We all know that Einstein's general theory of relativity is one of the cornerstones of modern physics, revealing the profound connection between spacetime and matter. General relativity tells us that space-time is not static, but a dynamic entity that changes with the distribution and movement of matter.
In many popular science articles, curved spacetime and warped spacetime are used interchangeably. But, in general relativity, the curvature and warping of spacetime are not the same thing. So, what is the difference between curved spacetime and warped spacetime?
What is time and space
First, we need to understand what space-time is. Simply put, space-time is our lifeA living four-dimensional world, which includes three spatial dimensions and one time dimension. We can use a coordinate system to describe any event or object in space-time, such as (x, y, z, t), where x, y, z represent the spatial position, and t represents the time.
Space-time is not an abstract mathematical concept, but a real physical entity. It can be measured and observed. For example, we can use light to probe the properties of spacetime. Light travels along a straight line in a vacuum, and this straight line is the shortest path in space-time, also called a geodesic. In this way, we can tell whether space-time is curved or warped by observing the deflection of light rays.
Curvature can describe the angular deviation between different directions in spacetime, which is expressed by the Riemann curvature tensor. The Riemann curvature tensor is a fourth-order antisymmetric tensor field, which is defined as: R(X,Y,Z,W)=g(R(X,Y)Z,W). where X,Y,Z,W are arbitrary vector fields, gis a metric on the Riemannian manifold, and R(X,Y)Z is a vector field that represents the amount of change in the Z vector after a parallel movement along the X and Y directions.
The Riemann curvature tensor measures the anti-commutativity of covariant derivatives, i.e. the effect of the order of parallel shifts on the result. If the Riemann curvature tensor is zero, then the covariant derivative is commutative, i.e. the order of the parallel shifts does not matter. In this case, the manifold is flat.
Now, we describe the geometric meaning of curvature in a more understandable way. When there is curvature in space-time, after a vector translates along the closed curve for one week, it does not coincide with the original vector, but differs by an angle. It is necessary to add another rotation before they can overlap, and this additional rotation is just the geometric effect produced by the curvature (bending) of space.
In physics, the Riemann curvature tensor can describe the gravitational field or matter-energy distribution existing in space-time, which will make space-time bend. For example, in general relativity, the gravitational field equation is a function of the Riemannian curvatureEquations for tensors and energy-momentum tensors, which reflect the effects of matter and energy on the curvature of spacetime. In this theory, spacetime is curved.
Distorted time and space
Distortion can describe the translational or rotational deviation between different points in spacetime, which is expressed by the torsion tensor. The torsion tensor is a third-order antisymmetric tensor field, which is defined as: T(X,Y)=XYYX[X,Y]. Where X,Y is an arbitrary vector field, is an arbitrary affine connection, and [X,Y] is the Lie brackets of the vector field.
The torsion tensor measures the asymmetry or non-metric nature of the connection, that is, the covariant derivative does not agree with the exchange of the vector field. If the torsion tensor is zero, then the connection is symmetric or metric, that is, the covariant derivative agrees with the commutation of the vector field. In this case, the connection is the Levi-Civita connection, which is the only certain metric connection on the Riemannian manifold.
Likewise, we describe the geometric meaning of torsion in understandable language. There are two vectors for the empty point O and they are OQ and OQ respectively. OQ translates to point Q along the direction of OQ to obtain vector OP; OQ translates to point Q along the direction of OQ to obtain vector OP. If there is no torsion in the space, then point P and point P coincide; if there is torsion in the space, then the two points do not coincide, and a movement must be added to coincide. And this additional movement is the geometric effect of torsion (distortion).
In physics, the torsion tensor can describe the spin-spin interaction or spin-orbit interaction existing in space-time, which will make space-time warp. For example, in the Einstein-Kaltan theory, the gravitational field equation includes the torsion tensor as a source term, which reflects the spin density of matter. In this theory, space-time is not only curved, but warped.