation that the kinematic dilation is the inverse of the gravitational dilation and therefore conclude that: $$t_0=t_f$$
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It is well-known that measurements of the velocity profile in galaxies are not compatible with Newtonian laws. A way to circumvent the problem is to assume that galaxies are surrounded by a spherical halo of Dark Matter. The density of mass of this Dark Matter can then be computed from the velocity profile with Newton's laws.   My question is the following: the formation of a galaxy involves essentially only the gravitational interaction. According to Newton (and General Relativity too), the trajectory of a particle does not depend on its mass. As a consequence, I do not understand how usual matter and Dark Matter could have followed different trajectories leading to such different mass distributions nowadays. Can somebody explain this (apparent) paradox ?
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I tried doing  just for fun. It is reproduced as follows:          (I'm not allowed to post the entire problem here since I'm a new user.)  Here are my equations:  $$T - mg = ma$$ (equation 1)  $$Mg - T = Ma$$  (equation 2)  (Upper case "M"  used for infinite series of masses on the right arm of the first pulley. Lower case "m" for the small, single mass on the left arm.)  algebra is as follows:  $$Mg - (ma + mg) = Ma$$ $$g (M - m) = a (M + m)$$ $$a = g (M - m) / (M + m)$$   Here is my reasoning to get final answer:   The limit of $a$ for large $M$ goes to $g$. Therefore, acceleration of the top mass is $g$. However, the correct answer is $g/2$. Why does my reasoning get the wrong answer? Any ideas?    (Please excuse the formatting, I'm a newbie here).