# Four vector component. how to recognize it?

• bernhard.rothenstein
In summary, the simplest way to find out if a physical quantity is the component of a four vector is to use equations (1) and (2).
bernhard.rothenstein
What is the simplest way to find out if a physical quantity is the component of a four vector? There are opinions that the Lorentz contracted length is not, the Lorentz dilate length being.

bernhard.rothenstein said:
What is the simplest way to find out if a physical quantity is the component of a four vector? There are opinions that the Lorentz contracted length is not, the Lorentz dilate length being.

Consider that observers from I detect simultaneously the space coordinates of the ends of a moving rod, at rest in the I', located along the x' axis. The result is that the proper length of the rod L0 measured in I' and its contracted length L are related by the formula that accounts for the Lorentz contraction. The same formula could be derived without imposing the simultaneous detection of the space coordinates of the two ends.
The space coordinates of the ends measured in the stationary frame do not depend on the time when they are measured. Consider that observers from I' detect simultaneously the space coordinates of the stationary rod, the measurements being associated with the space-time coordinates (x'1,t') and (x'2,t'). Observers from I perform the Lorentz transformations for the space coordinates of the two ends obtaining
L=$$\gamma$$L0 (1)
which accounts for the Lorentz expansion (dilation?).
Equation (1) is derived by radar detection of the space coordinates of the moving rod.
Virtues of (1) are mentioned.
Which is the simplest way to show that (1) is the component of a four vector the length contracted length being not?

A set of quantites form a 4-vector if they tranform via the Lorentz tranform. Thus if $\textbf{X} = (ct, x, y, z)^T$ and $\textbf{X'} = (ct', x', y', z')^T$ are coordinates for the same event in two different inertial frames (in "standard configuration", i.e. with aligned spatial axes and in relative motion along their common x axis), we can say this is a 4-vector because

$$\textbf{X'} = \Lambda \, \textbf{X}$$​

where

$$\Lambda = \left[ \begin{array}{cccc} \gamma & -\gamma v & 0 & 0 \\ -\gamma v & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} \right]$$​

"Lorentz-contracted length" doesn't satisfy this because the length is measured between two events that are simultaneous in the frame where the measurement is made. When you change frames, you also change events. The Lorentz transform applies only when you measure the same pair of events in two different frames.

Other examples of 4-vectors are

Energy-momentum... $$\textbf{P} = (\frac{E}{c}, p_1, p_2, p_3)^T = m\frac{d\textbf{X}}{d\tau}$$
4-force...... $$\textbf{F} = \frac{d\textbf{P}}{d\tau}$$
4-current..... $$\textbf{J} = (\rho c, j_1, j_2, j_3)^T$$
electromagnetic 4-potential $$\Psi = (\phi, a_1 c, a_2 c, a_3 c)^T$$​

(See four-vector on Wikipedia.)

DrGreg said:
A set of quantites form a 4-vector if they tranform via the Lorentz tranform. Thus if $\textbf{X} = (ct, x, y, z)^T$ and $\textbf{X'} = (ct', x', y', z')^T$ are coordinates for the same event in two different inertial frames (in "standard configuration", i.e. with aligned spatial axes and in relative motion along their common x axis), we can say this is a 4-vector because

$$\textbf{X'} = \Lambda \, \textbf{X}$$​

where

$$\Lambda = \left[ \begin{array}{cccc} \gamma & -\gamma v & 0 & 0 \\ -\gamma v & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} \right]$$​

"Lorentz-contracted length" doesn't satisfy this because the length is measured between two events that are simultaneous in the frame where the measurement is made. When you change frames, you also change events. The Lorentz transform applies only when you measure the same pair of events in two different frames.

Other examples of 4-vectors are

Energy-momentum... $$\textbf{P} = (\frac{E}{c}, p_1, p_2, p_3)^T = m\frac{d\textbf{X}}{d\tau}$$
4-force...... $$\textbf{F} = \frac{d\textbf{P}}{d\tau}$$
4-current..... $$\textbf{J} = (\rho c, j_1, j_2, j_3)^T$$
electromagnetic 4-potential $$\Psi = (\phi, a_1 c, a_2 c, a_3 c)^T$$​

(See four-vector on Wikipedia.)
Thank you. Please have a look at the following lines.
Consider that in a 2D approach A is the scalar component of a "two vector" and let A0 be its proper (invariant) magnitude related to its relativitic magnitude as
A=$$\gamma$$(u)A0. (1)
Let A be the vector component of the same "two vector",related to A by
A=$$\gamma$$(u)A0u (2)
(1) and (2) ensuring the invariance of the relativistic interval.
As we see the Lorentz contracted length is not the component of a four vector
whereas the dilated length L=L0$$\gamma$$ is.
The dilated length could be obtained simultaneously detecting the ends of the rod in its rest frame. Did Einstein take into account that measurement variant?
Would you accept such an approach?

## 1. What is a four vector component?

A four vector component is a mathematical concept used in the field of physics and special relativity. It is a set of four numbers that represent the position and direction of an object in space and time.

## 2. How is a four vector component different from a regular vector?

A four vector component is different from a regular vector in that it includes time as a fourth component, while a regular vector only has three components representing space. This allows four vector components to accurately describe the position and motion of an object in four-dimensional space-time.

## 3. How can I recognize a four vector component?

A four vector component is usually denoted by a symbol with an arrow above it, such as A → or p →. It will also have four components, typically labeled as x, y, z, and t for space and time, respectively.

## 4. What is the significance of four vector components in physics?

Four vector components are important in physics because they provide a way to accurately describe the position and motion of objects in space and time. They are used in special relativity, electromagnetism, and other areas of physics to calculate and describe physical phenomena.

## 5. Can four vector components be used to describe any type of motion?

Yes, four vector components can be used to describe any type of motion, as long as it occurs in four-dimensional space-time. This includes both linear and rotational motion, as well as motion at any speed, including the speed of light.

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