What is the problem with the Minkowski relationship in special relativity?

[John Hobson]

The Minkowski relationship is an important aspect of special relativity.

(c.t)^2 - x^2 = k

For two points in spacetime, observers moving at different speeds observe different time and space differences between them. Neither is constant but the above relationship is. It is the minus sign that implies space time has a hyperbolic structure.

However if we use the other expressions from special relativity, as an observer moves, clocks slow by the factor gamma so the measured time difference increases by 1/gamma. Similarly rulers shrink by the factor gamma, so measured distances increase by 1/gamma. But if we put these into the above relationship we would get

(c.t)^2 - x^2 = (1/gamma)^2.k

Since gamma is not a constant, this is not the Minkowski relationship. What am I missing?

 

[Bill_K]

However if we use the other expressions from special relativity, as an observer moves, clocks slow by the factor gamma so the measured time difference increases by 1/gamma. Similarly rulers shrink by the factor gamma, so measured distances increase by 1/gamma. But if we put these into the above relationship we would get

(c.t)^2 - x^2 = (1/gamma)^2.k

Since gamma is not a constant, this is not the Minkowski relationship. What am I missing?

You're thinking that

Δx' = γ Δx
Δt' = γ Δt

whereas actually

Δx' = γ(Δx - v Δt)
Δt' = γ(Δt - v Δx)

 

[dvf]

Since gamma is not a constant, this is not the Minkowski relationship. What am I missing?

Measured time differences increase by a factor gamma, but c.t' = c.t.gamma only for x=0 (see Wikipedia Time dilation)
Measured distances decrease by a factor gamma, but x' = x/gamma only for t'=0 (see Wikipedia Length contraction)
Combining these two clearly distinct situations with two events can only be done for the trivial case where x=t=x'=t'=0.

 

[Dale]

However if we use the other expressions from special relativity, as an observer moves, clocks slow by the factor gamma so the measured time difference increases by 1/gamma. Similarly rulers shrink by the factor gamma, so measured distances increase by 1/gamma.

You have run into a very common problem for new students of relativity. Often the time-dilation and length-contraction formulas are presented as though they are general formulas which always apply as written without any additional terms. The truth of the matter is that they are both simplifications of the Lorentz transform which are only valid under certain specific circumstances.

My recommendation is to not use the length contraction or time dilation formulas at all. Simply use the Lorentz transform always. It will automatically simplify to the length contraction or time dilation formula wherever it is appropriate, but you will never run into situations like this where you misapply them to situations where the assumptions are not met.

 

[jtbell]

whereas actually

Δx' = γ(Δx - v Δt)
Δt' = γ(Δt - v Δx)

Just to make sure the OP doesn't get confused, this is in units where c = 1. Putting in the factors of c explicitly, the first equation is unchanged but the second becomes

Δt' = γ(Δt - v Δx / c^2)