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jmcvirgo
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Suppose you're given a 4 tuple and told that its scalar product with any 4-vector is a lorentz scalar. How do I show that this implies the 4-tuple is a 4-vector?
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I suggest that you be a bit more careful with the terminology. A 4-tuple can't ever be 4-vector. In order to define a 4-vector you must specify a 4-tuple for each coordinate system. It's the assignment of 4-tuples to coordinate systems that defines a 4-vector, not a single 4-tuple. The assignment is of course usually done by specifying the 4-tuple that you want to associate with a specific coordinate system, and then explicitly stating that the 4-tuples associated with all the other coordinate systems are given by the tensor transformation rule.jmcvirgo said:the 4-tuple is a 4-vector?
Fredrik said:I suggest that you be a bit more careful with the terminology. A 4-tuple can't ever be 4-vector. In order to define a 4-vector you must specify a 4-tuple for each coordinate system. It's the assignment of 4-tuples to coordinate systems that defines a 4-vector, not a single 4-tuple. The assignment is of course usually done by specifying the 4-tuple that you want to associate with a specific coordinate system, and then explicitly stating that the 4-tuples associated with all the other coordinate systems are given by the tensor transformation rule.
A 4-vector is a mathematical construct used in special relativity to describe the position and momentum of an object in 4-dimensional spacetime. It is represented by a 4-tuple, with four components representing the object's position in the three dimensions of space and its position in time.
The 4-vector character of a 4-tuple is determined by using the Lorentz transformation equations. These equations help to convert coordinates between different reference frames and allow us to calculate the 4-vector character of a 4-tuple in a given frame of reference.
Determining the 4-vector character of a 4-tuple is important in understanding the behavior of objects in special relativity. It allows us to accurately describe an object's position and momentum in different reference frames, which is crucial in making accurate predictions and calculations in this branch of physics.
Yes, there are many real-world applications of determining the 4-vector character of a 4-tuple. For example, it is used in particle physics to calculate the interactions between particles, and in astrophysics to study the behavior of objects in space. It is also used in the development of technologies such as GPS systems and satellite communication.
Yes, the 4-vector character of a 4-tuple is directly related to the concept of space-time. In special relativity, space and time are considered to be interconnected, and the 4-vector character of a 4-tuple allows us to describe an object's position and momentum in this interconnected space-time. This concept is essential in understanding the behavior of objects moving at high speeds and in different reference frames.