Solow model, time and steady state Suppose we have a Solow model:

$$
Y(t)=C(t)+I(t)
$$

$$
I(t)=sY(t)
$$

$$
\dot K=I(t)-δK(t)
$$

With a given Cobb-Douglas:

$$
Y(t)=Z(t)K^aL^{1-a}
$$

$$
y(t)=Y(t)/L(t)
$$

$$
k(t)=K(t)/L(t)
$$

$$
y=Zk^a
$$

We also know these:

$$
L=L(t), \dot L/L=n
$$

$$
Z=Z(t), \dot Z/Z=g
$$

Therefore we have:

$$
L(t)=L(0)e^{nt}
$$

$$
Z(t)=Z(0)e^{gt}
$$

We get to this point:

$$
\dot k=sZk^a-(n+δ)k
$$

In the steady state:

$$
k=k^*
$$

$$
k^*=\left(\frac{sZ}{n+δ}\right)^{1/(1-a)}
$$ 

The question has to do with the time dimension in Z. Time enters the final expression if we substitute Z with what we found above, but I'm not sure what to make of this.

$$
k^*=\left[\frac{sZ(0)e^{gt}}{n+δ}\right]^{1/(1-a)}
$$

Surely there's something this neophyte is mixing up. I thought that in a steady state capital per worker remains the same.
In the model with technological progress the capital per effective worker remains constant, implies that capital per worker grows at the rate of exogenous rate of technological progress. See Barro and Martin book, Chapter 1.