Nearly Lorentz Coordinate Systems: Is h a Tensor?

In summary, Schutz introduces Nearly Lorentz coordinate systems as having a metric that is a small deviation from the normal Minkowski metric. He then introduces Background Lorentz transformations and shows that the deviation, h, can be treated as a tensor in special relativity. However, some may question why this proof is necessary since h is simply the difference between two tensors. Additionally, Schutz clarifies that h is not a tensor, but just a part of the metric. To understand this, it is important to understand why ##\eta## is considered a tensor in the first place. It is a function that takes two vectors and produces a real number, behaving like a ##(0,2)## tensor when applied to a
  • #1
Silviu
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Hello! I am reading Schutz A first course in GR and he introduces the Nearly Lorentz coordinate systems as ones having a metric such that ##g_{\alpha\beta} = \eta_{\alpha\beta} + h_{\alpha\beta}##, with h a small deviation from the normal Minkowski metric. Then he introduces the Background Lorentz transformations (this is section 8.3 in the second edition) in which all the points are transformed as ##x^{\bar\alpha}=\Lambda_\beta^{\bar\alpha}x^\beta##. Applying this transformation to g he gets in the end that h transforms as ##h_{\bar\alpha\bar\beta}=\Lambda_\mu^{\bar\alpha}\Lambda_\nu^{\bar\beta} h_{\mu\nu}## and from here he says that we can treat h as if it was a tensor in SR and this simplify the calculations a lot. Can someone explain to me why ones need all these calculations for this? I am sure I am missing something but h is the difference between g and ##\eta## so isn't it a tensor, just because it is the difference between 2 tensors? Why do you need a proof for it? Moreover Schutz says that h "it is, of course, not a tensor, but just a piece of ##g_{\alpha\beta}##". So can someone explain to me why isn't h a tensor and why my logic is flawed? Thank you!
 
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  • #2
Why is ##\eta_{\alpha\beta}## a tensor?
 
  • #3
martinbn said:
Why is ##\eta_{\alpha\beta}## a tensor?
Well a tensor (in this case a ##(0,2)## tensor) is a function that turns 2 vectors into a real number. ##\eta## is a 4x4 matrix so it behaves like a ##(0,2)## tensor, when applied to a 4D vector (which is our case).
 

FAQ: Nearly Lorentz Coordinate Systems: Is h a Tensor?

1. What is a Nearly Lorentz Coordinate System?

A Nearly Lorentz Coordinate System is a coordinate system used in physics that approximates a Lorentz coordinate system, which is a coordinate system that obeys the principles of special relativity and is used to describe the spacetime of a flat, isotropic universe. A Nearly Lorentz Coordinate System is used when the spacetime is not exactly flat or isotropic, but is close enough to be approximated as such.

2. What is the difference between a Nearly Lorentz Coordinate System and a Lorentz Coordinate System?

The main difference between a Nearly Lorentz Coordinate System and a Lorentz Coordinate System is that a Nearly Lorentz Coordinate System is an approximation of a Lorentz Coordinate System. This means that a Nearly Lorentz Coordinate System may not perfectly adhere to the principles of special relativity, but is close enough to be an effective tool for describing the spacetime of a system. On the other hand, a Lorentz Coordinate System strictly adheres to the principles of special relativity and is used to describe the spacetime of a flat, isotropic universe.

3. Why is h considered a tensor in Nearly Lorentz Coordinate Systems?

In Nearly Lorentz Coordinate Systems, h is considered a tensor because it transforms like a tensor under coordinate transformations. This means that its components change in a specific way when the coordinate system is changed, making it a useful tool for describing the spacetime of a system.

4. How is h used in Nearly Lorentz Coordinate Systems?

In Nearly Lorentz Coordinate Systems, h is used as a metric tensor to describe the spacetime of a system. It is used to calculate the interval between two events, which is a measure of the distance in spacetime between the events. This interval is an important concept in special relativity and is used to determine the laws of physics in different reference frames.

5. What are the practical applications of Nearly Lorentz Coordinate Systems?

Nearly Lorentz Coordinate Systems have many practical applications in physics and engineering. They are used to describe the spacetime of systems that are not exactly flat or isotropic, such as in cosmology and general relativity. They are also used in the design and testing of high-speed vehicles, such as airplanes and spacecraft, where the effects of special relativity must be taken into account.

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