# Measuring Relative Speed of Frames Using Spacetime Interval

• russphelan
In summary, the concept of relative speed between two events depends on the reference frame and can be calculated using the invariance of spacetime interval. The velocity of an object traveling between the events can be determined by dividing the spatial difference by the time difference in the reference frame. However, events themselves do not have velocity as they are just points in spacetime. Additionally, there is no absolute rest frame and any inertial observer can consider themselves at rest.
russphelan
Hi all,

Say two events happen in the same place according to one observer (1). They are separated in time by 3 years.

According to another observer (2) that is moving relative to the first, the events are separated by 5 years. We can calculate, using the invariance of the spacetime interval, that the events are separated in space by 4 lightyears, according to the second observer.

When we want to calculate the speed of observer 2 relative to observer 1, it is necessary to use the time separation of events according to observer 2. So, the relative speed would be

$\frac{4yrs \cdot c}{5yrs} = \frac{4}{5}c$

Why does the time interval according to observer 2 give us relative speed of the frames? Isn't this time interval dilated due to the amount of time it takes light from the events to reach the traveling observer?

Thanks,
Russ

In order to get the velocity, you need to take a something at rest in one of the system and consider how it moves in the other. In this case, the spatial difference between the events was 4 lightyears and the time difference 5 years in the second frame. In order to get from one event to the other, something moving linearly would need to be moving with 4c/5. This is just the definition of velocity in the second system, difference in space/difference in time. That this is the velocity of the first system relative to the second follows from the assumption that the event ocurs at the same spatial position in the first.

russphelan
Thank you very much for the reply!

It makes sense that velocity of the second frame is defined in this way. According to the first frame, the events have not moved at all, so their speed is 0. This is the system in which the events are at rest.

russphelan said:
Thank you very much for the reply!

It makes sense that velocity of the second frame is defined in this way. According to the first frame, the events have not moved at all, so their speed is 0. This is the system in which the events are at rest.
This sounds a warning bell for me. Events do not have velocities. The velocity you get is the velocity of an object traveling with constant speed between the events. The events themselves are points in space-time. This might sound picky, but using correct terminology becomes very important and I have seen many people have misconceptions based on misinterpreting the nomenclature.

I meant to say that the speed of the frame in which the events happen in the same place is 0. The speed of the frame in which the events have a spatial separation is given by the distance measured between the events, divided by the time measured between the events, from that frame.

Does that make sense?

This brings up another point though: say a frame is moving with respect to your frame. Is it safe to say that events happening in this moving frame do not have a "speed"? I would guess that the place at which the event actually occurs does not continue to move with the moving frame.

Thanks so much for taking the time to consider these posts.
Russ

russphelan said:
I meant to say that the speed of the frame in which the events happen in the same place is 0. The speed of the frame in which the events have a spatial separation is given by the distance measured between the events, divided by the time measured between the events, from that frame.
To be more precise: The speed relative to the frame where the events occur at the same spatial point is given by this procedure.
russphelan said:
Is it safe to say that events happening in this moving frame do not have a "speed"? I would guess that the place at which the event actually occurs does not continue to move with the moving frame.
Events do not happen in a frame. All events occur in all frames, they just have different coordinates. No events have speed, they are just points in space-time. There is no such thing as "a place where the event actually occurs".

There is also no distinction whatsoever between frames. You cannot say that a frame is at absolute rest and all others are moving. Any inertial observers can consider themselves at rest, even if they are moving relative to each other. This is just Galilean relativity.

## 1. What is the concept of "spacetime interval" and how does it relate to measuring relative speed of frames?

The spacetime interval is a measure of the distance between two events in spacetime. It is a combination of both time and space components, and is independent of the observer. Measuring the spacetime interval between two events in different frames can give us information about the relative speed between those frames.

## 2. How is the spacetime interval calculated and what are the units of measurement?

The spacetime interval is calculated using the Pythagorean theorem, where the time component is squared and added to the squared distance component. The resulting value is the spacetime interval. The units of measurement for the spacetime interval are typically meters squared (m^2) in the metric system, or seconds squared (s^2) in the cgs system.

## 3. Can the spacetime interval be negative and what does that indicate?

Yes, the spacetime interval can be negative. A negative value indicates that the two events being measured are not causally connected, meaning that no information can travel between them. This is referred to as a "spacelike" interval.

## 4. How does the concept of "time dilation" relate to measuring relative speed of frames using spacetime interval?

Time dilation is a phenomenon that occurs when an observer in one frame measures time passing at a different rate compared to an observer in a different frame. This can be observed when measuring the spacetime interval between events in different frames, as the relative speed between frames can affect the measurement of time.

## 5. Is there a limit to the accuracy of measuring relative speed using the spacetime interval?

Yes, there is a limit to the accuracy of measuring relative speed using the spacetime interval. This is due to the uncertainty principle in quantum mechanics, which states that the more precisely we measure the position of an object, the less precisely we can measure its velocity. This means that at very small scales, the measurement of the spacetime interval may become less accurate.

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