Lorentz invariance and General invariance

In summary: In general relativity, the laws of physics are generally covariant, and hence Lorentz covariant, unless we want to use "generally invariant" quantities (ie. generally covariant scalars) where we are not allowed to use Christoffel symbols.In summary, Lorentz invariance/covariance refers to the "form" of the laws of physics and the fact that certain quantities (such as mass) are the same in all inertial reference frames. General invariance/covariance refers to the fact that the laws of physics maintain the "same form" under arbitrary coordinate changes and that certain quantities (such as generally invariant scalars) have the same value at any point in spacetime.
  • #1
mtak0114
47
0
Hi
I am confused about these two related but different terms
Lorentz invariance/covariance and General invariance/covariance

As I understand it a Lorentz invariant is a scalar which is the same in all inertial reference frames i.e. it acts trivially under a Lorentz transformation
an example would be rest mass [tex]p^\mu p_\mu = m^2[/tex], all observers would agree on the value of the mass. But is this true for all scalars say for example the inner product between two arbitrary 4-vectors [tex]V^\mu W_\mu = C[/tex] would all inertial observers agree on the value of C? A good example may be the inner product between the 4-velocity and the 4-acceleration [tex]u^\mu a_\mu = 0[/tex].

How does this change for general invariance

[tex]p^\mu p_\mu = m[/tex] it is true that all observers would agree on the mass but how about for:

[tex]V^\mu W_\mu = C[/tex]

is this still a constant C or is it a spacetime dependant quantity [tex]C(x^\mu)[/tex]?

What about [tex]u^\mu a_\mu = 0[/tex]
I understand that this is still true in general relativity but is this a special scalar?

still very confused hope you can help

thanks

Mark
 
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  • #2
A scalar g(u,v) is a generally invariant quantity, and hence a Lorentz invariant quantity. However, it can be a spacetime dependent quantity - ie. has a different value at different points in spacetime (spacetime dependence), but at any particular point in spacetime it has the same value under arbitrary coordinate changes (generally invariant).

However, a different use of Lorentz invariant is the "form" of the laws of physics. In special relativity, Lorentz covariance means the laws are supposed to maintain the "same form" under a Lorentz transformation, where the "same form" means that we are NOT allowed to use Christoffel symbols (covariant derivatives disallowed) to write the laws of physics.

If we are allowed to use Christoffel symbols (covariant derivatives allowed), then any law is generally covariant, and hence also Lorentz covariant.
 

1. What is Lorentz invariance?

Lorentz invariance is a fundamental principle in physics that states that the laws of physics should be the same for all observers in uniform motion. This means that the laws of physics should not change under a transformation between two inertial reference frames.

2. What is General invariance?

General invariance, also known as general covariance, is a more general principle that extends Lorentz invariance to include all types of reference frames, including accelerating frames. It states that the laws of physics should be the same for all observers, regardless of their reference frame.

3. How are Lorentz invariance and General invariance related?

Lorentz invariance is a special case of General invariance, where the reference frames are limited to those in uniform motion. General invariance is a more general principle that encompasses Lorentz invariance and includes all types of reference frames.

4. Why are Lorentz invariance and General invariance important in physics?

These principles are important because they provide a foundation for the laws of physics and enable us to make accurate predictions and measurements in different reference frames. They are also essential for the development of theories, such as Einstein's theory of relativity, that describe the behavior of objects in different frames of reference.

5. Are there any known violations of Lorentz invariance or General invariance?

So far, there have been no confirmed violations of Lorentz invariance or General invariance. However, some theories, such as string theory, suggest that these principles may break down at very small scales. Ongoing experiments and research continue to test these principles and their limits.

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