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- Thread starter redtree
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In summary, the convention for scalar products of vectors in general relativity is to use (+,+,+,+) instead of the standard (+,+,+,-,). This choice arises from the geometrical structure of spacetime, and there is no cross-over with the metric. Tensor calculus is often necessary to understand the structure of a space-time manifold, and GR allows for more complex coordinate systems than classical physics.f

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[itex]v_\mu = (v_0, v_1,v_2, ... v_3)[/itex] and when you raise the index with the metric, you get [itex]v^\mu = (v_0, -v_1, -v_2, ... -v_3)[/itex].

Hope that helps.

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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.

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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 =)

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.

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Because that's not what nature chose to give us. Also, you'll notice that there's no cross-over with that metric. In other words, there is no limiting speed like the speed of light.

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- #8

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That's not actually a "convention".

The signature arises from the geometrical structure of the space you are dealing with.

Often one deals with a Euclidean space... which is often implicit... but it's there.

In relativity, as others have pointed out, the geometrical structure of spacetime has a Lorentzian-signature metric tensor... with one sign different from the rest.

Whether it's (-,+,+,+) or (+,-,-,-) or (+,+,+,-) or (-,-,-,+) is the choice of convention. (It's not just the signs... it's also the use of x0 or x4. These days x0 is preferred to allow consideration of more or fewer spatial dimensions.) Pick your favorite, make it known, and use it consistently [and be prepared to translate if necessary].

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Because it would require complex coordinates, which is pretty irritating!

Pete

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