Quick check as to space-like and time-like

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In summary, the conversation covered the confusion regarding the terms used in spacetime intervals and their corresponding signatures. The equation -Δr2+c2Δt2=s2 was discussed, as well as the convention of listing the timelike component first in the signature. It was clarified that a positive s2 indicates a spacelike interval, while a negative s2 indicates a timelike interval. The concept of proper time was also mentioned, with a reminder that it is defined as the time measured by a clock moving along a particle's worldline. Overall, the conversation provided a better understanding of the conventions and terminology used in spacetime intervals.
  • #1
nomadreid
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I keep getting the terms confused:

1) if I have the equation
-Δr2+c2Δt2=s2
does one say that I am using the (-,-,-,+) signature?

2) Given the above , if s2 > 0 then the interval is space-like, and if s2 < 0 it is time-like, or is it vice-versa?

3) If I now use the proper time τ , is it the same (as far as signature, space-like and time-like) for -Δr2+Δτ2=R2?

4) If R is the space-time radius of an arc in the above, what would that arc be?

Thanks for any help.
 
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  • #2
hi nomadreid! :smile:
nomadreid said:
I keep getting the terms confused:

don't worry … so do i ! :rolleyes:

(000,t1) and (000,t2) have time-like separation: two events at the same position but separated in time

(000,t) and (x00,t) have space-like separation: two events at the same time but separated in space

1) if I have the equation
-Δr2+c2Δt2=s2
does one say that I am using the (-,-,-,+) signature?

yes
2) Given the above , if s2 > 0 then the interval is space-like, and if s2 < 0 it is time-like, or is it vice-versa?

s2 > 0 means it's mostly a difference in t, so that's time-like :wink:
3) If I now use the proper time τ , is it the same (as far as signature, space-like and time-like) for -Δr2+Δτ2=R2?

4) If R is the space-time radius of an arc in the above, what would that arc be?

not following you :confused:
 
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  • #3
nomadreid said:
1) if I have the equation
-Δr2+c2Δt2=s2
does one say that I am using the (-,-,-,+) signature?

Assuming that by ##\Delta r^2## you mean ##\Delta x^2 + \Delta y^2 + \Delta z^2##, then yes, although it's more customary to list the timelike component first, so that the signature would be (+,-,-,-).

(Note: coordinate differentials are usually written with a ##d##, like this: ##ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2##. Note that I put the timelike component first.)

nomadreid said:
2) Given the above , if s2 > 0 then the interval is space-like, and if s2 < 0 it is time-like, or is it vice-versa?

Vice versa. You have one plus and three minuses in the coordinate differentials, so positive ##dt^2## must be a timelike interval; since ##ds^2## has the same sign as ##dt^2##, a positive ##ds^2## must also be a timelike interval.

nomadreid said:
3) If I now use the proper time τ , is it the same (as far as signature, space-like and time-like) for -Δr2+Δτ2=R2?

What you'e written here isn't a spacetime interval; in fact I'm not sure what it's supposed to mean, physically. If you're writing the spacetime interval in terms of the metric, you use coordinate differentials, as above. If you write the interval along a particle's worldline in terms of its proper time, the interval is just ##\Delta \tau##; that's the definition of proper time. If you're trying to write other intervals using the proper time, you would have to set up coordinates such that the particle's proper time was the time coordinate; but in those coordinates the metric probably wouldn't look as simple.
 
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  • #4
Many thanks, tiny-time and PeterDonis. This was a huge help. As far as my last two questions, it turns out that you are both right, that those questions didn't make sense: I had come across the equation using tau in a context of Minkowski space, and automatically stupidly assumed that the author was referring to proper time, which is what tau usually means in this context. However, going over the paper again, I noticed tucked away in a paragraph that the author defined tau as ct. OK, my fault for not noticing this. So questions (3) and (4) were, as you both pointed out, senseless. But your answers to questions (1) and (2) were supremely helpful. Thanks again!
 
  • #5


I am happy to provide clarification on the concepts of space-like and time-like intervals.

1) Yes, if you are using the equation -Δr2+c2Δt2=s2, it means that you are using the (-,-,-,+) signature. This is also known as the Minkowski metric, which is commonly used in special relativity to describe the geometry of space-time. The minus signs in front of the spatial coordinates indicate that the interval is measured in units of time squared, while the plus sign in front of the time coordinate indicates that the interval is measured in units of time.

2) You are correct. If s2 > 0, the interval is space-like, and if s2 < 0, it is time-like. This is because in the Minkowski metric, a positive interval indicates that the two events are separated by space, while a negative interval indicates that they are separated by time.

3) No, the signature for -Δr2+Δτ2=R2 is different. In this case, the interval is measured in units of length squared, as indicated by the plus sign in front of the proper time Δτ. This is known as the Euclidean metric, which is used to measure distances in regular space. Therefore, the concept of space-like and time-like intervals does not apply in this case.

4) The arc in the above equation would be a circle with a radius of R in space-time. This is because the equation represents a circle in the Euclidean metric, which is the same as a circle in regular space.

I hope this helps to clarify the concepts of space-like and time-like intervals for you. If you have any further questions, please do not hesitate to ask. As scientists, it is important for us to have a clear understanding of these concepts in order to accurately describe and analyze the physical world around us.
 

What is the difference between space-like and time-like?

Space-like and time-like are two different types of intervals that describe the relationship between events in space and time. A space-like interval indicates that two events could not have a causal relationship, meaning that one event could not have influenced the other. A time-like interval indicates that two events could have a causal relationship, meaning that one event could have influenced the other.

How do you determine if an interval is space-like or time-like?

To determine if an interval is space-like or time-like, you can use the Pythagorean theorem in spacetime. If the interval is positive, then it is space-like. If the interval is negative, then it is time-like. If the interval is zero, then it is light-like (meaning that the two events are connected by a beam of light).

Why is it important to distinguish between space-like and time-like intervals?

Distinguishing between space-like and time-like intervals is important because it helps us understand the relationship between events in space and time. It also plays a crucial role in theories such as relativity, where the concept of causality is central. Additionally, knowing the type of interval can help us determine the speed of an object or the direction of its movement.

Can an interval be both space-like and time-like?

No, an interval cannot be both space-like and time-like. This is because the Pythagorean theorem in spacetime only allows for a positive, negative, or zero interval. However, an interval can be light-like, which means that it has a zero interval and is connected by a beam of light.

How does the concept of space-like and time-like intervals apply to everyday life?

While the concept of space-like and time-like intervals is often used in physics and mathematics, it also has practical applications in everyday life. For example, GPS systems use the concept of time-like intervals to accurately determine location, as the time it takes for a signal to travel between satellites and receivers can be used to calculate distance. Additionally, understanding the difference between space-like and time-like intervals can help us make sense of the cause and effect relationships in our daily experiences.

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