# Time dilation and Lorentz-Einstein transformations

• bernhard.rothenstein
In summary: SR all massive objects will have a different [local] simultaneity than what they would have in a Galilean spacetime.
bernhard.rothenstein
Discussing with a friend, I was told that using in a derivation the time dilation formula I implicitly use the Lorentz-Einstein transformations.. I mentioned that many Authors derive time dilation without using the LET considering that the transformation equations obscure the physics behind the studied problem. Others derive the addition law of velocities without using LET based on time dilation and length contraction. My oppinion is that as long as we do not perform the transformation of the space-time coordinates of the same event we do not use LET. Your opinion is highly appreciated. Thanks
At leasts in physics who is right should be right!

To me its more instructive to proceed from the standpoint that all objects move through spacetime at c - and that the spacetime interval is invariant in all inertial frames. From this you get time dilation and length contraction in one step just as you do with the light clock

time dilation let

yogi said:
To me its more instructive to proceed from the standpoint that all objects move through spacetime at c - and that the spacetime interval is invariant in all inertial frames. From this you get time dilation and length contraction in one step just as you do with the light clock
do you mean that the derivation does not involve LET?

You can derive Lorentz transforms from the requirement that something moving with speed c in any frame must appear to move with speed c in all other frames.

http://www.mth.uct.ac.za/omei/gr/chap1/frame1.html gives a pretty good outline of what must be done.

bernhard.rothenstein said:
Discussing with a friend, I was told that using in a derivation the time dilation formula I implicitly use the Lorentz-Einstein transformations..

Here is an analogous statement:
using in a derivation the "formula that says, in a right triangle, the length of the adjacent side is equal to cos(included angle)*(the length of the hypotenuse)" I implicitly use the "Euclidean rotation" transformations..

Jheriko said:
You can derive Lorentz transforms from the requirement that something moving with speed c in any frame must appear to move with speed c in all other frames.

http://www.mth.uct.ac.za/omei/gr/chap1/frame1.html gives a pretty good outline of what must be done.

thank you for your answer but my problem is that if I use in a paper the time dilation formula I use or I do not use explicitly the LET. In my oppinion time dilation has nothing to do with LET because in order to derive the formula that accounts for it it is not compulsory to use the LET. Of course LET accounts for it. I conider that I use LET only when I establish a relationship between the space-time coordinates of the same event.

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LET and time dilation

robphy said:
Here is an analogous statement:
using in a derivation the "formula that says, in a right triangle, the length of the adjacent side is equal to cos(included angle)*(the length of the hypotenuse)" I implicitly use the "Euclidean rotation" transformations..

Thank you for your answer. I am more interested in your oppinion about the fact that if I use in a derivation the time dilation formula or the length contraction I should use the LET.
sine ira et studio

bernhard.rothenstein said:
do you mean that the derivation does not involve LET?
The premise that all objects move at velocity c through spacetime is consequent to Minkowski - it is implied in LET - but not explicitly stated by either Einstein or Lorentz - so for me its an easier starting point than the constancy of light in all frames - better as a tutorial from my perspective since you get the time dilation and length contraction relationships in an easily to visualize diagram. As one goes through the derivation of the Lorentz transforms it is easy to lose track of the physical connection

Regards

Yogi

yogi said:
The premise that all objects move at velocity c through spacetime is consequent to Minkowski - it is implied in LET - but not explicitly stated by either Einstein or Lorentz - so for me its an easier starting point than the constancy of light in all frames - better as a tutorial from my perspective since you get the time dilation and length contraction relationships in an easily to visualize diagram. As one goes through the derivation of the Lorentz transforms it is easy to lose track of the physical connection

Regards

Yogi

That "all [massive] objects move at velocity c through spacetime" is merely the statement that we describe their 4-velocities as unit-timelike vectors (conventionally normalized to c).

However, that in itself does not fully characterize the situation in special relativity [since the same situation is true in a Galilean spacetime]. Somehow, you have to specify the location of all of the tips of those 4-velocities [a hyperboloid for SR, a hyperplane for Galilean], which is almost the same as specifying the metric. Alternatively, one can use the null cone, which is almost the same as the postulating the "constancy of a maximal signal speed".

robphy said:
That "all [massive] objects move at velocity c through spacetime" is merely the statement that we describe their 4-velocities as unit-timelike vectors (conventionally normalized to c).
I'm confused; there isn't anything to normalize for 4-velocity vectors: they are exactly equal to
d{coordinate position}/d{proper time}.​

Hurkyl said:
I'm confused; there isn't anything to normalize for 4-velocity vectors: they are exactly equal to
d{coordinate position}/d{proper time}.​

Yes, that's true. Maybe I didn't make my point clearly.
What I meant to say is that the statement "all [massive] objects move at velocity c through spacetime" by itself is empty without somehow specifying a metric.

one way to derive the time dilation formula is to use the invariance of the ST interval: t^2 - x^2 (assuming that you measure time in meters or distance in seconds - as long as everything is in the same units)is constant in all frames. for example, if there's a tunnel with length x in one frame and in that frame it takes time t for a rocket to go thru it then the ST interval between the first event (entering the tunnel) and the second (exiting) is t^2 - x^2. but in the rocket frame the distance between the events is 0 so: Trocket^2 = Trest^2 - Xrest^2.

another way to derive it is with the famous parallel mirrors clock.

time dilation and LET

daniel_i_l said:
one way to derive the time dilation formula is to use the invariance of the ST interval: t^2 - x^2 (assuming that you measure time in meters or distance in seconds - as long as everything is in the same units)is constant in all frames. for example, if there's a tunnel with length x in one frame and in that frame it takes time t for a rocket to go thru it then the ST interval between the first event (entering the tunnel) and the second (exiting) is t^2 - x^2. but in the rocket frame the distance between the events is 0 so: Trocket^2 = Trest^2 - Xrest^2.

another way to derive it is with the famous parallel mirrors clock.

