Why is ct used as the fourth dimension in Special Relativity?

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In summary: In a light-clock, one can declare a given separation of mirrors [without "measuring it"] to define a standard tick of this clock. (Allow two mirrors to travel inertially in the same direction. Changing the separation effectively changes the resolution of the clock.) Again, the key is [regular] periodicity of clock. Instead of the light-clock, one could crudely use your heartbeat to measure time [a la Galileo].The point is that a "wristwatch" time measurement, unlike a "meterstick" measurement that must be extended out, is local to the observer's worldline. This has conceptual advantages in relativity.From a more practical point of view, distances to remote objects are not
  • #1
NanakiXIII
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I'm looking for an explanation as to why a lot of results of Special Relativity are described using [tex]ct[/tex] as a fourth dimension instead of just [tex]t[/tex]. Now, I understand that using [tex]ct[/tex] in a Minkowski diagram with identically scaled axes will cause worldlines of light to angle at a nice 45 degrees, and I've seen how [tex]x-ct[/tex], if I remember correctly, is Lorentz invariant, as well as some other cases where it does seem useful to use ct, but the general nature of this usefulness eludes me. Could anyone tell me more about this?
 
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  • #2
NanakiXIII said:
I'm looking for an explanation as to why a lot of results of Special Relativity are described using [tex]ct[/tex] as a fourth dimension instead of just [tex]t[/tex].
Each component in the decomposition of spacetime in "space" and "time" components must have an identical dimensionality. Both the "space" and "time" components use a measure of "distance". Hence: ct = m/s * s = m

By the way the decomposition in "space" and "time" is observer dependent in relativity.
 
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  • #3
Also note that with a suitable choice of units, you can use simply t instead of ct in calculations. For example, measure time in seconds and distance in light-seconds (the distance light travels in one second).
 
  • #4
jtbell said:
Also note that with a suitable choice of units, you can use simply t instead of ct in calculations. For example, measure time in seconds and distance in light-seconds (the distance light travels in one second).
Yeah, but (ct,x,y,z) is easier to write and read than (t,x/c,y/c,z/c)
 
  • #5
jtbell said:
Also note that with a suitable choice of units, you can use simply t instead of ct in calculations. For example, measure time in seconds and distance in light-seconds (the distance light travels in one second).
Interestingly, in a way we already do that since the meter is defined as the length of the path traveled by light in vacuum during a time interval of 1/299,792,458 of a second. :)
 
  • #6
DaleSpam said:
Yeah, but (ct,x,y,z) is easier to write and read than (t,x/c,y/c,z/c)

Conceptually, I prefer the latter.
In some sense, measuring a time seems more fundamental than measuring a length. (Radar measurements use a clock.)
In addition, the latter form lends itself more easily to the analyzing the Galilean limit.
 
  • #7
MeJennifer said:
Each component in the decomposition of spacetime in "space" and "time" components must have an identical dimensionality.

Why?

I don't see the (general) need for any such requirement - but scaling time by c does provide a convenient notation within Minkowski space.

Regards,

Bill
 
  • #8
Antenna Guy said:
Why?

I don't see the (general) need for any such requirement
One would not be able to construct a Minkowski metric or more advanced metrics in curved spacetimes when the dimensions do not match.

Antenna Guy said:
- but scaling time by c does provide a convenient notation within Minkowski space.
Time is not scaled by c, since both dimensions are different, e.g. s * m/s.
 
  • #9
robphy said:
Conceptually, I prefer the latter.
In some sense, measuring a time seems more fundamental than measuring a length. (Radar measurements use a clock.)


Eh? I think you've got that precisely the wrong way round, imo. I can't think of a single example of how one could measure an interval of time without *first* having to measure a spatial length.
 
  • #10
shoehorn said:
Eh? I think you've got that precisely the wrong way round, imo. I can't think of a single example of how one could measure an interval of time without *first* having to measure a spatial length.

