# Two Observers Moving Opposite Dir Rel to Each Other

• I
• Sang-Hyeon Han
In summary, the postulates of relativity state that an object cannot move faster than the speed of light, regardless of the observer's frame of reference. When observer A sees observers B and C, they will appear to be moving at a speed less than or equal to the speed of light. However, when B and C see each other, their relative speed can be greater than 0.7c but less than c, approximately 0.94c. This can be calculated using the relativistic velocity addition formula and manifestly covariant objects such as proper four-velocities.
Sang-Hyeon Han
Hello guys. I have a question for one of postulates of relativity. Consider there are three observers (called A, B, and C) in x-direction only. A is at rest. B is moving to the left relative to A with velocity 0.7c. C is to the right relative to A with velocity 0.7c. Then when A sees B or C, they can not move faster than c. It's correct right? However, when B and C see each other, is it possible that their (relative) velocities are faster than c (maybe 1.4c) or not?

No. When B and C look at each other, they will see the other moving away at a speed grater than 0.7c but less than c. It will be about 0.94c.

Last edited:
Sang-Hyeon Han
.Scott said:
No. When B and C look at each other, they will see the other moving away at a speed grater than 0.7c but less than c. It will be about 0.94c.
How can we get that value??

Sang-Hyeon Han said:
How can we get that value??
Using the 'relativistic velocity addition' formula:

You will find that, no matter what values you use for the two spaceships, the receding velocity will not exceed c.

eg: If you set v=-0.7c and u' as 0.7c, the result will be 0.94c.

Note: this is not just for relativistic velocities. It will give accurate results at any speed (even stoned koala speed) it's just at - at anything less than relativistic velocities - the denominator becomes one plus (nearly) zero and we get the common v+u' velocity we know and love.

#### Attachments

• 1560952265900.png
934 bytes · Views: 218
Last edited:
Sang-Hyeon Han
Ahh I understand. many thanks guys!

The relative speed of two objects is the speed of one object in the rest frame of the other object. It's best to work with manifestly covariant objects. In this case these are the proper four-velocities, which are in your case
$$u_A=\begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}, \quad u_B=\frac{1}{\sqrt{1-0.7^2}} \begin{pmatrix}1 \\ -0.7 \\ 0 \\ 0 \end{pmatrix}, \quad u_C=\frac{1}{\sqrt{1-0.7^2}} \begin{pmatrix}1 \\ 0.7 \\ 0 \\ 0 \end{pmatrix}.$$
Now you don't need a Lorentz transformation to get the relative speed of each observer since you can get this with Minkowski products between these four-velocities.

For observer A, at rest in the computational frame, it's easy to see that you get the time-component of the four-velocities of B and C just by the Minkowski product of their four-velocities with ##u_A##:
$$(\gamma_B)_A =u_A \cdot u_B=\frac{1}{\sqrt{1-0.7^2}}.$$
This is the ##\gamma## factor of B measured by A who is at rest in the computational frame. From the ##\gamma## factor you get back ##\beta=|\vec{\beta}|=|\vec{v}/c|## simply by
$$(\beta_B)_A=\sqrt{1-\frac{1}{(\gamma_B)_A^2}}=0.7.$$
That's trivial, but the magic of covariant treatments is that since it's working with invariants, it's a general valid formula, i.e., to get the speed of ##B## as measured by ##C## you simply calculate the ##\gamma## factor of ##B## as measured by ##C## via the Minkowski product of the four-velocities,
$$(\gamma_B)_C=u_C \cdot u_B=\frac{1}{1-0.7^2}(1+0.7^2)=149/51$$
and thus the relative speed
$$(\beta_B)_C=\sqrt{1-1/(\gamma_B)_C^2} \simeq 0.940.$$

Sang-Hyeon Han

## What is the concept of "Two Observers Moving Opposite Dir Rel to Each Other"?

The concept of "Two Observers Moving Opposite Dir Rel to Each Other" refers to two observers who are moving in opposite directions relative to each other. This means that they are both in motion, but in opposite directions, and their movements are being observed by each other.

## What is the significance of this concept in science?

This concept is significant in science because it helps us understand the concept of relative motion. It also plays a role in the theory of relativity, which states that the laws of physics are the same for all observers, regardless of their relative motion.

## How do the observations of the two observers differ?

The observations of the two observers differ because they are in different frames of reference. This means that they are experiencing different velocities and accelerations, which can affect their perception of time, distance, and other physical phenomena.

## What factors can affect the observations of the two observers?

There are several factors that can affect the observations of the two observers, such as their relative velocities, the distance between them, and the presence of any external forces or accelerations. These factors can impact their perception of time, space, and the laws of physics.

## How does the concept of "Two Observers Moving Opposite Dir Rel to Each Other" relate to everyday life?

This concept relates to everyday life in many ways, such as in the way we perceive motion and distance. It also plays a role in technologies like GPS, which rely on the principles of relativity to accurately determine location and time. Additionally, this concept is important in fields like astronomy and aviation, where precise measurements and calculations of relative motion are crucial.

• Special and General Relativity
Replies
25
Views
766
• Special and General Relativity
Replies
12
Views
913
• Special and General Relativity
Replies
84
Views
4K
• Special and General Relativity
Replies
43
Views
2K
• Special and General Relativity
Replies
33
Views
2K
• Special and General Relativity
Replies
16
Views
904
• Special and General Relativity
Replies
20
Views
1K
• Special and General Relativity
Replies
57
Views
4K
• Special and General Relativity
Replies
18
Views
1K
• Special and General Relativity
Replies
15
Views
1K