Metric Transformation b/w Inertial Frames: Analyzing Effects

In summary, the metric tensor in an inertial frame with only 1-D space is ## \eta = diag(-1, 1)##. After a coordinate change, the metric transformation rule is given by $$ g_{\mu \nu} = \frac{\partial x^\alpha}{\partial x'^{\mu }} \frac{\partial x^\beta}{\partial x'\nu } \eta_{\alpha \beta} $$. With the specific form of ## \eta ##, the element ##g_{00}## changes after a Lorentz boost, resulting in a different value from the original ## \eta_{00}## due to mixing units where ##c \neq 1##.
  • #1
Jufa
101
15
TL;DR Summary
Found something weird when calculating the transformation due to a boost.
The metric tensor in an inertial frame is ## \eta = diag(-1, 1)##. Where I amb dealing with only 1-D space. The metric tranformation rule after a crtain coordinate chane is the following:

$$ g_{\mu \nu} = \frac{\partial x^\alpha}{\partial x'^{\mu }} \frac{\partial x^\beta}{\partial x'\nu } \eta_{\alpha \beta} $$
with ##x^0 = t## and ##x^1= x##

Given the particular form of ## \eta ## we obtain for ## \mu = \nu = 0 ## :

$$ g_{00} = -\Big(\frac{\partial t}{\partial t' }\Big)^2 + \Big(\frac{\partial x}{\partial t' }\Big)^2 = -\gamma ^2 + v^2\gamma^2 = \frac{v^2-1}{1-v^2/c^2} \neq -1 = \eta_{00}$$

So I get that after a Lorentz boost one of the metric's elements has changed.
Where am I wrong?
 
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  • #2
You seem to be mixing units where ##c = 1## with units where ##c \neq 1##.
 
  • #3
Orodruin said:
You seem to be mixing units where ##c = 1## with units where ##c \neq 1##.
Oh yes. It is definitely that. Many thanks.
 

Related to Metric Transformation b/w Inertial Frames: Analyzing Effects

1. What is a metric transformation between inertial frames?

A metric transformation between inertial frames is a mathematical tool used to analyze the effects of changes in reference frames on physical measurements. It involves transforming the coordinates and measurements of an event from one inertial frame to another, while taking into account the differences in the frames' relative motion.

2. Why is it important to analyze the effects of metric transformation between inertial frames?

It is important to analyze the effects of metric transformation between inertial frames because it allows us to accurately compare and understand physical measurements made in different reference frames. This is crucial in fields such as physics and engineering, where precise measurements and calculations are necessary.

3. What are some common effects of metric transformation between inertial frames?

Some common effects of metric transformation between inertial frames include changes in the perceived length, time, and velocity of an event. These changes are a result of the relative motion between the frames and can be described using mathematical equations.

4. How is metric transformation between inertial frames related to the theory of relativity?

Metric transformation between inertial frames is a fundamental concept in the theory of relativity. It is based on the principle that the laws of physics should be the same in all inertial frames, and that measurements made in one frame should be consistent with measurements made in another frame through a metric transformation.

5. Are there any real-world applications of metric transformation between inertial frames?

Yes, there are many real-world applications of metric transformation between inertial frames. For example, it is used in GPS systems to accurately determine the position and time of an object, taking into account the effects of different reference frames. It is also used in space exploration, where precise measurements and calculations are necessary for successful missions.

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