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redtree
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What is the rationale for the sign convention in the space-time 4-vector? How is it related to the sign convention in the energy-momentum 4-vector, if at all?
Relativists seem to prefer the latter, while field theorists seem to prefer the former. Which sucks when you're both a relativist and a field theorist =)jtbell said:I assume you mean, why do we write [itex]ds^2 = (cdt)^2 - dx^2 - dy^2 - dz^2[/itex] rather than [itex]ds^2 = dx^2 + dy^2 + dz^2 - (cdt)^2[/itex]?
For one thing, with this choice, we get a positive number for a "causally connectable" spacetime interval. Also, if we use the same convention for all four-vectors, then for energy-momentum [itex]E^2 - (p_x c)^2 - (p_y c)^2 - (p_z c)^2 = (m_0 c^2)^2[/itex] which is also a positive number.
We could do it the other way, but it seems to me that this way, we get fewer minus signs associated with the invariant quantities that we're usually interested in.
redtree said:My real question is the following: why we don't use the standard convention for scalar products of vectors (+,+,+,+) in GR?
redtree said:My real question is the following: why we don't use the standard convention for scalar products of vectors (+,+,+,+) in GR?
Because it would require complex coordinates, which is pretty irritating!redtree said:My real question is the following: why we don't use the standard convention for scalar products of vectors (+,+,+,+) in GR?
The sign convention in the space-time 4-vector refers to the convention used to describe the direction and magnitude of a vector in space and time. It is used in special relativity to represent four-dimensional spacetime, where the first three dimensions represent space and the fourth dimension represents time.
The sign convention in the space-time 4-vector is significant because it allows for the consistent representation of events in space and time. It helps to describe the relationship between space and time and enables calculations in special relativity to be performed accurately.
The sign convention is represented by the use of plus and minus signs in the components of the vector. The first three components represent the spatial dimensions and are denoted by x, y, and z, while the fourth component represents time and is denoted by ct, where c is the speed of light. A positive sign indicates motion in the positive direction, while a negative sign indicates motion in the negative direction.
Yes, there are two common sign conventions used in the space-time 4-vector. The first is the Minkowski or (-,+,+,+) convention, which is commonly used in theoretical physics and relativity. The second is the mostly plus or (+,-,-,-) convention, which is commonly used in particle physics and cosmology. Both conventions are valid and can be used in calculations, but it is important to be consistent within a specific context.
The sign convention does not affect the interpretation of the space-time 4-vector. The interpretation remains the same regardless of which convention is used. However, the sign convention does affect the numerical values of the components in the vector and the resulting calculations. It is important to be aware of the sign convention being used in order to interpret the results accurately.