How can space time be bent? [duplicate]

Question

Einstein’s theory of general relativity states that Gravity is not a force per say but it is the bending of the space-time grid making a dent by the object it is holding that creates a gravitational pull! But Space is space, IT’S EMPTY, how can it possibly behave as a surface made up of grids that bends when it holds a celestial body?

What I don’t get is how can empty space time with absolutely nothing bent while holding celestial bodies? I am well aware of the trampoline analogy but Trampoline is ‘made up’ of something, i.e. rubber which is made up of atoms and molecules i.e. matter and has this “property” to stretch! Space time is not made up of anything! So how can it be bent and stretched?

Besides doesn’t it contradict the special theory of relativity which states that space-time are FLAT! How can both of them be true at the same time?

Answer 1

But Space is space, IT’S EMPTY.

Ok. But, there is no cause for alarm.

how can it possibly behave as a surface made up of grids that bends when it holds a celestial body? Besides doesn’t it contradict the special theory of relativity which states that space-time are FLAT!

General relativity is different from special relativity. You are correct that spacetime is flat in special relativity. But this is not true in general relativity.

This is the whole point of general relativity. Namely, to account for gravity via the introduction of curvature to the metric.

The "metric" can be thought of as a 4x4 matrix that instructs us how to perform the dot product of four-vectors. In special relativity the metric is constant and diagonal. In general relativity the metric is not, and this is what we mean when we say spacetime is curved.

How can both of them be true at the same time?

As discussed above, they are not really true "at the same time." Special relativity can be thought of as a special case of general relativity. Though both can be thought of as "true" in their realm of applicability.

This as analogous to how Newtonian dynamics is a special case of relativistic dynamics. They aren't true "at the same time." But Newtonian dynamics is still very useful for most calculations.

As another analogy, consider the surface of the earth. If you walk along a street in your home town, it seems pretty flat. But then someone tells you the earth is round. How can they both be true "at the same time." They are not really applicable to the same situation: One deals with a local perspective and one deals with a global perspective.

Answer 2

"Stretch" is just kind of an analogy.   What mass and energy do is change rules of geometry in their vicinity, which govern how points in space and time are connected to each other.  For instance, in normal Cartesian Coordinates, the formula for distance $ds$ between two points very close together is

$ds^2 = dx^2 + dy^2 + dz^2$

This is simply Pythagorean Theorem.
In special relativity, we need to add time to this to define a "space time distance" and for physical reasons it is the opposite sign to spatial coordinates, so we get:

$ds^2 = –c^2dt^2 + dx^2 + dy^2 + dz^2$

This is a "flat" spacetime. Now- going a step further, in general relativity, we have to account for the effect of curvature caused by mass and energy, so the distance between two nearby points becomes:

$ds^2 = g_{00} dt^2 + g_{11} dx^2 + g_{22} dy^2 + g_{33} dz^2 +...$ (and possibly cross terms like $g_{01} dt\ dx$...)

These $g$ coefficients form a 4x4 matrix called the spacetime metric, which is the main thing that physicists use Einstein's equations to solve for in a given scenario, given a certain distribution of mass and energy.    The coefficients themselves can be functions of $x, y, z, t$

So in a nutshell, it's not really "bending and stretching," it is altering geometry which forces the trajectories of objects to change, or "bend."