# Constant Jacobian transformation of an inertial frame

• hwl
In summary, the conversation discusses the apparent field created by a constant Jacobian transformation of a special relativity inertial frame. This apparent field has a constant metric that is not the same as the SR metric, resulting in relative acceleration of separation. However, the geodesic-metric equation shows that the acceleration vector should be zero since the metric is constant with zero partial derivatives. This creates a puzzle as to how the acceleration vector can be nonzero. The only explanation is that some of the first partial derivatives of the constant metric must be nonzero.
hwl
Suppose we do a constant Jacobian transformation (which is not Lorentz) of a SR (inertial)
frame, by using four linear change of variables equations. This defines an apparent field with a
constant metric (which is not the SR metric) in which there is relative acceleration of separation.
From the geodesic - metric equation we see that the acceleration vector depends on the first
partial derivatives of this constant metric and so at least some of these derivatives must be
non-zero. How can this be true?
Can anyone shed light on this puzzle?

hwl said:
the acceleration vector depends on the first partial derivatives of this constant metric and so at least some of these derivatives must be non-zero.
Why would the first partial derivatives of a constant metric be nonzero?

The acceleration vector in this field is NON-ZERO. But according to the geodesic-metric equation it should be
ZERO because the metric is constant with (presumably !) zero partial derivatives. The only way we can
reconcile these two conflicting values is if these derivatives were non-zero. How else can we explain this

## 1. What is a constant Jacobian transformation of an inertial frame?

A constant Jacobian transformation of an inertial frame is a mathematical concept used in the field of physics to describe the relationship between two coordinate systems. It is a transformation that preserves the shape of an object or system in one coordinate system when it is translated to another.

## 2. How is a constant Jacobian transformation different from other transformations?

A constant Jacobian transformation is different from other transformations because it preserves the shape of an object or system, whereas other transformations may change the shape or orientation. It is also unique in that it has a constant Jacobian matrix, which is a matrix of partial derivatives used to represent the transformation.

## 3. What is an inertial frame?

An inertial frame is a reference frame in which Newton's first law of motion holds true. This means that an object in a state of rest will remain at rest and an object in motion will continue to move in a straight line at a constant speed, unless acted upon by an external force. In physics, inertial frames are used as a point of reference to describe the motion of objects.

## 4. Why is a constant Jacobian transformation important in physics?

A constant Jacobian transformation is important in physics because it allows us to easily convert between different coordinate systems while preserving the shape of an object or system. This is especially useful in fields such as mechanics, where the position and motion of objects are described using different coordinate systems.

## 5. Can a constant Jacobian transformation of an inertial frame be applied to all situations?

No, a constant Jacobian transformation of an inertial frame can only be applied in situations where the underlying physical laws and principles are the same in both coordinate systems. In other words, the transformation must be done within the same frame of reference and not in a non-inertial frame, where Newton's first law does not hold true.

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