Refer to "The Interval" chapter (around p.70) in Geroch's General Relativity from A to B
http://books.google.com/books?id=Uk...ty+a&sig=ZwKt1KmtQ_UUElxeSrA1d0iDiTI#PPA72,M1
(see p.72)

The key to a clock is a periodicity.
To perform radar measurements, one makes use of light rays and clocks.
 
  • #11
robphy said:
Refer to "The Interval" chapter (around p.70) in Geroch's General Relativity from A to B
http://books.google.com/books?id=Uk...ty+a&sig=ZwKt1KmtQ_UUElxeSrA1d0iDiTI#PPA72,M1
(see p.72)

The key to a clock is a periodicity.
To perform radar measurements, one makes use of light rays and clocks.

Care to give me an example of a measurement of an interval of time that can be made without measuring a spatial interval then?
 
  • #12
shoehorn said:
Care to give me an example of a measurement of an interval of time that can be made without measuring a spatial interval then?

In a light-clock, one can declare a given separation of mirrors [without "measuring it"] to define a standard tick of this clock. (Allow two mirrors to travel inertially in the same direction. Changing the separation effectively changes the resolution of the clock.) Again, the key is [regular] periodicity of clock. Instead of the light-clock, one could crudely use your heartbeat to measure time [a la Galileo].

The point is that a "wristwatch" time measurement, unlike a "meterstick" measurement that must be extended out, is local to the observer's worldline. This has conceptual advantages in relativity.

From a more practical point of view, distances to remote objects are not done by extending a ruler out to meet it. Radar using light-rays is often used.
 
  • #13
As strange as it sounds... If t was used instead of ct, then the maximum velocity would be 1 instead of c. Of course, I think this is actually done when the speed of light is arbitrarily set to 1 in geometrized units. This follows from the fact that the magnitude of the 4-velocity vector must equal c (or 1) at all times. Where v = 0, t = 1, and so the entire magnitude is due to motion through time (ct = c). At v = c, t = 0, and so the entire magnitude is due to motion through space (ct = 0).
 
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  • #14
shoehorn said:
Care to give me an example of a measurement of an interval of time that can be made without measuring a spatial interval then?

Watch a flashing light.
Record the time of flash one.
Record the time of flash two.
You just recorded a time interval with no spatial measurements.

This also gets to the essence of time measurement.
It's matching world events to clock (reference) events.

Like matching an object to a ruler to measure length.
 
  • #15
shoehorn said:
Care to give me an example of a measurement of an interval of time that can be made without measuring a spatial interval then?

I think the duration of a photon (single complete wave [clarification: passing through a plane of reference]) would constitute both a time and space interval measured by a clock. f=1/t, l=c*t

Conversely, the spatial measurement of one period of a standing wave (segment of a continuous stream of photons) can be used to calculate t.

Regards,

Bill
 
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1. What is the difference between ct and t in scientific notation?

In scientific notation, ct represents the speed of light, which is approximately 299,792,458 meters per second. On the other hand, t represents time, which is a unit of measurement for duration.

2. Why is ct commonly used instead of t in scientific equations?

Ct is commonly used instead of t in scientific equations because it helps to simplify complex equations and make them more manageable. It also allows for easier comparison and calculation of values related to the speed of light.

3. Can ct be used interchangeably with t in scientific calculations?

No, ct cannot be used interchangeably with t in scientific calculations. Ct is specifically used to represent the speed of light, while t is used to represent time in general. Using them interchangeably can result in incorrect calculations and conclusions.

4. How does the usage of ct instead of t impact scientific research?

The usage of ct instead of t has a significant impact on scientific research, especially in the fields of physics and astrophysics. It allows for more accurate and precise calculations and helps to better understand the fundamental principles of the universe.

5. Are there any limitations to using ct instead of t in scientific equations?

Yes, there are limitations to using ct instead of t in scientific equations. Ct can only be used in equations that involve the speed of light, and it may not be applicable in other scenarios. Additionally, some equations may not be easily convertible from using t to using ct.

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