# Why is the time dimension multiplied by c^2?

1. Aug 28, 2008

### redtree

Why is the time dimension multiplied by c^2?

2. Aug 28, 2008

### HallsofIvy

Staff Emeritus
Re: space-time

Where have you ever seen the "time dimension multiplied by c^2"?

3. Aug 28, 2008

### redtree

Re: space-time

The invariance of the interval in space-time:

S$$^{2}$$ = x$$^{2}$$ + y$$^{2}$$ + z$$^{2}$$ - (ct)$$^{2}$$

4. Aug 28, 2008

### Staff: Mentor

Re: space-time

The time dimension is multiplied by c so that you wind up with units of length.

5. Aug 28, 2008

### tiny-tim

So that you can swap over length and time when nobody's looking.

6. Sep 1, 2008

### petm1

Re: space-time

Then why don’t we think of energy as being equal to the speed of light times the speed of time which is c. Both are used in describing motion and every one knows that nothing can go faster than light, so the speed of light squared never has made sense to me.

7. Sep 1, 2008

### JesseM

Re: space-time

The speed of light squared is not itself a speed. Speeds always have units of meters/second in the metric system, whereas c^2 has units of meters^2/second^2.

8. Sep 1, 2008

### tiny-tim

speed of time

Hi petm1!

Where did you read that the speed of time is c?

The speed of time would be in units of time/time, in other words dimensionless.

Its value would be 1, or 1/√(1 - v2/c2).

9. Sep 1, 2008

### petm1

Re: speed of time

What else does ct look like other than assigning a speed to time.

10. Sep 1, 2008

### ZikZak

Re: speed of time

It looks like a unit conversion. Does "2x" look like assigning the value 2 to the variable x?

11. Sep 1, 2008

### tiny-tim

Welcome to PF!

Hi ZikZak! Welcome to PF!
Nicely put!
c just converts the time coordinate into a distance coordinate (as ZikZak said).

c converts years into light-years.

but that doesn't make it the "speed of time".

if there was a "speed of time", it would be in units of time/time, wouldn't it?

if there is a "speed of time", what is it measured relative to? itself?

12. Sep 1, 2008

### Staff: Mentor

Re: speed of time

It looks like what it is: assigning a distance to time ("the length of time").

13. Sep 1, 2008

### petm1

Re: speed of time

This “length of time”, ct; how long of a duration does it correspond to?

14. Sep 2, 2008

### tiny-tim

“length of time"

One year corresponds to one light-year!

This is so that the separation along the path of something moving at the speed of light, c2dt2 - dx2 - dy2 -dz2, is 0.

(As I indicated earlier, ct lets you mix up length and time.)

15. Sep 2, 2008

### petm1

Re: “length of time"

So one second, is associated With the length of c? It still looks like time must be dilating at the rate of c to make this possible, or I am still missing something.

16. Sep 2, 2008

### ZikZak

Re: “length of time"

No, c is not a length. c has the units of, say, meters per second. But c is more than just a velocity. It is a universal constant of nature on whose value everyone agrees, regardless of their reference frame. Just because it appears in an equation doesn't mean anything is traveling at that speed. It appears in all sorts of equations that have nothing whatsoever to do with motion, such as the equation that started this discussion.

The equation that started all of this was the metric equation, in which we attempt to calculate the interval between two events in spacetime. We do this basically by drawing right triangles and using the Pythagorean theorem. But you can't add a meter to a second. We MUST use the same units of measurement to measure time as we do length, width, and breadth. But because we commonly use a stupid system of units, we bizarrely measure length in meters but time in seconds. That is dumb. We have to fix that. How do you convert seconds to meters? By using a universal constant everyone agrees on.

c's weird numerical value, $c=299792458\, m/s$ arises precisely because we chose those stupid units to measure it with in the first place. I mean, come on, who in the universe uses meters and seconds? Certainly no one who has never heard of Earth, on which those units are based. We really should be using a system of units more natural to the universe, and c is a universal constant. So set $c=299792458\, m/s \equiv 1$ and you have just defined a much more natural system of units for the universe in which $c=1$. Solve for "seconds" in that equation and you have the recipe for converting units between meters and seconds: $299792458\, m = s$.

That is what c is doing in that equation: it is converting from a stupid system of units in which lengths and times are measured with different units into one in which they are measured in the same units, so that we can add them together.

Time isn't doing anything. Nothing is moving in this equation. There is no motion involved. AT ALL. We have taken two events in spacetime, and we are measuring the interval between them. In order to do so, we must convert units from the meter-second system to some system like second-second or meter-meter where everything is measured in meters, with the universal, everyone agrees on it, constant c as the conversion factor.

Last edited: Sep 2, 2008
17. Sep 2, 2008

### petm1

Re: “length of time"

I must admit that this is all imaginary, and it is not what I see the world to be like, but this best describes how I feel a "length of time" to be.