Special relativity and apparent, actual and synchronized positions

In summary, the conversation discusses a particle moving with speed V in the positive direction of the OX axis in the inertial reference frame I. At time t=0, the particle is located at point Ma and a light signal is emitted from the origin O in the positive direction of the OX axis. The arrival of the light signal at point Ma is associated with event E1, while the particle reaching its actual position at event E2 is taken into account that it has advanced with Vxa/c during the time interval xa/c. The Lorentz transformation of event E2 to the rest frame of the moving particle is performed, resulting in equations (1) and (2) for the space and time coordinates. Neglecting second
  • #1
bernhard.rothenstein
991
1
Consider a particle that moves with speed V in the positive direction of the OX axis of the inertial reference frame I. At a time t=0 it is located at a point Ma characterized by a apace coordinate xa. Using a terminology proposed by Deissler1 we call that position aparent. At t=0 a light signal is emitted from the origin O in the positive direction of the OX axis. Its arrival at the point Ma is associated with event E1 characterized by a space coordinate xa and by a time coordinate xa and by a time coordinate xa/c. At that very time the moving particle arrives at its actual position associated with an event E2 characterized by a space coordinate xa(1+V/c) taking into account that during the time interval xa/c the moving particle has advances with Vxa/c, and by a time coordinate xa/c. The two events are simultaneous in I.
Performing the Lorentz transformation of the space-time coordinates of event E2
to the rest frame I' of the moving partcle the results are
x'=gxa=gcta (1)
and
t'=gta[1-V/c(1+V/c)]=gxa[1-V/c(1+V/c)] (2)
g standing for the Lorentz factor. If we neglect second order effects (2) becomes
t'=ta[(1-V/c)/(1+V/c)]1/2 (3)
Let E be the event associated with the fact that the world lines of the moving particle and of the propagating light signal intersect each other. Let x be its space coordinate. From
x=xa+Vx/c (4)
we obtain that
x=xa/(1-V/c) (5)
event E being characterized by a time coordinate
 
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  • #2
bernhard.rothenstein said:
Consider a particle that moves with speed V in the positive direction of the OX axis of the inertial reference frame I. At a time t=0 it is located at a point Ma characterized by a apace coordinate xa. Using a terminology proposed by Deissler1 we call that position aparent. At t=0 a light signal is emitted from the origin O in the positive direction of the OX axis. Its arrival at the point Ma is associated with event E1 characterized by a space coordinate xa and by a time coordinate xa and by a time coordinate xa/c. At that very time the moving particle arrives at its actual position associated with an event E2 characterized by a space coordinate xa(1+V/c) taking into account that during the time interval xa/c the moving particle has advances with Vxa/c, and by a time coordinate xa/c. The two events are simultaneous in I.
Performing the Lorentz transformation of the space-time coordinates of event E2
to the rest frame I' of the moving partcle the results are
x'=gxa=gcta (1)
and
t'=gta[1-V/c(1+V/c)]=gxa[1-V/c(1+V/c)] (2)
g standing for the Lorentz factor. If we neglect second order effects (2) becomes
t'=ta[(1-V/c)/(1+V/c)]1/2 (3)
Let E be the event associated with the fact that the world lines of the moving particle and of the propagating light signal intersect each other. Let x be its space coordinate. From
x=xa+Vx/c (4)
we obtain that
x=xa/(1-V/c) (5)
event E being characterized by a time coordinate x/c. Performing the Lorentz transformations to the rest frame of the moving particle we obtain
x'=gxa
t'=gxa/c (6)
Do you consider that the derivations above are correct? Are they a simple exercise in handling the Lorentz transformations or there is some physics behind them.
Thanks in advance for your help.
 
  • #3
t=ta(1-V/c).

In this scenario, we can see the effects of special relativity on the apparent, actual, and synchronized positions of the moving particle. The apparent position, Ma, is where the particle appears to be located at time t=0. However, due to its motion with speed V, it has already advanced to its actual position, E2, by the time it takes for the light signal to reach Ma. This is because the particle's motion affects the measurement of time and space in the inertial reference frame I.

The synchronized positions of the two events, E1 and E2, are also affected by the particle's motion. In the rest frame I', the Lorentz transformation equations (1) and (2) show that the space and time coordinates of E2 are related to those of E1 by a factor of 1/(1+V/c). This means that the two events are no longer simultaneous in the rest frame I'. This is a consequence of the relativity of simultaneity, which states that the concept of simultaneous events depends on the observer's reference frame.

Finally, the event E, where the world lines of the particle and the light signal intersect, is also affected by the particle's motion. The space coordinate of E, x, is given by equation (5), which shows that it is dependent on the particle's speed V. This means that the intersection of the world lines will also be different depending on the observer's reference frame.

In conclusion, special relativity has a significant impact on the apparent, actual, and synchronized positions of objects in motion. It highlights the relativity of space and time and emphasizes the importance of considering different reference frames in understanding the physical world.
 

1. What is special relativity?

Special relativity is a theory in physics that explains the relationship between space and time. It was developed by Albert Einstein and states that the laws of physics are the same for all observers in uniform motion, regardless of their relative velocity.

2. What is the difference between apparent, actual, and synchronized positions?

Apparent position is the location of an object as perceived by an observer, taking into account the time it takes for light to reach the observer. Actual position is the true location of the object, independent of any observer. Synchronized positions refer to the positions of two or more objects that are moving relative to each other but are in sync in terms of their motion and time measurements.

3. How does special relativity affect the concept of time?

Special relativity introduces the idea of time dilation, meaning that time passes differently for different observers depending on their relative motion. This means that time is not absolute and can vary for different observers, which can have significant implications in areas such as space travel.

4. Can special relativity be applied to all types of motion?

Special relativity is only applicable to motion that is uniform, meaning that the speed and direction are constant. It does not apply to acceleration or non-uniform motion.

5. How does special relativity impact our everyday lives?

Although the effects of special relativity are not noticeable in our everyday lives, the theory has had significant impacts on technology and our understanding of the universe. It has allowed for the development of technologies such as GPS, and has led to advancements in fields such as astrophysics and cosmology.

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