Suppose we try to imagine a 1+1D analogy of spacetime (1 dimension for space + 1 dimension for time). Lets say there is a point mass $m$ at $x = 5$. So the world line would be the line $x = 5$ in $x$-$t$ spacetime.
Now let's assume there is a point mass ($M >> m$) at the origin. So my question is, how would the spacetime bend exactly (what's the exact mathematical description of bending), that would make it seem like the object is being accelerated at $\frac{-GM}{r^2}$ when its at $x=r$? And what would be the universal "postulate" here, which direction do objects travel in the space time here and at what speed. Some reasoning would also be great as to why is this the case.
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3$\begingroup$ Related: physics.stackexchange.com/q/303999 $\endgroup$– AmitCommented 2 days ago
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2$\begingroup$ Also related: physics.stackexchange.com/q/1417/2451 $\endgroup$– Qmechanic ♦Commented 2 days ago
1 Answer
It will depend on the nature of the mass $M$, but there is a good mathematical description of this.
The math behind the relativity
Consider the mass $m$ whose worldline is clearly curved by the larger mass’s presence. For simplicity’s sake, let’s say that the large mass is a Schwarzschild black hole (we could use the weak-field approximation, but in my opinion it’s a bit overcomplicated), so our spacetime is described adequately by the Schwarzschild metric,
$$c^2\text{d}\tau^2=k_S\text{d}t^2-k_S^{-1}\text{d}r^2-\text{d}\theta^2-r^2\sin^2(\theta)\text{d}\phi^2$$
where $k_S=1-\frac{r_s}{r}=1-\frac{v_e^2}{c^2}=1-\frac{2GM}{rc^2}$ is a factor related to the spacetime. In your 1+1D case, this would be
$$c^2\text{d}\tau^2=k_S\text{d}t^2-k_S^{-1}\text{d}x^2.$$
This spacetime describes a gravitating black hole; the reason it “gravitates”, in GR, is that worldlines that are “straight” actually are curved in coordinate spacetime, by an amount prescribed by the geodesic equation:
$$\frac{\text{d}^2X^\alpha}{\text{d}\tau^2}=-\Gamma^\alpha_{\mu\nu}\frac{\text{d}X^\mu}{\text{d}X^\nu},$$
where $X$ is coordinate position, $X=(t,r,\theta,\phi)$ or $X=(t,x)$, and where $\tau$ is the proper time, an affine parameter describing the amount of time an observer following the curve experiences. Note that the Lorentz factor is actually defined as $\frac{\text{d}t}{\text{d}\tau}$, so in the low-velocity weak-field approximation we can take that to be 1. We also implicitly sum over repeated indices, so on the RHS, there’s an implied $\sum_\mu\sum_\nu$; that’ll be the case for my other equations too.
The $\Gamma$ terms are the connection coefficients, which are derived from the metric and exactly describe how much geodesics, i.e. the curves things follow in curved space, actually deviate “straight lines” drawn as one normally would. We also call them the Christoffel symbols in a holonomic coordinate basis, which we’ve chosen, but those are details that aren’t totally relevant; in such a basis, they’re defined as
$$\Gamma_{\mu\nu\lambda}=\frac{1}{2}(g_{\mu\nu,\lambda}+g_{\mu\lambda,\nu}-g_{\nu\lambda,\mu}),$$
where $g_{\mu\nu}=e_\mu\cdot e_\nu$ are the components of the metric tensor such that the expression for the metric is $c^2\text{d}\tau^2=g_{\mu\nu}\text{d}X^\mu\text{d}X^\nu$those components being expressible as an $n\times n$ matrix in $n$-dimensional space, and where $g_{\mu\nu,\lambda}$ is shorthand for $\frac{\text{d}g_{\mu\nu}}{\text{d}X^\lambda}$. You get the mixed-index version of the symbols that you use in the geodesic equation, $\Gamma^\mu_{\nu\lambda}$, by the process of index raising: $\Gamma^\mu_{\nu\lambda}=g^{\mu\rho}\Gamma_{\rho\nu\lambda}$. There, we take the inverse metric $g^{\mu\nu}$ to be the matrix inverse of the regular metric.
For the Schwarzschild metric, you eventually calculate that there are Christoffel symbols that lead to a negative radial acceleration, i.e. $\frac{\text{d}r}{\text{d}\tau}\approx\frac{\text{d}r}{\text{d}t}<0$. The question of calculating those symbols has already been dealt with here.
The explanation behind the math
The concept of metrics and connection coefficients/Christoffel symbols isn’t an invention of relativity; it is a set of tools that are part of differential geometry. GR simply postulates that spacetime can be treated as a 3+1-dimensional Lorentzian manifold; every other prediction is just looking at particular properties of such a manifold.
The exact curvature that leads to GR gravity is pretty much said in the connection coefficients. You could say that a non-flat metric (i.e. a metric with nonzero components in the Riemann tensor or others, those being other differential geometry tools) implies gravitation, but not all non-flat metrics have what we think of as “gravity” in them. In the simple case of the Schwarzschild metric, the curvature encoded in the very simple spherically-symmetric solution produces an inverse-square gravity when time dilation is not significant.