Space bends relative to what?

Question

We all are aware of gravitational waves, as it bends space and time, black hole squeeze space, but the squeezing, bending, expanding happens reference to what? Since the observable universe is the universe existing within itself, so it bends in reference to whose perspective?

Answer 1

The easiest way to understand this is to understand the notion of extrinsic versus intrinsic curvature.

Extrinsically curved things follow non-straight lines relative to the space they are contained in.

Intrinsically curved things, however, obey non-Euclidean laws on their own surfaces.

Now, if you fit an intrinsically curved space into an enveloping Euclidean space, it will also be extrinsically curved, but the opposite is not true -- you can turn a sheet of paper into a cylinder and back and not distort it, but you cannot do this with an orange peel.

Now, let's get back to general relativity -- it explains gravity as curvature of spacetime, but this curvature is an intrinsic curvature -- it makes sense in spacetime, without having to refer to some larger space it is contained in. You can tell you're in a curved spacetime just by observing that Euclid's parallel postulate is violated. Most dramatically, this is true, because if you start out with two objects high above earth, initially at rest (and therefore, initially travelling in parallel paths through spacetime), their paths will intersect at the center of the Earth (at least, if the earth somehow didn't stop them).

Answer 2

This is a fundamental question explored in non-Euclidean geometry. Here are two easy to imagine consequences of the bending of space-time, imagined as just effects on space.

First, as is used in the expanding universe model, the size of the universe, it's scale factor, can change with time. In that scenario the wavelength of waves get stretched when measured compared to quantities that arise from oscillations that do not involve spatial movement, like the radius of an atom. The Bohr radius, for example, is fixed by the mass of the electron ($m_e$), the speed of light ($c$), the conversion between energy and frequency (Planck's constant, $h$), and the strength of the interaction between photons and electrons (the fine structure constant, $\alpha$).

Second, is to answer the question: what is the relationship between the radius of a circle and it's circumference? When space is flat, unbent, the answer is $C = 2\pi r$. If space becomes bent, though, the measured value can be changed by a small amount in a way that depends on the size of the circle. This scenario is easiest to visualize by considering two dimensional surface from the outside. In a geometry like the surface of a sphere, the circumference of circles will be smaller than $2\pi r$, by more and more the bigger the circle. In the extreme case, if $R$ is the radius of the sphere then when $r = \pi R$ the circumference of the circle is $0$. The general formula for the surface of a sphere is $C = 2\pi R \sin\left(\frac{r}{R}\right) \approx 2\pi r \left(1 - \frac{r^2}{6 R^2}\right).$ The quantity measured by WMAP and Planck that is proportional to $R$ is known as the spatial curvature density, $\Omega_k$.

So, it is possible to measure the curvature of space-time without reference to any external standards.

Answer 3

You do not need to have an "external reference" to see consequences of the bending of spacetime. For instance, light travels on a straight line if spacetime is flat. If spacetime is locally bended by a very massive object, a ray of light will follow a curved path when travelling close to this object.