Shabazzle asked in Science & MathematicsPhysics · 5 months ago

# what is the relationship between space, time and gravity, how do they 'work' together, and could they be the same thing ?

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• neb
Lv 7
5 months ago

Gravity is a metric tensor field defined at each point of spacetime. The metric tensor - and how it changes with position/time - define the geometry of spacetime. This is the conventional view of the relationship between gravity and spacetime.

A deeper view comes from the principle of active general covariance. Einstein’s field equations define an invariant relationship between the metric tensor and the sources of gravity. This invariance under active transformations of space and time, is called a diffeomorphism. This means that Einstein’s field equations are independent of a background spacetime. As long as we continuously deform spacetime - as well as the gravitational field and sources of gravity - the relationships remain invariant. It would appear then that ‘spacetime‘ has no independent existence or meaning without gravity.

• 5 months ago

No, they are not the same thing but are closely related.

The most common mathematical model on which special relativity is formulated is a combination of three-dimensional Euclidean space and time so that they make up a four-dimensional manifold in which intervals between events are independent of the inertial frames of reference. This mathematical model is developed by Hermann Minkowski initially for Maxwell's equations of electromagnetism and is known as the "Minkowski spacetime". In 1907, Minkowski showed that special theory of relativity (Einstein was a former student of Minkowski) could be understood geometrically, utilizing four-dimensional spacetime.

In popular culture this spacetime is often confused with four-dimensional Euclidean space, popularized by Charles Howard Hinton, starting in 1880. Because it treats time differently than the 3 spatial dimensions, this space differs from four-dimensional Euclidean space which treats 4 spatial dimensions equally.

In spacetime, 3 spatial dimensions and time are a different thing but are interwoven.

In a Euclidean space, the interval between two points is measured by the purely spatial distance between them. In spacetime, the interval separating two events takes into account not only the spatial distance between them but also their distance in time. The interval between two events is:

s² = Δr² - c²Δt²

where c is the speed of light, and Δr and Δt are differences in space and time coordinates, respectively, between the events.

General relativity published by Einstein in 1915 is a continuation of special relativity that introduced the geometric concept of gravitation. It upgrades Newton's law of universal gravitation, which is special case of GRT. Rather than Euclidean geometry, GRT uses curvilinear coordinates in which the angles between axes can change from point to point.

Think of the surface of the Earth. Maps frequently show meridians and parallels as a simple rectangular grid, in reality, meridians are not just curved but moreover meet at the north and the south pole. This is because Earth is round. The closest distance for travel between two cities is not a straight line but instead a circular arc.

In general relativity, the presence of energy and mass changes the curvature of spacetime. This curvature causes the gravitational force. A popular analogy is a heavy object on a stretched rubber sheet, which bends the sheet downward and deforms the grid drawn on the sheet, just like a massive star bends the space in its vicinity.

The math here is more mind-boggling than in spherical geometry, as it deals with 4 dimensions (3 spatial+time) of curved coordinates instead of three.