Understanding Invariance of Spacetime Intervals

In summary, the conversation discusses the concept of invariance under coordinate transformations and how it applies to measuring spacetime intervals. An example is given where Alice is moving away from two observers, Bob and Charlie, at different velocities. It is pointed out that the measurements of Alice's spacetime interval by Bob and Charlie are not between the same two points, leading to the conclusion that invariance is not applicable in this scenario.
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nomadreid
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Category of simple questions

Obviously I am misunderstanding how an interval of space- time can be invariant under coordinate transformations. The following elementary (but obviously incorrect) calculation will illustrate my difficulty.

Alice is leaving her two boyfriends, Bob and Charlie. Bob sees Alice going at a constant velocity of 3/5 the speed of light away from him, so after a second, Bob measures Alice’s space-time change as (using the (+,-,-,-) convention) (9.0 x 1016 m2- 5.4 x 1016 m2 = 3.6 x 1016 m.

Charlie also sees Alice going away from him at a constant velocity, but at 4/5 the speed of light, so after a second, Charlie measures Alice’s space-time change as

9 x 1016 m2- 7.2 x 1016 m2 = 1.8 x 1016 m.

I would be grateful for corrections.
 
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  • #2
A second in Charlie's time is not the same as a second in Bob's time, and their hyperplanes of simultaneity are different.
Assuming Alice, Bob and Charlie are next to one another at t=0, with Alice heading west at 0.6c and Charlie heading East at 0.2c (approx.) the spacetime location that Bob identifies as Alice's position after one (Bob) second is not the same as the spacetime location that Charlie identifies as Alice's position after one (Charlie) second.

So the spacetime intervals being measured are not between the same two spacetime points. So invariance is not applicable.
 
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  • #3
Ah, I hadn't taken that into account. Thanks, andrewkirk. Back to the drawing board.
 

1. What is the concept of invariance of spacetime intervals?

The invariance of spacetime intervals refers to the fact that the distance between two events in spacetime is independent of the reference frame in which it is measured. This means that the time and space components of the interval remain the same, regardless of the observer's motion or perspective.

2. Why is it important to understand the invariance of spacetime intervals?

Understanding the invariance of spacetime intervals is crucial in the study of special relativity. It allows us to make accurate predictions and calculations about the behavior of objects moving at high speeds. It also helps us to reconcile seemingly contradictory observations and phenomena in physics.

3. How does the invariance of spacetime intervals relate to the speed of light?

The invariance of spacetime intervals is closely related to the speed of light, which is a fundamental constant in the universe. According to Einstein's theory of special relativity, the speed of light is the same for all observers, regardless of their relative motion. This means that the time and space components of the interval must also remain the same for all observers, leading to the concept of invariance.

4. Can the invariance of spacetime intervals be demonstrated in real-life situations?

Yes, the invariance of spacetime intervals has been confirmed through numerous experiments and observations. One example is the famous Michelson-Morley experiment, which demonstrated that the speed of light is the same in all directions, regardless of the observer's motion. This supports the concept of invariance and led to the development of special relativity.

5. Are there any exceptions to the invariance of spacetime intervals?

The invariance of spacetime intervals holds true for all inertial reference frames, which are frames of reference that are not accelerating. However, in non-inertial reference frames, such as those experiencing acceleration or gravity, the invariance of spacetime intervals may not hold. This is accounted for in the theory of general relativity, which expands upon the principles of special relativity to include non-inertial frames of reference.

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