Thank you. I figured it out in connection with distant clock synchronization. Please give me some references where interval invariance is used in order to derive time dilation.
The best things a physicist can offer to another one are information and criticism

The standard "light clock" uses the invariance of the interval (declaring that a round trip by the light signal in an identically constructed light clock is one unit interval).

light clock and other clocks

robphy said:
The standard "light clock" uses the invariance of the interval (declaring that a round trip by the light signal in an identically constructed light clock is one unit interval).

I think that the light clock involves in its rest frame two clocks located on the two distant mirrors respectively. In the reference frame relative to which it moves, it involves a clock located where the first light signal is emitted, a clock located where the light signal arrives at the upper mirror and a clock located where the reflected light signal returns. Time dilation is the result of the synchronization of the mentioned clocks in theirs reat frames. That fact is not allways mentioned. Do you think that it is worth to mention it in the teaching process?
sine ira et studio

A light clock is formed from two inertial mirrors and a light ray that bounces back and forth between them. You may (for consistency) introduce other clocks... but, in my opinion, that is unnecssary for the operation of the light clock as a clock [at its natural resolution].

light clock

robphy said:
A light clock is formed from two inertial mirrors and a light ray that bounces back and forth between them. You may (for consistency) introduce other clocks... but, in my opinion, that is unnecssary for the operation of the light clock as a clock [at its natural resolution].
Thanks. I think that it is good to mention the other clocks when we consider the light clock out from its rest frame.

robphy said:
That "all [massive] objects move at velocity c through spacetime" is merely the statement that we describe their 4-velocities as unit-timelike vectors (conventionally normalized to c).

However, that in itself does not fully characterize the situation in special relativity [since the same situation is true in a Galilean spacetime]. Somehow, you have to specify the location of all of the tips of those 4-velocities [a hyperboloid for SR, a hyperplane for Galilean], which is almost the same as specifying the metric. Alternatively, one can use the null cone, which is almost the same as the postulating the "constancy of a maximal signal speed".

It is true you need to be more rigorous to recover the complete transforms - but as a visual tutorial to display time dilation simply draw the t axis vertical, the space axis horizontal and construct a first quadrant arc of length ct centered on the origin. The intercept on the t axis is the spacetime interval ct (which is invarient from one frame to the other) in the frame selected to be at rest, the length vt along the space axis is the distance traveled by a clock in the moving frame during the time t, the vertical line drawn from the point vt to intercept the arc gives the length (ct') logged by the moving clock. The line from the origin to the intercept is the hypotenuse (also ct), therefore

(ct)^2 = (vt)^2 + (ct')^2

time dilation and let

bernhard.rothenstein said:
Discussing with a friend, I was told that using in a derivation the time dilation formula I implicitly use the Lorentz-Einstein transformations.. I mentioned that many Authors derive time dilation without using the LET considering that the transformation equations obscure the physics behind the studied problem. Others derive the addition law of velocities without using LET based on time dilation and length contraction. My oppinion is that as long as we do not perform the transformation of the space-time coordinates of the same event we do not use LET. Your opinion is highly appreciated. Thanks
At leasts in physics who is right should be right!
most of the answers did neglect my original question: If I use in a derivation the time dilation formula and I am told that I use implicitly the Lorentz-Einstein transformation, is my discussion partner right or wrong?

## FAQ 1: What is time dilation?

Time dilation is a phenomenon predicted by Einstein's theory of relativity, where time appears to pass slower for objects in motion compared to stationary objects. This is due to the fact that time and space are intertwined and can be affected by an object's velocity and gravitational field.

## FAQ 2: How does time dilation relate to Lorentz-Einstein transformations?

Lorentz-Einstein transformations are mathematical equations used to describe the effects of time dilation and length contraction in special relativity. These transformations allow us to calculate the differences in time and space measurements between two reference frames in relative motion.

## FAQ 3: Can time dilation be observed in everyday life?

Yes, time dilation has been observed in various experiments and is also taken into account in everyday technologies such as GPS systems. Even at relatively low speeds, time dilation is present and can be measured with highly accurate clocks.

## FAQ 4: What is the difference between time dilation in special relativity and general relativity?

Time dilation in special relativity is caused by differences in relative velocities, while in general relativity it is caused by differences in gravitational forces. In general relativity, time dilation also takes into account the curvature of spacetime.

## FAQ 5: Is time dilation a proven phenomenon?

Yes, time dilation has been extensively tested and confirmed through various experiments and observations. Its predictions have also been used to develop technologies such as GPS systems and particle accelerators.

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