What dimension does spacetime curve into?

four dimensions
Space is indeed curved — in four dimensions. Many people think the fourth dimension is simply time, and for some astronomical equations, it is. Einstein used time as a fourth dimension to describe a coordinate system called space-time.

What is the difference between extrinsic and intrinsic curvature?

Intrinsic curvature comes from the parallel translation of a vector tangent to the path of translation. If a vector is translated around a loop and it fails to come back onto itself that is intrinsic curvature. Extrinsic curvature is computed by the parallel translation of a vector normal to a surface or space.

What is the curvature of a circle at any point on it?

At every point on a circle, the curvature is the reciprocal of the radius; for other curves (and straight lines, which can be regarded as circles of infinite radius), the curvature is the reciprocal of the radius of the circle that most closely conforms to the curve at the given point (see figure).

How is curvature defined?

Definition of curvature 1 : the act of curving : the state of being curved. 2 : a measure or amount of curving specifically : the rate of change of the angle through which the tangent to a curve turns in moving along the curve and which for a circle is equal to the reciprocal of the radius.

What causes the curvature of spacetime?

Gravity is the curvature of spacetime Gravity is the curvature of the universe, caused by massive bodies, which determines the path that objects travel. That curvature is dynamical, moving as those objects move. In Einstein’s view of the world, gravity is the curvature of spacetime caused by massive objects.

Is Gaussian curvature intrinsic?

Gaussian curvature is an intrinsic measure of curvature, depending only on distances that are measured on the surface, not on the way it is isometrically embedded in Euclidean space. Gaussian curvature is named after Carl Friedrich Gauss, who published the Theorema egregium in 1827.

Is mean curvature intrinsic?

Gaussian curvature
Curvature is in general an extrinsic property of a surface. For example, mean curvature H is extrinsic because it depends on how the surface is embedded in a 3 (or higher) dimensional coordinate system. In contrast, the Gaussian curvature K is an intrinsic property of the surface.

Does curvature have direction?

Intuitively, the curvature describes for any part of a curve how much the curve direction changes over a small distance travelled (e.g. angle in rad/m), so it is a measure of the instantaneous rate of change of direction of a point that moves on the curve: the larger the curvature, the larger this rate of change.

Why curvature is reciprocal of radius?

The center of curvature of the curve at parameter t is the point q(t) such that a circle centered at q which meets our curve at r(t), will have the same slope and curvature as our curve has there. We will see that the radius of curvature, which is a length is exactly , the reciprocal of the curvature.

Can curvature be negative?

A surface has positive curvature at a point if the surface curves away from that point in the same direction relative to the tangent to the surface, regardless of the cutting plane. A surface has negative curvature at a point if the surface curves away from the tangent plane in two different directions.

What does the intrinsic curvature of a surface mean?

The other benefit of the intrinsic based metric tensor is that it is applicable in all GR calculations, (invariance). Exactly what you wrote. It means that a surface appears curved when looked from the space that contains it. Take the circle, S 1, for example. The intrinsic curvature would be what people call Riemann tensor.

What is the extrinsic curvature of paper?

If we consider a plane sheet of paper (whose intrinsic curvature is zero) rolled into a cylindrical shape, then we say that its extrinsic curvature is non-zero. So how can I visualize the extrinsic curvature? I read somewhere that the extrinsic curvature indicates how the 2D surface is embedded in 3D space. So what does it mean?

What is the curvature of the surface of a cylinder?

So the surface of a cylinder has no intrinsic curvature, but it does have extrinsic curvature when embedded in the obvious way into 3-space. Thanks for contributing an answer to Mathematics Stack Exchange!

What is the difference between DTDs and curvature in physics?

In the limit dTds will be in the direction N and the curvature describes the speed of rotation of the frame. This is the magnitude of the acceleration of the particle and the vector dTds is the acceleration vector. Geometrically, the curvature κ measures how fast the unit tangent vector to the curve rotates.