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## Main Question or Discussion Point

Hi all, I tried searching for this but failed to find an answer to my question. I am having an issue with properly interpreting the equations for time dilation and length contraction. Let's assume that I am standing still and a train is passing by next to me (moving with uniform velocity). Let:

Δx = a length of an object on the train measured by me

Δx' = the length of the same object on the train measured by a person on the train

Δt = a time interval measured by a clock I'm holding in my hand

Δt' = the same time interval measured by a clock held by a person on the train

v = velocity of the train

Then we have the following equations (the Lorentz transformations):

Δx' = [itex]\frac{Δx-vΔt}{\sqrt{1-v^2/c^2}}[/itex] (1)

Δt' = [itex]\frac{Δt-vΔx/c^2}{\sqrt{1-v^2/c^2}}[/itex] (2)

First, let's assume that I measure the length Δx instantaneously so that Δt=0. Then eq. 1 becomes:

Δx' = [itex]\frac{Δx}{\sqrt{1-v^2/c^2}}[/itex]

Second, let's assume that in a new experiment I measure a time interval. Since I'm not moving, Δx=0, so the second equation becomes:

Δt' = [itex]\frac{Δt}{\sqrt{1-v^2/c^2}}[/itex]

Given that the denominator is some number between zero and one, it turns out that both the length Δx that I measure, as well as the time interval Δt I measure are smaller than those measured by a person on the train (Δx' and Δt', respectively).

Now, I understand the example with the simple "light clock" which demonstrates time dilation. The light goes up and down and every cycle is 1 unit of time, for the person on the train. For me, it takes longer for light to complete a cycle, since it moves diagonally, so if I compared a clock I hold in my hand to the light clock on the train, I would conclude that the clock on the train is running slower than mine. But that's not what the equation above is telling me. It tells me that whatever Δt (the time interval I measured) is, Δt' will be higher. So, if I measured 1 second, the person on the train will measure, say, 2 seconds.

Similarly, if I measure Δx to be 1 meter, the person on the train will say Δx' is 2 meters.

So, it seems like the equations are telling me that time is actually running faster for the person on the train and the lengths are extending, not contracting.

There clearly is some inconsistency in how I interpret the terms, or I'm making some other type of an error. Please help me resolve this issue :)

Δx = a length of an object on the train measured by me

Δx' = the length of the same object on the train measured by a person on the train

Δt = a time interval measured by a clock I'm holding in my hand

Δt' = the same time interval measured by a clock held by a person on the train

v = velocity of the train

Then we have the following equations (the Lorentz transformations):

Δx' = [itex]\frac{Δx-vΔt}{\sqrt{1-v^2/c^2}}[/itex] (1)

Δt' = [itex]\frac{Δt-vΔx/c^2}{\sqrt{1-v^2/c^2}}[/itex] (2)

First, let's assume that I measure the length Δx instantaneously so that Δt=0. Then eq. 1 becomes:

Δx' = [itex]\frac{Δx}{\sqrt{1-v^2/c^2}}[/itex]

Second, let's assume that in a new experiment I measure a time interval. Since I'm not moving, Δx=0, so the second equation becomes:

Δt' = [itex]\frac{Δt}{\sqrt{1-v^2/c^2}}[/itex]

Given that the denominator is some number between zero and one, it turns out that both the length Δx that I measure, as well as the time interval Δt I measure are smaller than those measured by a person on the train (Δx' and Δt', respectively).

Now, I understand the example with the simple "light clock" which demonstrates time dilation. The light goes up and down and every cycle is 1 unit of time, for the person on the train. For me, it takes longer for light to complete a cycle, since it moves diagonally, so if I compared a clock I hold in my hand to the light clock on the train, I would conclude that the clock on the train is running slower than mine. But that's not what the equation above is telling me. It tells me that whatever Δt (the time interval I measured) is, Δt' will be higher. So, if I measured 1 second, the person on the train will measure, say, 2 seconds.

Similarly, if I measure Δx to be 1 meter, the person on the train will say Δx' is 2 meters.

So, it seems like the equations are telling me that time is actually running faster for the person on the train and the lengths are extending, not contracting.

There clearly is some inconsistency in how I interpret the terms, or I'm making some other type of an error. Please help me resolve this issue :)

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