What did Einstein mean by-matter curves and warps spacetime?

Question

In general relativity one keeps hearing that. Please tell what is really means and its connection to reality we see around us?

Answer 1

The curvature of spacetime describes tidal gravity, that is gravitational effects that change from one event to another in spacetime. Let’s get a little background.

Let’s exclude gravity for a minute and consider the idea of spacetime without gravity. Spacetime means that we have a 4D space where three dimensions are space and one is time. In this framework a point particle traces out a “worldline” where each “event” on the line represents the location of the particle at a given time. Two intersecting worldlines represent two particles that collide, and two parallel worldlines represent particles that are at rest with respect to each other.

One nice thing about this view is that Newton’s first and second laws can be expressed geometrically. Newton’s first law says that free particles have straight worldlines, and the second law says that the force is proportional to how tightly the worldline bends. So an accelerometer directly measures the amount that a worldline bends.

So to recap: a particle’s spacetime worldline represents the position at each moment in time, if an attached accelerometer reads 0 then the worldline is straight and parallel worldlines represent particles at rest with respect to each other and intersecting worldlines represent particles colliding.

Now, let’s consider gravity. Suppose that we have two objects far outside of a planet but at the same altitude, and let’s say that the planet and the objects only interact gravitationally, so like neutrinos they can just pass through each other. Now, if those two objects start out at rest then that means their worldlines are parallel to each other. Gravity does not activate an accelerometer, so according to our earlier discussion that means that the worldlines are straight (also called geodesics). However, if we wait long enough then they will collide at the center, meaning their worldlines will intersect.

Geometrically, that means that we have two initially parallel straight lines that intersect. This is not possible in flat geometry, but it is possible in curved geometry. For example, consider two nearby longitude lines, longitude lines are great circles, so they are geodesics, or the spherical equivalent of straight lines. At the equator they are parallel, but they intersect at the poles. This is because the geometry on a sphere is curved rather than flat.

This means that curved spacetime geometry represents the effects of gravity. With curvature in spacetime you can have initially parallel straight lines intersect, representing particles initially at rest that collide despite never having accelerometers detect any forces bending their worldlines.

This is what it means for matter to curve spacetime. Tidal gravity, that is gravity that changes from one place to another, has effects that cannot be explained using flat spacetime geometry but can be explained using curved spacetime geometry. The amount of curvature is related to the strength of tidal gravitational effects, and this is what is described by the Einstein field equations.

Taken seriously this idea also predicts gravitational effects on time, which have been measured to very high precision. So it appears to be more than just an odd metaphor. To date this approach has passed all experimental tests while ruling out many alternatives.

Answer 2

I would like to complement Dale’s answer with a bit of the mathematics of General Relativity.

It isn’t only matter that warps spacetime; radiation and other forms of energy do also. Spacetime curvature is actually caused by the density and flow of energy and momentum, represented by the energy-momentum tensor $T_{\mu\nu}$ on the right side of the Einstein field equation,

$$G_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}.$$

On the left side of the equation is another tensor, $G_{\mu\nu}$, measuring the curvature of spacetime. Einstein realized that the two are simply proportional!

Everyone can intuitively understand that a two-dimensional surface, such as a sphere, can have curvature. But how can three-dimensional space, or four-dimensional spacetime, have curvature?

Dale described a curved space as one in which initially parallel lines don’t stay parallel. Another way to think about it is that a curved space is one in which the Pythagorean Theorem doesn’t hold. On the curved surface of a sphere, there is no coordinate system in which

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

as on a plane. Similarly, although you are used to thinking that

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

in 3D space and

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

in 4D Minkowski spacetime, these equations actually apply only when the space or spacetime is “flat” because of the absence of energy and momentum. When you generalize the notion of infinitesimal separation between spatial points or spacetime events so that $ds^2$ can be an arbitrary quadratic form like

$$\begin{align}ds^2&=g_{xx}(x,y,z)dx^2+g_{yy}(x,y,z)dy^2+g_{zz}(x,y,z)dz^2\\&+2g_{xy}(x,y,z)dx dy+g_{xz}(x,y,z)dx dz+g_{yz}(x,y,z)dy dz\end{align}$$

for a 3D space, you have what is called Riemannian geometry instead of Euclidean geometry. The six functions $g_{ij}(x,y,z)$ are components of what is called the “metric tensor” because this tensor tells you how to measure distances in the space.

You can imagine a similar expression with more terms for spacetime.

The curvature tensor $G_{\mu\nu}$ that measures how “non-Euclidean” space is, or how non-Minkowskian spacetime is, is an expression that depends on the metric tensor $g_{\mu\nu}$ and its first and second derivatives. When there is no energy and momentum present, the metric is “flat”, all the components of the metric tensor are just 1 or 0 (in Cartesian coordinates), and all components of the curvature tensor are zero.

This is boring. What is interesting is when the presence of energy and momentum cause the metric to be non-Pythagorean. Gravity -- from apples falling, to black holes, to the expanding universe -— is just the consequences of the metric being allowed to be non-Pythagorean!