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edpell
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How many numbers does it take to describe a point in space in GR?
hamster143 said:Just the location? Four.
diazona said:there is a lot of extra information that goes into Einstein's equation that is not built into the 4 coordinates of a point - like the energy density at the point, the curvature at the point, etc.
edpell said:How many numbers does it take to describe a point in space in GR?
Nabeshin said:Indeed, one can completely determine the metric at any point in spacetime with 10 numbers. If one allows for coordinate transformations, a clever person could get this down to only six numbers.
edpell said:Not being smart I will stick with 10.
So each point can be described by four numbers for the coordinate and 10 numbers for one symmetric 4x4 tensor a total of 14 numbers.
hamster143 said:A well posed question is half the answer.
The question "where is it" can be answered with four numbers, x, y, z, t, presupposing the coordinate system.
The question "what's the relationship between distances and the coordinate chart in the vicinity of that point" can be answered with ten numbers, ten components of the metric tensor, again presupposing the coordinate system.
The question "what's the geometry and the curvature in the vicinity of that point" can be answered with twenty additional numbers, due to the fact that the most general Riemann tensor has twenty independent components.
The question "what's the physics over there" requires four to infinity numbers, four in the simplest case of non-interacting dust, physically interesting cases will, at the very least, require pressure, viscosity strains and temperature, for the total of 11 (10 independent components of the stress-energy tensor + temperature). If you have multiple fields in the same region of space, you need the complete description of all internal degrees of freedom for each.
The answer to this question depends on the specific context and coordinate system being used. In GR, points in space are typically described using four numbers, known as coordinates. These coordinates can be represented in various coordinate systems, such as Cartesian or spherical, each of which may require a different number of numbers to describe a point.
GR is based on the theory of relativity, which states that space and time are intertwined and cannot be considered separately. Therefore, points in space are described using four numbers, known as spacetime coordinates, to account for both space and time dimensions. This allows for a more complete and accurate description of the universe.
Yes, there are situations where more or less than four numbers may be needed to describe a point in space in GR. For example, in certain coordinate systems or spacetimes, more than four numbers may be required to accurately describe a point. Additionally, in simplified models or approximations, fewer than four numbers may be used to describe a point in space.
No, a point in space cannot be accurately described using only one number in GR. As mentioned earlier, a point in space is typically described using four numbers to account for both space and time dimensions. Attempting to describe a point using only one number would oversimplify the complex nature of space and time in GR.
The numbers used to describe a point in space in GR are related to the curvature of spacetime through the theory's fundamental equations, known as the Einstein field equations. These equations describe how matter and energy cause spacetime to curve, and the coordinates of a point in space are used to map out this curvature and its effects on the surrounding spacetime.